Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors
by
Oscar Karl Johannes Henriksson
B.S., Uppsala University, 2011
M.S., University of Colorado Boulder, 2013
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Physics
2017 This thesis entitled: Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors written by Oscar Karl Johannes Henriksson has been approved for the Department of Physics
Prof. Oliver DeWolfe
Prof. Senarath de Alwis
Prof. Thomas DeGrand
Prof. Andrew Hamilton
Prof. Paul Romatschke
Date
The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii
Henriksson, Oscar Karl Johannes (Ph.D., Physics)
Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors
Thesis directed by Prof. Oliver DeWolfe
We apply holographic duality to the study of strongly interacting quantum matter. The cor- respondence between the four-dimensional = 8 gauged supergravity and the three-dimensional N superconformal ABJM quantum field theory allows us to study the latter theory by performing computations in the former. Asymptotically anti-de Sitter spacetimes satisfying the classical su- pergravity equations of motion are interpreted as states of strongly interacting ABJM theory. If such a spacetime sources an electric field, the dual state is at non-zero charge density.
Interesting observables of such states include spectral functions of fermionic operators — we compute these by solving Dirac equations in a variety of spacetimes. In a family of extremal charged black holes, we find Fermi surface singularities with non-Fermi liquid characteristics. In a special
“three-charge” black hole, an interval appears in the spectral functions within which the fermionic excitations are perfectly stable. We then study three different domain wall spacetimes dual to zero-temperature states with a broken U(1) symmetry. In these “holographic superconductors”, we find features similar to conventional superconductors such as the development of a gap in the fermionic spectra.
Finally, we investigate the question of how bosonic properties, for example susceptibilities, are affected by fermionic properties, such as Fermi surface singularities, in holographic states of matter. We do this by computing the static charge susceptibility in the three-charge black hole state. Our results reveal singularities at complex momenta, with a real part approximately equal to the largest Fermi momentum in the state. Dedication
To my parents, Ingvar and Marita Henriksson, with gratitude. v
Acknowledgements
During my years in Boulder it has been a true pleasure to work with my advisor Oliver
DeWolfe. I have benefited from his great knowledge in physics, his clear explanations and his kindness from the first class I took from him and throughout the writing of this dissertation.
In similar ways, Christopher Rosen has been a fantastic collaborator and a source of help and motivation for most of the work herein. I have also gained much from fruitful collaborations with Steven Gubser, Paul Romatschke and Chaolun Wu. Various other physicists at CU Boulder, including Shanta de Alwis, Andrea Carosso, Anqi Cheng, Tom DeGrand, Dan Hackett, Anna
Hasenfratz, Takaaki Ishii, Will Jay, Ethan Neil and Greg Petropoulos, taught me a great deal about physics, as well as other topics, during lectures, office chats and 10:30 coffee breaks.
I was lucky to share the wonderful and strange experience of graduate school with a strangely wonderful group of fellow students. I hope to enjoy conversations and adventures with Andrew and
Helen, Jack, Nico and Dani, Paige, Scott, Tyler, and many others, for many years to come.
During my time in the U.S., the friendships of (and constant stream of messages from) my friends from the Aland˚ Islands — Christoffer, Fredrik, Jonas, Jonathan and Kim — have made the long distance home feel a lot shorter.
My girlfriend Marion Boulet has been a wonderful, beautiful and reliable source of support, laughter and tasty cookies throughout our time together — she truly has a Heart of Gold.
Finally, this dissertation is dedicated to my parents for the unrelenting support and encour- agement they have provided throughout my life, making all of this possible. Contents
Chapter
1 Introduction and Overview 1
1.1 A Brief Tour of String Theory...... 2
1.2 An Introduction to AdS/CFT...... 5
1.2.1 The Dictionary...... 8
1.3 Applications of AdS/CFT...... 9
1.3.1 Condensed Matter Physics and AdS/CFT...... 10
1.4 Summary of Dissertation...... 13
1.4.1 Top-down Non-Fermi Liquids and a Special Black Hole...... 13
1.4.2 Fermions in Supergravity Superconductors...... 14
1.4.3 Correlations between Correlators: Charge Oscillations and Fermi Surfaces.. 16
2 The Two Theories 18
2.1 4D = 8 Gauged Supergravity...... 18 N 2.1.1 Bosonic Sector...... 19
2.1.2 Quadratic Fermion Action...... 23
2.2 The M2-brane ABJM Theory...... 26
2.2.1 A Simplified Description...... 27
3 Fermi Surface Behavior in the ABJM M2-brane Theory 29
3.1 Introduction and Summary...... 29 vii
3.1.1 Holographic Realizations of Non-Fermi Liquids...... 29
3.1.2 Fermionic Response in the M2-brane Theory...... 32
3.2 Black Branes and Dirac Equations in Maximal Gauged Supergravity...... 35
3.2.1 Black Brane Solutions...... 35
3.2.2 Linearized Dirac Equations...... 38
3.3 Fermionic Green’s Functions...... 41
3.3.1 Solving the Dirac Equation...... 41
3.3.2 Quantization of Fermi Fields and Green’s Functions...... 43
3.4 Regular Black Holes and Non-Fermi Liquids...... 48
3.4.1 Regular Black Holes and Non-Fermi Liquids...... 50
3.4.2 The 3+1-Charge Black Hole...... 53
3.4.3 The 2+2-Charge Black Hole...... 62
3.5 The Extremal Three-Charge Black Hole and the Gap...... 69
3.5.1 Near Horizon Analysis of the 3QBH...... 70
3.5.2 Connection with Extremal (3+1)-Charge Black Holes...... 72
3.5.3 Fermion Fluctuations and Fermi Surfaces...... 75
3.6 RG Flow Backgrounds: 2QBH and 1QBH...... 78
3.6.1 The 1-Charge Black Hole...... 80
3.6.2 The 2-Charge Black Hole...... 82
4 Fermionic Response in Finite-Density ABJM Theory with Broken Symmetry 86
4.1 Introduction...... 86
4.2 The Bosonic Background Geometries...... 89
4.2.1 The SO(3) SO(3) truncation...... 89 × 4.2.2 The Domain Wall Backgrounds...... 90
4.2.3 Holographic Interpretation...... 95
4.2.4 Conductivities...... 100 viii
4.3 Fermion Response in the Domain Wall Solutions...... 102
4.3.1 Coupled Dirac equations and holographic operator map...... 102
4.3.2 Solving the Dirac equations and spinor Green’s functions...... 106
4.3.3 Fermion Normal Modes...... 110
4.3.4 Spectral Functions...... 113
4.3.5 Modifying Couplings...... 118
4.4 Discussion...... 122
5 Gapped Fermions in Top-down Holographic Superconductors 126
5.1 Overview...... 126
5.2 The SU(4)− Flow...... 129
5.2.1 The SU(4)− Truncation...... 129
5.2.2 SU(4)−-invariant Domain Wall Solutions...... 130
5.2.3 The Fermionic Sector...... 132
5.2.4 Fermion Response...... 135
5.2.5 Field Theory Operator Matching...... 141
5.3 The H = SO(3) SO(3) Flows...... 142 × 5.3.1 The Fermionic Sector...... 143
5.3.2 Fermion Response...... 147
5.3.3 Field Theory Operator Matching...... 151
5.4 Lessons for Strongly Coupled Systems...... 152
5.4.1 Top-down vs. Bottom-up Fermion Response...... 152
5.4.2 Extremal AdSRN and Effects of Broken Symmetry...... 153
5.4.3 Stability in Supergravity and Zero Temperature Response...... 157
6 “1kF ” Singularities and Finite Density ABJM Theory at Strong Coupling 160
6.1 Overview...... 160
6.2 The Three-Charge Black Hole...... 166 ix
6.2.1 Thermodynamics and Field Theory Interpretation...... 167
6.3 Fluctuations and Linear Response...... 171
6.3.1 Fermion Spectral Functions...... 171
6.3.2 The Holographic Static Susceptibility Setup...... 173
6.4 Results...... 177
6.5 Discussion...... 180
6.5.1 Singularities and Scaling Exponents...... 180
6.5.2 Commentary...... 184
7 Concluding Comments 189
Bibliography 191
Appendix
A Lifting the Three-Charge Black Hole to Five Dimensions 199
B Further Notes on the Static Susceptibility Computation 202
B.1 Finite Counterterms and the Alternate Quantization...... 202
B.2 Boundary Conditions and Numerical Methods...... 204 x
Tables
Table
2.1 4D supergravity modes and their dual ABJM operators...... 27
3.1 The 16 independent fermion eigenvectors that do not mix with the gravitini..... 40
3.2 A summary of the results for fermion modes in the 3QBH...... 78 Figures
Figure
1.1 A cartoon showing dispersion relations of stable fermionic modes in purple, all ex-
isting within stable regions whose edges are shown in blue. As the stable modes hit
the blue edge, they acquire a finite lifetime. Three different geometries are shown:
The regular black holes (left), where the only stable modes are right at the chemical
potential; the 3QBH (right), where a stable momentum-independent interval opens
up; and a superconducting domain wall (middle), where the stable region lies outside
the emergent lightcone...... 15
3.1 A cartoon of the parameter space of black holes we consider...... 49
3.2 Class 1 fermions for the (3+1)QBH. Fermi surface singularities are shown as blue
dots, while zeroes are marked by empty circles. The green hatched region is the
“oscillatory region” characteristic of an infrared instability towards pair production
in the bulk. The solid blue contours bound the region of Fermi surfaces with non-
Fermi liquid-like excitations...... 55
3.3 Class 2 fermions for the (3+1)QBH. These modes are unique in that they exhibit
multiple Fermi surfaces for small µR...... 57
3.4 Class 3 fermions for the (3+1)QBH. The poles end at the oscillatory region just
before µR = 1...... 59
3.5 Class 4 fermions for the (3+1)QBH. For the net-neutral modes, there is a novel
transition at the 4QBH state from Fermi surface singularities to zeroes...... 59 xii
3.6 Class 5 fermions for the (3+1)QBH. Unlike their net-charged brethren, there exists
no oscillatory region for the net neutral modes, but a single “oscillatory point” at
the pole/zero transition...... 61
3.7 Poles and zeros of the retarded Green’s function for the class I fermion from 0 <
µ˜R < 1 (left) and for the class II fermion from 1 < µ˜R < 0 (right). Viewed together
the two plots depict the entire range 0 < µ˜R < for class I or > µ˜R > 0 for class ∞ ∞ II...... 64
3.8 Poles and zeros of the retarded Green’s function for the class III fermion from 0 <
µ˜R < 1 (left) and for the class IV fermion from 1 < µ˜R < 0 (right), or the entire
range 0 < µ˜R < for class III or > µ˜R > 0 for class IV...... 65 ∞ ∞ 3.9 Poles and zeros of the retarded Green’s function for the class V fermion from 0 <
µ˜R < 1 (left) and for the class VI fermion from 1 < µ˜R < 0 (right), or the entire
range 0 < µ˜R < for class V or > µ˜R > 0 for class VI...... 66 ∞ ∞ 3.10 Class 1 fermions for the 3QBH. There is both a line of poles throughout the stable
region ∆ < ω < ∆, and a pair of poles nucleating very close to ω = ∆ before − ending on the oscillatory region (green)...... 74
3.11 Class 2 fermions for the 3QBH. Here there is a line of poles only...... 76
3.12 Class 3 fermions for the 3QBH. No line of poles through ω = 0 exists, but a pair of
poles nucleate near ω = ∆ and end on the oscillatory region...... 76
3.13 Class 4 fermions for the 3QBH, with a line of poles only...... 77
3.14 Class 5 fermions for the 3QBH, with a pair of poles nucleating near ω = ∆..... 79 −
3.15 Class II fermions for the (2+2)QBH, with k normalized relative to µ2 instead ofµ ˜2.
The Fermi surfaces all lie at kF µ2/√8...... 84 ≈ xiii
4.1 The Massive Boson background. The dashed lines in the plot of G/r2 are at 14/3 and
2, indicating the values obtained in the IR and UV AdS4 fixed points respectively.
The ratio of the speed of light in the UV CFT compared to that of the IR theory is
λ1/2 n = 26.900, and the non-vanishing scalar fall-off is 2 0.0308...... 94 ΨUV ≈ 4.2 The Massive Fermion background. The dashed lines in the plot of G/r2 are at 14/3
and 2, indicating the values obtained in the IR and UV AdS4 fixed points respectively.
This geometry is characterized by n = 1.861 and λ1 1.227...... 94 ΨUV ≈ 4.3 In units where the UV light cone is 45o (dotted black), we compare the Massive
Boson (solid blue) and Massive Fermion (dashed red) IR light cones...... 96
4.4 The real (left) and imaginary (right) AC conductivity in Massive Boson (darker) and
Massive Fermion (lighter) backgrounds. The imaginary part of the conductivity has
been multiplied by ω to highlight the 1/ω pole at low energies giving rise to the delta
function in Re σ...... 101
4.5 The band structure of fermion normal modes in the Massive Boson background.
The shaded blue triangle is the stable wedge where it is possible for normal modes
to appear, and the solid blue curves are the locations of fermion normal modes of
the bulk theory, as determined by solving (4.61). The intersection of the dashed
line with one band indicates the presence of a gapless mode. This band appears
to terminate where it reaches the top boundary of the shaded region, but follows it
closely along the bottom edge as far as our numerics allow us to compute...... 112
4.6 The band structure of fermion normal modes in the Massive Fermion background,
determined by solving (4.61). Again, there is a gapless mode at finite momentum.
At higher momentum the gapped band approaches the ungapped band, but appears
to meet the IR lightcone before the two bands coincide. As in the Massive Boson
background, the ungapped band traces the bottom edge of the stable region as far
as our numerics can reliably follow it...... 112 xiv
4.7 Plots of G†G for the Massive Boson background. Within the wedge marked by red
edges, all excitations are stable...... 115
4.8 Plots of G†G for the Massive Fermion background. Within the wedge marked by red
edges, all excitations are stable...... 116
2 4.9 The squared amplitude cI of each ABJM operator participating in the “Fermi | | surface” zero-energy mode in the AdS-RN background with no scalar (left), and the
Massive Boson (middle) and Massive Fermion (right) backgrounds. The normalized
Fermi momentum kF vUV/µ in the three cases are 0.53 (AdS-RN), 0.58 (MB), and
0.48 (MF). Note that in all cases the contributions from χ and χ are insignificant.118 O 2 O 0 4.10 Plots of the modulus squared of the Green’s function for massless probe fermions of
various charges...... 119
4.11 Normal mode structure in the Massive Boson background with the off-diagonal cou-
plings turned off (left), compared to the full top-down result of figure 4.5 (right).
With the off-diagonal couplings turned off there are three bands, associated to χ2
(orange),χ ¯2 (yellow), andχ ¯0 (red), intersecting in three places. Turning on the
couplings between the different fermions induces repulsion between the bands, as
described in the text...... 121
5.1 The AdS4 to PW flow. The flow is characterized by an index of refraction n 3.78 ≈ 2 and a scalar vev proportional to ξ2/φ 0.33...... 133 UV ≈ 5.2 Spectral function for fermionic operators in the 20. The red lines mark the IR
lightcone, while the blue lines show the lightcone of the UV theory. The right figure
shows a close-up around the origin for ω < 0. Superimposed on the right figure
are black dashed lines, showing the lines of maxima of the spectral weight; black
dots, marking the point of closest approach to the ω = 0 axis (k?); and white dots,
showing the location of the Fermi surface singularities in the normal phase (kF ).
These special points will be discussed in more detail in section 5.4...... 139 xv
5.3 Spectral function for fermionic operators in the 20 as a function of frequency at
various momenta. At left, k = 0 and the dashed purple line shows the maximally
symmetric AdS4 result as given by (5.30). The inset details the falloff at low frequen-
cies, which asymptotes to a power law with exponent 2∆IR 3 = √3/2 as shown by − the pink line. At non-zero momenta (right), the spectral function develops a hard
gap. For momenta in the vicinity of k k? there is a narrow quasiparticle-like peak ≈ just below the gap, as well as a more diffuse hump at larger ω/µ as dictated by the | | UV conformal theory...... 140
5.4 An illustration of the level repulsion induced by the chiral Majorana coupling in the
(ω, k)-plane. Left: Without a Majorana coupling, (one of the two spinor components
α of) a fermion operator will generically display lines of normal modes (purple)
crossing the dashed ω = 0 line, leading to a Fermi surface singularity. Center:
Looking at the conjugate fermion, and switching to the other spinor component,
gives an identical normal mode line flipped across ω = 0. Right: Turning on the
chiral Majorana coupling mixes these two energy bands, causing them to repel.... 146
5.5 The band structure of fermion normal modes in the Massive Boson (type 1) back-
ground. The normal mode is shown in purple. The inset zooms in on the beginning of
this band, emphasizing that it very nearly coincides with the edge of the IR lightcone.149
5.6 The spectrum in the Massive Boson background. The red and blue lines mark the IR
and UV lightcones, respectively, and the white lines show the location of the line of
normal modes, corresponding to a line of delta function peaks in the spectral weight.
The white dots at ω = 0 show the Fermi momentum in the normal phase...... 149
5.7 The spectrum in the Massive Fermion background. The red and blue lines mark the
IR and UV lightcones, respectively; for spacelike IR momenta the spectral weight is
zero everywhere. The white dots at ω = 0 show the Fermi momentum in the normal
phase...... 150 xvi
5.8 Illustration of gapped fermionic excitations in BCS theory and holography. In the
left panel, the BCS dispersion relation in the superconducting (normal) phase is
plotted in blue (dashed black). The parameters are arbitrarily chosen such that
vF = kF = 1 and ∆ = 2. In the holographic fermion spectral function (cartoon, | | right), the boundaries of the IR lightcone determines the stability of the fermionic
excitations, but the gapping is qualitatively similar...... 155
6.1 The spectral function A(ω, k) computed from “Class 2” fermions in the 3QBH back-
ground. The solid lines indicated delta function peaks in the spectral weight. This
weight broadens in the shaded bands, and has been omitted from the plot for clarity. 174
6.2 The low temperature static susceptibility along the real kˆ k/µ axis (right) and in ≡ the complex kˆ–plane (left). The susceptibilities pictured are computed at a temper-
ature Tˆ = 10−5. In the rightmost plot, a point at kˆ = 0 shows the location of the
uniform static susceptibility computed from the background, in excellent agreement
with the fluctuation analysis...... 178
6.3 The location of the branch points with Re kˆ = 0 as a function of temperature.... 179 6 6.4 The linearized charge density response due to the introduction of a charged impurity
at low temperature , Tˆ = 10−5 (left). No oscillations are easily discernible at large
distances. To observe the effects of the branch points with Re kˆ? = 0, it is helpful 6 to remove the exponential damping from the branch point along the imaginary axis
(right)...... 180
0 − 6.5 The analytic structure of ν|| (left) and ν|| (right) in the complex momentum plane. The plots show the argument of the scaling exponent, which clearly reveals the
existence of two classes of branch points. Both of these classes are present in the
susceptibility. Numerically, they occur at kˆ? 0.87 i and kˆ? 0.75 1.15 i in ≈ ± ≈ ± ± excellent agreement with the numerically computed susceptibility...... 182 Chapter 1
Introduction and Overview
There can be no doubt that quantum field theory (QFT) captures fundamental properties of nature. This is powerfully demonstrated in the field of high energy physics, where the QFT known as the Standard Model has reigned supreme for around four decades, precisely predicting outcomes of uncountable experiments. High energy physicists, however, do not have a monopoly on this theoretical framework. At lower energies, QFT is employed in condensed matter physics and atomic physics, where the quanta of various fields successfully describe not only “fundamental” particles such as electrons, but also emergent quasiparticles and collective excitations. Phenomena ranging from normal metallic behavior to superconductivity to the quantum Hall effect can be explained by such low-energy effective field theories.
Despite the numerous stories of success, QFT still holds many mysteries; most of these are encountered at moderate or strong interactions, when the usual perturbative expansions fail. Of course, the issue of strongly interacting QFT (SIQFT) has been around for a while, and several points of attack have been established; perhaps foremost being numerical methods such as lattice
field theory. These have produced many impressive results and are a cornerstone of our current understanding of QFT. However, such methods also have limitations — notably, the fermion sign problem is a major obstacle to doing numerical computations at finite density. This is one problem that we will attempt to attack in this dissertation.
To make progress, we will employ a different tool that can be used to learn more about QFT, namely duality. A duality is a statement that two physical theories describe the same physics; 2 this requires the existence of some kind of dictionary that explains how different quantities in the two theories are related. In some cases, the two theories might be the same one at a different values of the interaction strength; the theory is then said to be self-dual. In other cases, such as the ones discussed in the following, the two theories can be vastly different. An important category for practical applications are strong/weak dualities — dualities where, as one of the theories is strongly interacting and thus difficult to compute in, its dual theory is weakly coupled and permits a controlled perturbative treatment.
In this dissertation, a specific duality will be employed which is both strong/weak and well- suited to our needs: the anti-de Sitter/conformal field theory correspondence, or AdS/CFT for short. This surprising duality establishes a dictionary between two naively very different physical systems: a specific conformal quantum field theory (CFT), and a string theory in an anti-de Sitter
(AdS) spacetime. In the rest of this chapter, we will provide a brief introduction to string theory and AdS/CFT, discuss its possible applications to strongly interacting quantum matter, and give an overview of the rest of the chapters in this dissertation.
1.1 A Brief Tour of String Theory
Like a number of other fascinating dualities, AdS/CFT traces its roots to string theory, and so this is where we begin. Famously, string theory started its life in the heads of theorists as a description of hadrons, with mesons being envisioned to be connected by the strings. It was eventually realized that string theory a) doesn’t do very well as a theory of hadrons, but b) does surprisingly well as a quantum theory of gravity. In particular, the spectrum of states naturally includes a massless spin-two particle which can couple to a stress tensor, hence earning the name graviton.
In the following years, much work was done developing string theory into an impressive framework of ideas. Along the line, the hope that string theory was in fact a theory of everything was born, since it seemed able to provide both a consistent quantum theory of gravity and a UV- completion of the Standard Model. As one might guess, this is a tall order; quantum gravity effects 3 are inherently hard to study experimentally, and hopes that the Standard Model (without other unwanted light fields) might be directly derived from string theory have been called into question by the discovery of string theory’s vast “landscape” of vacua. Regardless of these difficulties, however, string theory has already established its importance in another way: by providing a fruitful new perspectives on the study of QFTs. It is this form of “applied” string theory that this dissertation is concerned with.
First, however, let us go over some basic features of string theory – this will necessarily be a bit rushed. To first approximation, string theory is a quantum theory of one-dimensional ele- mentary strings. As such a string propagates through spacetime, it sweeps out a two-dimensional worldsheet. The theory is formulated in terms of a QFT living on this worldsheet, with a set of scalar fields representing the embedding of the string in spacetime.1 To compute physical observ- ables such as S-matrix elements, one sums over the different types of worldsheets that contribute to a certain process. This sum is a perturbative expansion in a dimensionless parameter gs, called the string coupling constant; the expansion is controlled only when gs is small. A complete, non-perturbative formulation of string theory is still unknown. However, the perturbative definition already has many fascinating consequences, some of which we now list:
Mathematical consistency requires the dimensionality of spacetime to be 26. As troubling • as this might seem from a phenomenological point of view, there are various methods of
getting the effective number of dimensions down to something more familiar, most notably
Kaluza-Klein compactifications.
A string can be either open or closed. For each possibility, one can readily compute the • resulting spectrum. Important (low-energy) modes in the closed string sector include the
promised graviton as well as a scalar known as the dilaton, while the open string sector
contains, for example, gauge bosons.
String theory contains no free dimensionless parameters. While it appear at first that gs • 1 In other words, a (non-)linear sigma model with spacetime as its target space. 4
would be one such parameter, it turns out to be fixed in terms of the expectation value of
the dilaton field.
To include fermions in the spectrum, the theory must be augmented with supersymmetry. • This changes the critical dimension of spacetime from 26 for the bosonic string to 10 for
the superstring.
There exist five different ways of setting up superstring perturbation theory. These are • called (for reasons we will not discuss) type I, type IIA, type IIB, Heterotic SO(32), and
Heterotic E8 E8. However, it has been realized that there exists a web of dualities × relating these different theories. Therefore, it is now believed that if a full, non-perturbative
definition of string theory is found, it will contain all of the perturbative superstring theories
as different limits.
From the duality web just mentioned, the existence of an 11-dimensional theory known as • M-theory can be inferred in a particular strong coupling limit.
The low-energy limit of string and M-theory gives rise to theories of gravity, obeying Ein- • stein’s equations plus calculable corrections. Incorporating supersymmetry, this results in
field theories of supergravity.
The most important realization for our purposes, however, is that string theory is not only a theory of strings. The theory naturally contains non-perturbative objects of varying dimensionality called branes. These are dynamical objects that can act as sources of fields such as the metric and various gauge fields. A brane that extends in p spatial dimensions is called a p-brane — the fundamental string is thus a type of 1-brane. Most famous among the other types of branes are
D-branes; these are objects that fundamental strings can end on (the “D” stands for Dirichlet, referring to the boundary conditions of strings that end on them). M-theory contains another important type called M-branes, which will be discussed in the next section.
We are now close to being able to introduce the AdS/CFT correspondence; it originates in 5 the existence of two quite different but complementary ways of describing the dynamics of branes in string theory. This is most easily seen for D-branes. On the one hand, from the point of view of the open string sector, D-branes appear in open string perturbation theory as surfaces where strings can end. One can show that the dynamics of these open strings, whose spectrum contains gauge bosons, gives rise to an effective worldvolume description, taking the shape of a (p + 1)- dimensional U(1) gauge theory. Now generalize to N D-branes. If these are spatially separated, each brane gives rise to its own U(1) gauge theory, plus a set of massive excitations corresponding to open strings connecting the branes. If the N D-branes become coincident, however, those massive excitations become massless, and the U(1)N symmetry is enhanced to U(N).
On the other hand, from the point of view of the closed string sector containing gravity, the stack of N D-branes will curve spacetime. In the low-energy supergravity limit, the stack of a large number of these D-branes is well described by a black brane, a generalization of the familiar black hole solutions of general relativity, being extended in p spatial dimensions.
These two points of view are useful in different regimes. For the open strings, the perturbative expansion is controlled by gsN (the string coupling times the number of branes); the expansion breaks down if gsN 1. But this is exactly when curvatures are small, ensuring the validity of the supergravity approximation in the closed string description. This complementarity is a cornerstone of the AdS/CFT correspondence, to which we now turn.
1.2 An Introduction to AdS/CFT
We will now add some detail to the discussion at the end of the previous section, and sketch how AdS/CFT follows from string/M-theory. A large part of the work on AdS/CFT has been concerned with the duality between = 4 Super-Yang-Mills (SYM) and type IIB string theory N 5 on AdS5 S , which originates in the dynamics of D3-branes. Here, on the other hand, we will × instead study the similar case of M2-branes, which are (2+1)D objects in (10+1)D M-theory. The resulting duality will be used throughout most of this dissertation.
The low-energy limit of M-theory is 11D supergravity. Besides the graviton and its super- 6 partner the gravitino, this theory also contains a three-form gauge field field. Such a field naturally couples to objects extended in two spatial dimension — these are known as M2-branes. Just like in the case of D-branes, a stack of N M2-branes houses a non-abelian gauge theory on its world- volume. And just like with D-branes, such a stack will curve the spacetime around it. Starting from the point of view of the worldvolume gauge theory, it is interesting to note that a complete formulation of this theory was unknown when AdS/CFT was first proposed in 1997 — its existence was in some sense a prediction of string theory [1]. A Lagrangian for this theory, which is also the low-energy limit of (2+1)D SYM, was suggested about 11 years later by Aharony, Bergman,
Jafferis and Maldacena [2], and the resulting theory is now known as ABJM theory. This is a
(2+1)D superconformal Chern-Simons-matter field theory with gauge group U(N) U(N). It will × be introduced in greater detail in section 2.2.
Now let us look at these branes in the supergravity limit; a large number N of them will significantly curve spacetime, giving rise to the metric
2 −2/3 µ ν 1/3 i i ds = H(r) ηµνdx dx + H(r) dx dx , (1.1) where ηµν is the Minkowski metric; Greek indices labeling the worldvolume of the M2-brane range from 0 to 2; latin indices labeling the other dimensions range from 4 to 9; and
2 6 32π Nlp H(r) = 1 + , r2 = xixi . (1.2) r4
Taking the near-horizon r 0 limit (while keeping dimensionful quantities in the worldvolume → gauge theory constant), this metric reduces to 2 2 1 2 2 ds = L ds + ds 7 , (1.3) 4 AdS4 S
2 2 7 where dsAdS4 and dsS7 are the metrics of (planar) AdS4 and S , respectively. The curvature length scale L in Planck units is given by L = (32π2N)1/6 , (1.4) lp which tells us that small curvature, and validity of the supergravity limit, requires N 1. We can further simplify: in this background the 11D supergravity admits a consistent truncation down to 7
7 2 4D = 8 gauged supergravity on AdS4 (this essentially gets rid of the S ). This theory will N be described in greater detail in section 2.1.
Maldacena [3] argued that at low energies both this near-horizon sector and the worldvolume theory should decouple from the rest of the spacetime. Thus, he conjectured that these two theories are in fact equivalent, describing the same physics. This is very surprising; for one thing, even if you discount the compact dimensions of the S7, the supergravity theory lives in a spacetime of one more dimension than the worldvolume theory! However, the evidence in favor of the conjecture is by now quite impressive. And it turns out that the extra holographic dimension in fact has a fairly natural interpretation as an energy scale of the dual QFT, with the UV (high energies) at the AdS boundary, and the IR (low energies) in the deep interior. For pure AdS space, there is an isometry that moves points along this dimension — this corresponds to the field theory being conformal, looking the same at all energies. Of course, in a CFT we can break conformality by turning on sources for various relevant operators. This corresponds to deforming the boundary conditions of
AdS, as we will see in more detail shortly.
Let us make some remarks on terminology: The AdS spacetime where the string/supergravity theory lives is often referred to as the bulk, while the worldvolume gauge theory is referred to as the boundary since it can roughly be viewed as living on the boundary of the AdS spacetime.
Furthermore, the term AdS/CFT is often used interchangeably with the terms gauge/gravity duality and holography. Strictly, the latter two should probably be thought of as more general
— gauge/gravity could refer to a duality between a gauge theory that is not a CFT and a gravity theory in a non-AdS spacetime, while the term holography could refer to any description of a
(gravitational) theory by another theory in a lower-dimensional spacetime.
2 “Consistent” here means that solutions to the 4D supergravity are guaranteed to be solutions to the full 11D supergravity as well. 8
1.2.1 The Dictionary
Having specified the two theories of our correspondence, we must now get quantitative — how can we relate results of supergravity computations to information about the gauge theory?
The most significant entry into the dictionary of AdS/CFT was formulated by Gubser, Klebanov and Polyakov [4], and by Witten [5]. This GKPW prescription tells us that we should identify the partition functions on both sides of the duality. Moreover, the source (x) of any particular J operator (x) in the CFT should be identified with the boundary conditions of a corresponding field O φ(x, r) in AdS space. Now, the partition function of string theory in a general curved background is not very well understood. But in the supergravity limit, which we will work in throughout this dissertation, it can safely be replaced by the exponential of the on-shell supergravity action, with the same boundary conditions imposed. Thus, the supergravity action acts as a generating function for correlation functions in the dual CFT:
R 4 d xJ (x)O(x) −SAdS[ φ(x,r→∞)=J (x)] e CFT = e (1.5) h i
Which supergravity fields source which gauge theory operators can often be determined from sym- metries; a scalar field sources a scalar operator, a gauge field sources a conserved current, the metric sources the boundary stress tensor, and so on.
To make this somewhat more concrete, let us consider a specific scalar field φ in the super- gravity, dual to a specific operator in the gauge theory. Solving the Klein-Gordon equation for O φ in AdS space, we find two independent solutions whose asymptotic behaviors are
φ±(x) φ(x, r)± + ... for r , (1.6) → r∆± → ∞ where the ellipsis denotes terms of higher order in r. The GPKW prescription now instructs us
3 to identify either φ+(x) or φ−(x) with the source for the dual operator. Thus by fixing the source we impose one boundary condition. We need one more to fully fix a solution of the second
3 Usually the leading order term corresponds to the source, but not always. The subleading term acting as a source is called “alternate quantization”, and will be encountered later in this dissertation. Also note that we are neglecting subtleties due to holographic renormalization. 9 order equation; in a Euclidean AdS spacetime this is done by demanding regularity throughout the bulk.4 With these boundary conditions given a solution can be found, and by taking n functional derivatives of the source (x), n-point correlation functions can be computed. As an example, if J we identified the source with φ+(x), we would find (x) φ−(x). hO i ∼
1.3 Applications of AdS/CFT
We are now ready for the meat and bones of this dissertation, namely what we can do with the holographic tools that we have introduced. First of all, the reader will hopefully agree that an exact dynamical duality between two systems with such differing characteristics is fascinating with or without direct applications, as it reveals a very non-trivial and surprising aspect of QFT. As such, it is easy to argue for the study of this duality for its own sake — it is almost guaranteed to provide insights into the deeper workings of QFT, as well as string theory and quantum gravity in general.
However, we will choose a slightly more ambitious route here, by attempting to apply
AdS/CFT to problems inspired by experimental phenomena. There are two possible approaches to this. The one we will mostly follow is referred to as top-down; this means that the duality under study can be explicitly embedded in string theory (or possibly some other consistent theory of quantum gravity). This includes the “AdS4/ABJM” duality outlined in the previous section, the
“AdS5/ = 4 SYM” duality, and several others. In fact, since string theory admits a plethora of N vacua, many of which are asymptotically AdS, we expect many different holographic dualities to exist. Thus from our current point of view, we see that the infamous landscape problem of string theory vacua is more of a feature than a bug.
With the knowledge that AdS/CFT dualities are plentiful, one might feel brave enough to go one step further: By postulating a classical gravity theory, the GKPW prescription could be used to compute any desired physical observable of its presumed dual theory (even allowing for a
4 In a spacetime with Minkowski signature, there are often several possible boundary conditions — this is related to the more complicated causality structure. If the end goal is to compute retarded two-point functions, as is mainly the case in this dissertation, infalling boundary conditions are imposed in the bulk. 10 comparison with experimental data), all without specifying anything but the broadest features (such as symmetries) of the dual! This approach is known as bottom-up, and has been very popular in the applied AdS/CFT community. In part this is due to convenience; the known dualities coming from string theory tend to involve complicated gravitational theories with unrealistic properties such as supersymmetry. By instead focusing on universal sectors of classical gravity theories (for example, Einstein-Maxwell-scalar theories) one can hope to draw general conclusions which will be true for generic QFTs admitting a gravity dual. A prominent example of this is the proposed bound on the ratio of shear viscosity to entropy density [6]. On several occasions in our top-down forays, we will discover features anticipated from bottom-up constructions arising naturally in top-down embeddings, sometimes in a generalized form. This shows that the two approaches can complement each other very well.
Having now discussed two different ways in which holographic duality can be applied, we must next discuss to what it can be applied. Many applications fall into one of two categories:
finite density and non-equilibrium. Not coincidentally, these are the two main areas where numerical approaches such as lattice field theory run into problems. Non-equilibrium applications are often motivated by experiments at heavy ion colliders. They can also be useful when describing quenches in smaller laboratory experiments. Many interesting results have been obtained in this area [7]; however, we will not spend time detailing these, since our main interest will be in the other category of finite density QFT. These applications are generally inspired by problems from condensed matter physics, and are sometimes termed AdS/CMT, for “condensed matter theory”.
This is what we turn to next.
1.3.1 Condensed Matter Physics and AdS/CFT
Let us start by asking what problems involving strong interactions exist in the field of con- densed matter physics. An important example is that of non-Fermi liquids, also called strange metals. These notably arise in the phase diagram of many high-temperature superconductors, as the ordered phase from which the superconducting condensate forms. As such, understanding this 11 phase is likely vital to understanding the mechanism giving rise to the high critical temperature.
To see what is “strange” about these metals, let us first recall what would be considered normal. Most conventional metals are well-described by Fermi liquid (FL) theory. From a modern effective field theory point of view, this is simply the statement that the free Fermi gas is an attrac- tive low energy fixed point [8]. A theory of fermions at finite density is therefore generally expected to be described by a ground state consisting of a Fermi surface, together with excitations consist- ing of electron/hole-type quasiparticles that are asymptotically stable at low energies. Among the predictions that result from this is a resistivity that scales with temperature as T 2. The Fermi surface ground state is not perfectly stable; notably, the BCS mechanism causes pairing between electrons at low temperature, giving rise to (conventional) superconductivity.
The strange metals still display Fermi surfaces. However, the excitations around the surface are not well-defined quasiparticles, and the measured resistivity is linear in temperature, often over a large range of temperatures. A large amount of theoretical work has yet to reach a fully satisfactory description of these materials. Proposed explanations have included coupling the Fermi surface excitations to a gapless bosonic mode, such as an order parameter near a critical point or an emergent gauge field, which can shorten the lifetime of the fermionic excitations [9]. Interestingly, we will later arrive at some qualitatively similar interpretations arising naturally through holographic models.
To study a finite density of matter, and to look for holographic strange metals, we need some conserved charge in the CFT, and we need to know its dual in the gravity theory. In relativistic theories, the charge density operator is the time-component of a conserved four-current — as mentioned already, these naturally couple to spin-1 gauge fields on the gravity side. Thus, we need to turn on something like an electric field in the bulk gravity theory; from its falloff at the
AdS boundary, we can then read off the dual QFT’s charge density. To turn on an electric field we need a density of charge to source it — this is could be achieved in many ways, for example by introducing a charged black hole. This gives an AdS-Reissner-Nordstr¨om(AdSRN) spacetime, which has been something of a workhorse of the applied holography community. 12
In fact, introducing a black hole provides another feature for free. Namely, it introduces a non-zero temperature — the Hawking temperature — in the dual field theory. Thus, by tuning two parameters of the black hole, say its horizon radius and charge, we are able to tune the temperature and chemical potential of the dual state. Utilizing the GKPW prescription, various physical observables of the dual state can then be calculated. In particular, [10] computed fermionic two-point functions from a bottom-up point of view, showing that as the fermion couplings are tuned, the AdSRN state can display Fermi surface singularities (poles in the fermionic two-point function at zero energy and non-zero momentum) with either FL or non-FL behavior. They further argued that a great deal of the low-energy properties of the AdSRN geometry, as well as more general black holes we will encounter later, can be explained by the near-horizon region — this is natural if we remember to think of the holographic radial direction as an energy scale with the IR being towards the interior of the bulk. For extremal (zero-temperature) AdSRN black holes, the near
D−2 horizon region takes the form of AdS2 R , where D is the total spacetime dimension of the × bulk. Holographically, AdS2 should be dual to a 1D CFT, and indeed it can be seen to imply the emergence at low energies of a particular scaling symmetry under which time scales but space does not. The whole AdSRN bulk can then be viewed as a renormalization group flow connecting the conformal UV theory “living” near the AdS boundary with an emergent theory with this symmetry in the IR.
Having seen that AdS/CFT can describe phases with similarities to strange metals, one might naturally ask if it can also display superconductivity. In fact, it can; as argued in [11], charged black holes in AdS coupled to scalars are sometimes unstable to the formation of black hole “hair”, that is, a non-trivial profile for the scalar field outside the horizon. The asymptotic value of this scalar field is dual to a non-zero symmetry-breaking condensate, giving rise to superconductivity
[12] (or more precisely superfluidity, since the broken symmetry of the CFT is global). In practice, one finds two different black hole solutions to the equations of motion plus boundary conditions, one with the scalar turned on and one with it turned off. By computing the free energy of the dual states, we can check which one is thermodynamically favored at different temperatures. As one 13 might hope from analogy with the real world, the symmetric solution without the scalar turned on dominates at high temperature; as the temperature is lowered, a critical temperature is encountered below which the solutions with a condensate takes over.
1.4 Summary of Dissertation
In the next chapter we will discuss in some detail the ABJM M2-brane theory, as well as the 4D = 8 gauged supergravity we will use to study it. First, however, we will summarize the N original work of this dissertation, as detailed in chapters3–6. We will see both non-FL behavior and superconductivity appearing in ABJM theory through our holographic lens, sometimes in a generalized form, and sometimes singling out specific behaviors.
1.4.1 Top-down Non-Fermi Liquids and a Special Black Hole
Chapter3 starts from 4D = 8 gauged supergravity, which as discussed is dual to ABJM N theory. This theory admits solutions generalizing the AdSRN geometry, with four independent charges corresponding to the four diagonal U(1)s contained in the theory’s SO(8) gauge symme- try. By solving linearized Dirac equations (derived from the full supergravity Lagrangian) in the extremal limits of these backgrounds, we are able to compute fermionic two-point functions, uncov- ering a plethora of Fermi surface singularities. Interestingly, the spectrum of excitations around all of these Fermi surfaces is exclusively of non-FL type. As this was true in previous top-down studies as well [13], it raises the question of whether this will always be true when the bulk solution can be embedded in string theory (or more generally, in any consistent theory of quantum gravity).
The large class of black hole solutions studied in chapter3 contains a special case where one of the four charges is set to zero while the rest are non-zero, termed the three-charge black hole
(3QBH). This case is shown to be interesting for several reasons. In the zero-temperature limit the horizon area of most of the charged black holes stays finite, implying a non-zero ground state entropy in the dual QFT state.5 The 3QBH horizon, however, disappears at zero temperature,
5 This likely indicates that these charged black holes are not true ground states of the dual theory; instead they 14 removing this issue. Furthermore, we find a qualitatively different behavior of the fermionic two- point functions in the 3QBH state. While for the generic black holes, the fermions are unstable at all energies away from the Fermi surface, spectral functions computed in the 3QBH show stability within an extended interval of energies. A plausible interpretation of this is offered: The emergent
AdS2 IR sector, which for the regular black holes facilitated the decay of fermionic excitations, has here developed a gap. Attractively, this provides an explanation for both the stable interval and the vanishing of the ground state entropy.
1.4.2 Fermions in Supergravity Superconductors
Having discussed examples of holographic ordered states, chapters4 and5 then goes on to introduce symmetry breaking. By considering two consistent truncations of the full 4D gauged supergravity action, the bosonic sector is reduced to a more manageable one involving the metric, a U(1) gauge field, and a charged scalar. All the necessary ingredients for a holographic supercon- ductor are thus present, and indeed it has been shown that both regular AdSRN and hairy black holes exist as solutions in both truncations. Again we will be interested in computing fermionic spectral functions in the dual states of these black hole geometries. This is most conveniently done at zero temperature (for one thing, if there exists a Fermi surface, it will only be sharply defined at T = 0). The zero temperature limits of hairy black hole geometries is less straightforward than for regular black holes, partly because the solutions are only known numerically. However, it has been argued to correspond to an interpolation from the maximally symmetric AdS4 in the UV, to another less symmetric AdS4 in the IR. These domain wall geometries have the interesting holo- graphic interpretation of a conformal QFT, that when perturbed by a chemical potential embarks on a renormalization group flow; on the way it develops a non-zero expectation value for a scalar operator, and finally settles down into an IR where conformal invariance has been fully restored.
This complete conformality in the IR turns out to have some interesting effects on the fermion spectrum. As the IR theory has Lorentz symmetry (being a subset of the conformal symmetry) we may represent an intermediate energy phase [14]. ω ω ω 15
k k k
Figure 1.1: A cartoon showing dispersion relations of stable fermionic modes in purple, all existing within stable regions whose edges are shown in blue. As the stable modes hit the blue edge, they acquire a finite lifetime. Three different geometries are shown: The regular black holes (left), where the only stable modes are right at the chemical potential; the 3QBH (right), where a stable momentum-independent interval opens up; and a superconducting domain wall (middle), where the stable region lies outside the emergent lightcone.
might expect to find some kind of lightcone structure, and this is indeed what happens. Inside this lightcone, we find the IR again acting as a “bath” into which the fermions can decay. Outside the lightcone, the fermions are kinematically isolated from this bath; similarly to the 3QBH, they then become perfectly stable.
At this point, we can already notice a general theme: Some important properties of the fermionic spectra are determined entirely by the emergent holographic IR theory deep in the bulk.
Particularly, if the fermion mode is kinematically unable to decay into the IR theory, it is perfectly stable. This is visualized for three different types of IR geometries in figure 1.1, where “stable regions” and dispersion relations of stable fermionic modes are sketched.
Fermionic spectra have previously been computed in bottom-up constructions of holographic superconductors. For a minimal bulk Dirac equation, it was found that the spectrum in general supports Fermi surfaces [15]. This can be contrasted with the Bardeen-Cooper-Schrieffer mechanism of conventional superconductors, where elementary fermions near the Fermi surface pair together into bosons, destroying the Fermi surface. In the interest of reproducing this feature, Faulkner et al. [16] engineered a particular coupling that forces the fermionic spectrum to be gapped, essentially through level repulsion. Interestingly, in one of the two truncations under study we find precisely this type of coupling, albeit in a slightly more complicated form, naturally occurring in 16 the supergravity Dirac equations. Chapter4 first considers a simplified version of these top-down couplings, as well as several other bottom-up Dirac equations, and finds that Fermi surface zero- modes in general still occur. Then, chapter5 considers the full top-down Dirac equations, and now a gap appears in all spectral functions. In this chapter, the other truncation is also considered; in that case the special coupling does not occur, but it turns out that the specific values of the fermion charge and mass imposed by supergravity gives a gapped spectrum anyway. Thus, all the top-down fermions in all of the symmetry-breaking background under study display gapped spectra, analogously to the case of conventional superconductors. It is tempting to speculate that this is a generic feature of top-down holographic superconductivity.
1.4.3 Correlations between Correlators: Charge Oscillations and Fermi Surfaces
In chapter6, we address a question which one might think should have a straightforward answer: How do the fermionic excitations studied in the earlier chapters affect the transport prop- erties of the finite density states? In a conventional, weakly coupled system it is obvious that there should be a connection; the fermionic sector, consisting of e.g. electrons, is what “makes up” the material. Changes in the spectra of these fermions is certain to cause changes in conductivities, for example; most dramatically, the (non-)existence of a Fermi surface would be expected to leave a large imprint.
However, in a holographic bottom-up construction, this gets turned on its head. Here, one typically begins by specifying the bosonic Lagrangian, and by solving the resulting equations of motion for various geometries that are asymptotically AdS. From these geometries, one can then compute various “bosonic” properties such as conductivities, susceptibilities, and viscosities — all without saying a word about fermions! A bottom-up holographer can then decide to include fermions, by writing down an arbitrary Dirac equation. Solving this, a Fermi surface may or may not be found; regardless, the previously computed transport properties will not change at all. This is a strange feature of holography, which can be traced back to the fact that it provides a classical limit of a QFT, even though the form of the classical theory is highly unexpected. Fermions are 17 inherently quantum mechanical, and thus appear to affect bosonic properties only at subleading order in N, when quantum effects in the bulk come into play.
On the other hand, in a top-down construction the full Lagrangian is specified right from the start, including both bosonic and fermionic parts (often related by supersymmetry). One can therefore ask if the requirement of a UV completion might, in some subtle way, enforce a relationship between fermionic and bosonic properties. Chapter6 studies the static charge susceptibility in the 3QBH geometry. In a weakly interacting theory with a Fermi surface, this quantity should exhibit non-analytic behavior at a momentum of 2kF , where kF is the Fermi momentum. This is not observed for the state dual to the 3QBH. Non-analyticities do appear, but at a complex momentum whose real part is approximately equal to 1kF . We further observe that a top-down embedding of the AdSRN geometry exhibits a similar “1kF ” relationship. Thus, our observations hint at a possible relationship between fermionic and bosonic properties, although no link as clear as at weak coupling is visible.
We conclude the dissertation in chapter7. In the appendices some further relevant material is collected: In appendixA we show how to lift of the 3QBH discussed in chapters3 and6 to five dimensions, and in appendixB we give some further details on the computation of the static charge susceptibility in chapter6. Chapter 2
The Two Theories
In this chapter, we will review some key properties of four-dimensional = 8 gauged super- N gravity and three-dimensional superconformal ABJM theory. The discussion here is based largely on material from [17, 18, 19, 20].
2.1 4D = 8 Gauged Supergravity N
As explained in chapter1, the AdS/CFT dual of the M2-brane theory is given by the near-
7 horizon limit of a stack of M2-branes, which is M-theory on an AdS4 S background with N units × of 4-form flux on AdS4 [3,2]. In the large- N limit, M-theory reduces to eleven-dimensional super- gravity. The Kaluza-Klein reduction of eleven-dimensional supergravity on S7 [21, 22, 23] includes an infinite tower of supersymmetry multiplets; the theory of the modes sharing the multiplet of the four-dimensional massless graviton is four-dimensional = 8 (maximal) gauged supergravity N [24, 25], which represents a consistent truncation of the higher-dimensional theory [26]. This is the theory we will discuss in some detail in the rest of this section, and the one we will do calculations in for the majority the dissertation.
Let us start by discussing symmetries. Four-dimensional ungauged = 8 supergravity1 N
contains an global, noncompact E7(7) symmetry, whose maximal compact subgroup is SU(8).
There is also a local SU(8) symmetry; this will have associated indices i, j = 1 ... 8, with indices up in the fundamental and indices down in the antifundamental. The gauged supergravity which
1 The ungauged version of the maximal supergravity can be obtained by dimensionally reducing from eleven dimensions on a seven-torus instead of a seven-sphere. 19 we are interested in gauges an SO(8) subgroup of E7(7), with indices I,J = 1,... 8 in the 8s and no distinction made between upper and lower. This gauging breaks the full global E7(7) symmetry, leaving us with local SO(8) SU(8) invariance. Naturally, there will be gauge fields associated × with both SO(8) and SU(8); however, as we will see only the former will have kinetic terms and correspond to dynamical degrees of freedom, while the latter will be determined in terms of the physical fields of the theory through a constraint equation.
In notation common throughout the litterature, an object is defined with SU(8) indices in a particular up/down configuration, and then raising/lowering all the indices corresponds to complex conjugation. Thus if we define Xi, we have
i ∗ Xi (X ) . (2.1) ≡
For SO(8) indices, no distinction is made between up or down indices, though they may be con- ventionally raised and lowered along with the SU(8) indices.
2.1.1 Bosonic Sector
µ IJ The bosonic fields are the vierbein e µˆ, 28 gauge fields Aµ in the adjoint of SO(8), and 35 complex scalars, with the real parts parity-even and the imaginary parts parity-odd. The vierbein
µ field e µˆ corresponds to the graviton, which is a singlet under both SO(8) and SU(8). From the
IJ [IJ] SO(8) gauge fields Aµ = Aµ , we define the field strengths
IJ IJ IJ IK KJ F ∂µA ∂νA 2gA A . (2.2) µν ≡ ν − µ − [µ ν]
It is useful to define imaginary (anti-)self-dual field strengths F ± in terms of the field strength F and the dual field strength F˜:
± IJ 1 IJ IJ IJ 1 ρσIJ F F iF˜ , F˜ µνρσF , (2.3) µν ≡ 2 µν ± µν µν ≡ 2 which obey the imaginary (anti-)self-dual and conjugation relations,
F ± IJ = iF˜± IJ , (F ± IJ )∗ = F ∓ IJ , (2.4) µν ± µν µν µν 20 with the duals F˜± defined analogously to F˜. The inverse relations are
F + IJ + F − IJ = F IJ ,F + IJ F − IJ = iF˜ IJ . (2.5) µν µν µν µν − µν µν
The scalar fields parameterize an E7(7)/SU(8) coset space and can be written in the form of a 56 56 matrix coset representative (sechsundf¨unfzigbein) × IJ u vijKL ij = . (2.6) V klIJ kl v u KL Here each pair IJ or ij is antisymmetric, and may be thought of as a single composite index running from 1 to 28, decomposing the 56 56 coset representative into 28 28 blocks corresponding to the × × u- and v-tensors. The coset representative transforms by the local SU(8) on the left, and the global
E on the right. There are 133 independent degrees of freedom in , and 63 may be removed by 7(7) V SU(8) transformations, leaving 70 degrees of freedom matching the number of real scalars degrees of freedom we want. The inverse of is V ij u IJ vklIJ −1 = − , (2.7) V vijKL u KL − kl which transforms by SU(8) on the right and E7(7) on the left, and the fact that it is the inverse requires relations between u and v:
IJ kl klKL kl KL IJ uij u IJ vijKLv = δij , vijKLukl uij vklIJ = 0 , − − (2.8) ij IJ klIJ IJ ij kl u u vklKLv = δ , u vijKL vklIJ u = 0 , KL ij − KL IJ − KL −1 −1 where the first two come from = I and the second two from = I, and we have defined VV V V 1 δIJ (δI δJ δI δK ) . (2.9) KL ≡ 2 K L − L J
Let µ be the SU(8)- and SO(8)-covariant derivative. On an SO(8) index (up or down doesn’t D matter) we get
I I IJ J µX = µX gA X , (2.10) D ∇ − µ where includes the spin connection and g is the gauge coupling, while for SU(8) indices we have ∇
i i 1 i j 1 j µY = µY + Y , µZi = µZi Zj , (2.11) D ∇ 2Bµ j D ∇ − 2Bµ i 21 which contains the “composite connection” i , which is antihermitian and traceless, Bµ j
i = i , i = 0 . (2.12) Bµ j −Bµj Bµ i
Thus we have covariant derivative relations like
IJ IJ k IJ K[I J]K µu = ∂µu + u 2gA u . (2.13) D ij ij Bµ [i j]k − µ ij
As already mentioned, i does not represent separate dynamical degrees of freedom; instead it Bµ j will be determined by imposing the constraint 0 −1 1 µijkl µ = A , (2.14) D V·V −2√2 mnpq µ 0 A resulting in
i 2 ik IJ ikKL µ j = u IJ Dµujk v DµvjkKL B 3 − (2.15) 2 ik IJ ikKL IJ ik JK ikIK = u ∂µu v ∂µvjkKL 2gA u u v vjkJK , 3 IJ jk − − µ IK jk − ijkl while the off-diagonal entries define µ : A
IJ IJ µijkl = 2√2 vklIJ Dµuij ukl DµvijIJ A − (2.16) IJ IJ IJ JK IK = 2√2 vklIJ ∂µu u ∂µvijIJ 2gA vklIK u u vijJK . ij − kl − µ ij − kl
Here we have introduced Dµ, which is covariant only with respect to SO(8),
I I IJ J i i DµX = µX gA X , DµY = µY , DµZi = µZi , (2.17) ∇ − µ ∇ ∇ and so
IJ IJ K[I J]K Dµu = ∂µu 2gA u , (2.18) ij ij − µ ij and analogously for DµvijIJ . We note that µijkl is totally antisymmetric and self-dual up to A conjugation,
ijkl [ijkl] 1 mnpq = , ijkl = ijklmnpq . (2.19) Aµ Aµ A 4! A
It turns out that µijkl is the proper notion of a covariant derivative of the scalars, and will appear A in the action. 22
The action also contains a number of tensors built out of the scalars, which start with the
T-tensor,
jkl kl klIJ JK jm jmKI T = (u + v )(u u vimJK v ) , (2.20) i IJ im KI − which obeys the identities
jkl j[kl] ikl [ij]k [kij] ikj jki Ti = Ti ,Ti = Tk = Tk = 0 ,Tk = Tk . (2.21)
The T-tensor is composed of two independent pieces, called the A-tensors,
[jkl] 2 [k 3 [jkl] 3 [k l]j T jkl = T + δ T l]mj A + δ A (2.22) i i 7 i m ≡ −4 2i 2 i 1 where we have defined
4 4 [jkl] Aij = T ikj ,A jkl = T , (2.23) 1 21 k 2i −3 i which obey
ij ji jkl [jkl] A1 = A1 ,A2i = A2i . (2.24)
Finally, the Pauli terms require the S-tensor, which is defined in terms of the equation
ij ijIJ IJ,KL ij (u IJ + v )S = u KL . (2.25)
Using 2.8 one can show that
SIJ,KL = SKL,IJ . (2.26)
In practice, it is convenient to evaluate the scalar sector in a fixed gauge for SU(8). A convenient gauge is 1 0 φIJKL = exp , (2.27) V − √ 2 2 φMNPQ 0 where the 70 scalars are described by the complex φIJKL, which are totally antisymmetric and
(complex conjugate) self-dual,
1 MNPQ φIJKL φ , φIJKL = IJKLMNP Qφ . (2.28) ≡ [IJKL] 4! 23
The real parts of φIJKL are parity-even and the imaginary parts parity-odd. In this gauge, SU(8) and SO(8) indices are indistinguishable, and we use I,J for both; raising/lowering indices still corresponds to complex conjugation.
Having introduced all the necessary formalism, we can now write down the bosonic super- gravity Lagrangian: −1 1 1 ijkl µ 2 3 1 2 1 2 2 e bosonic = R + g A A (2.29) L 2 − 96Aµ Aijkl 4| ij| − 24| ijkl| 1 IJ,KL F + 2SIJ,KL δIJ F +µν + F − 2S δIJ F −µν (2.30) − 8 µνIJ − KL KL µνIJ − KL KL
We can immediately learn a few things by setting all of the scalar fields in this Lagrangian to zero.
Besides canonical kinetic terms for the gauge fields, this gives the usual Einstein-Hilbert term and a cosmological constant:
R + 12g2 + ... , (2.31) L ∝ which has AdS4 solutions with characteristic scale L given by
1 L = . (2.32) √2g
This maximally symmetric solution is dual to the conformal vacuum of ABJM theory.
2.1.2 Quadratic Fermion Action
i The fermionic content of the supergravity theory is 8 spin-3/2 gravitini ψµ in the 8s of SU(8),
ijk [ijk] and 56 spin-1/2 dilatini χ χ in the 56s of SU(8) (or SO(8) after the above mentioned gauge ≡ fixing). While these are Majorana, they transform in a complex representation of SU(8), and it is natural to write the action in terms of chiral projections of Majorana spinors. We thus add a
ijk subscript M clarifying when we are dealing with a Majorana spinor, e.g. χM , and define chiral projections as
ijk ijk ijk χ PRχ and χijk PLχ . (2.33) ≡ M ≡ M
Note the index placement; in a basis where Majorana spinors are real, Γ5 is imaginary, and rais- ing/lowering indices again becomes complex conjugation. 24
In this dissertation, we will only solve linearized Dirac equations (this is sufficient for com- puting two-point functions holographically). We will therefore only need the terms of the full supergravity Lagrangian that are quadratic in fermion fields. Moreover, to simplify we will only consider the spin-1/2 fields that decouple from the gravitini. Which spin-1/2 fields decouple depends on the bosonic background geometry, and can often be determined by symmetry considerations, as discussed in more detail in chapters3,4 and5. With this in mind we drop terms involving gravitini; the relevant terms of the Lagrangian involving fermions are then [25]
−1 i ijk µ i ijk µ e χχ¯ = χ¯ Γ µχijk χ¯ ←−µΓ χijk L 12 D − 12 D 1 F + SIJ,KLO+µνKL + h.c. (2.34) − 2 µνIJ √2 ijklmnpq r + g A χ¯ χpqr + h.c. , 144 2 lmn ijk with the fermion tensor O+ defined through
ij +µνIJ √2 ijklmnpq µν u O = χ¯ Γ χnpq . (2.35) IJ 288 klm
Let us carefully process this into a more convenient form. First, we make the hermitian conjugate terms explicit, obtaining
−1 i ijk µ e χχ¯ = χ¯ Γ µχijk L 6 D √2 + IJ,KL −1 KL ijklmnpq µν FµνIJ S (u ) ij χ¯klmΓ χnpq − 576 (2.36) √2 − IJ,KL ∗ −1 ij klm µν npq F (S ) (u ) ijklmnpqχ¯ Γ χ − 576 µνIJ KL √2 ijklmnpq r lmn ijk pqr + g A χ¯ χpqr + A χ¯ χ , 144 2 lmn ijk ijklmnpq 2r where we integrated by parts to combine the two parts of the kinetic term (discarding boundary terms), and
−1 IJ ij IJ −1 ij KL KL (u ) iju KL = δKL , (u )IJ uij = δIJ . (2.37)
Let’s now make the chiral projectors of (2.33) explicit. Note that in all spinor bilinears, the projector coming from theχ ¯ can be moved to the right until it encounters the one coming from the χ; by this point they will be the same (if they were opposite the bilinear would vanish), and 25 since the projectors square to themselves we can get away with simply writing the projector for the χ field. Now all the fermions are Majorana, with indices up; we’ve broken the meaning of the up/down index structure. Thus instead of indicating the conjugate of the scalar objects by raising/lowering indices, we show it explicitly with the conjugation symbol. We obtain
−1 i ijk µ ijk e χχ¯ = χ¯ Γ µPLχ L 6 M D M √2 klm µν h + IJ,KL −1 ij ∗ − IJ,KL ∗ −1 ij i npq ijklmnpqχ¯ Γ F S ((u ) ) PL + F (S ) (u ) PR χ − 576 M µνIJ KL µνIJ KL M √2 ijk lmn ∗ lmn pqr + g χ¯ (A ) PL + A PR χ . 144 ijklmnpq M 2r 2r M (2.38)
lmn IJ −1 ij Here we have used the notation that A2r and uij (and hence (u )IJ ) are the “original” functions and given the others as their conjugates.
The kinetic terms can be processed further. The covariant derivative on the chiral spinor is
1 l 1 l 1 l µχijk = µχijk iχljk jχilk kχijl . (2.39) D ∇ − 2Bµ − 2Bµ − 2Bµ Plugging this in, we arrive at
−1 i ijk µ ijk i ijk µ l l ∗ l l ∗ ljk e χχ¯ = χ¯ Γ µχ χ¯ Γ + ( ) Γ5( ( ) ) χ L 12 M ∇ M − 16 M Bµ i Bµ i − Bµ i − Bµ i M √2 klm µν h IJ,KL −1 ij ∗ IJ,KL ∗ −1 ij i npq ijklmnpqχ¯ Γ FµνIJ S ((u ) ) PL + (S ) (u ) PR χ − 576 M KL KL M √2 ijk lmn ∗ lmn pqr + g χ¯ (A ) PL + A PR χ , 144 ijklmnpq M 2r 2r M (2.40) where we also simplified the term involving the field strengths using
µν + − µν Γ FµνPL + FµνPR = Γ Fµν . (2.41)
Equation (2.40) will be our starting point when writing down Dirac equations in chapters4 and5.
In the special case where all the scalar tensors are real, it simplifies to
−1 i ijk µ ijk i l ijk µ ljk e χχ¯ = χ¯ Γ µχ χ¯ Γ χ L 12 M ∇ M − 8Bµ i M M √2 IJ,KL −1 ij klm µν npq FµνIJ S (u ) ijklmnpqχ¯ Γ χ (2.42) − 576 KL M M √2 + g ijklmnpqA lmnχ¯ijkχpqr , 144 2r M M which will be our starting point in chapter3. 26
2.2 The M2-brane ABJM Theory
The four-dimensional gauged supergravity theory discussed in the previous section holograph- ically describes a set of low-dimension operators in the maximally superconformal theory living on a stack of N coincident M2-branes. For a single M2-brane the theory is 8 free scalars in the 8v and 8 free spinors in the 8c of the SO(8) R-symmetry; for N > 1, however, the theory becomes interacting. While it can be characterized as the IR limit of three-dimensional Super-Yang-Mills theory, it is most explicitly formulated as ABJM theory.
ABJM theory ([27, 28, 29, 30,2]; for reviews see [31, 32]) is a 3D U(N) U(N) Chern- × Simons theory at levels2 (k, k) coupled to bifundamental matter. The full Lagrangian will not − be essential to us here; it can be found in [33], for example. The manifest supersymmetry is = 6 N and the manifest global symmetry is SU(4) U(1)b. For general k this represents the theory of N × M2-branes at a Zk orbifold singularity. However, for the cases k = 1, 2 there is an enhancement to = 8 supersymmetry and SO(8) R-symmetry; the decomposition of the eight-dimensional N representations of SO(8) into SU(4) U(1)b are ×
8v 41 4¯−1 , 8c 4¯1 4−1 , 8s 60 12 1−2 . (2.44) → ⊕ → ⊕ → ⊕ ⊕
Here we are interested in the k = 1 case, corresponding a stack of N M2-branes with no orbifold.
The bifundamental matter may be written as four complex scalars Y A, A = 1 ... 4 in the 4 of
SU(4) and four complex spinors ψA in the 4¯; both sets of fields are in the N N of U(N) U(N) × × and neutral under U(1)b. Alone these fields do not assemble into complete SO(8) representations; however, they combine with monopole operators, representing the scalars dual to the gauge fields, into gauge-invariant objects with proper SO(8) transformation properties. We will denote by eqτ the monopole operator with U(1)b charge q in the q-fold tensor product of N N; monopole × 2 The Chern-Simons level enters the action as the coefficient of Chern-Simons terms of the form k Z „ 2i « S = d3xµνρ Tr A ∂ A − A A A . (2.43) CS 4π µ ν ρ 3 µ ν ρ
Consistent quantization requires k ∈ Z. 27 Table 2.1: 4D supergravity modes and their dual ABJM operators.
a i IJ SUGRA Mode eµ ψµ Aµ χijk Re φijkl Im φijkl SO(8) Rep 1 8s 28 56s 35v 35c Dual ABJM Operator T µν µ J µ Y ψ Y 2 ψ2 S R Conformal dimension ∆ 3 5/2 2 3/2 1 2
operators are neutral under SU(4). We then have gauge-invariant operators such as
A τ ¯ † −τ ¯ τ †A −τ 41 : Y e , 4−1 : YAe , 41 : ψAe , 4−1 : ψ e , (2.45) assembling into complete 8v and 8c representations according to (2.44). It is these combinations that are analogous to the free bosons and fermions in the N = 1 case; the ABJM presentation fractionalizes the symmetry carriers into ordinary matter charged under SU(4) and monopole operators charged under U(1)b, which bind into gauge-invariant “composite” bosons and fermions.
The supergravity modes discussed in the previous section are dual to such gauge-invariant operators. These are described in table 2.1, with the dual ABJM operators indicated schematically.
The first three sets of operators in the table are the energy-momentum tensor, supercurrents and
SO(8) R-symmetry currents. The 28 R-symmetry current operators include 15 SU(4) currents,
±2τ one U(1)b current, and 12 additional operators including e monopoles, corresponding to the decomposition
28 150 10 62 6−2 . (2.46) → ⊕ ⊕ ⊕
We note that while some operators include monopoles and some do not, the enhancement to full
SO(8) symmetry means that they are all treated on equal footing. Indeed, one can imagine distinct embeddings of SU(4) U(1)b inside SO(8) where a monopole operator in one case becomes a non- × monopole operator in the other.
2.2.1 A Simplified Description
While the complete Chern-Simons-matter theory above is the proper description of coincident
M2-branes, when identifying gauge-invariant operators we will sometimes find it sufficient to think 28 simply about taking 8 scalar fields X and 8 Majorana fermions λ describing the case for a single
M2-brane (N = 1), and generalizing these to N N matrices; this oversimplified way of describing × the theory neglects important aspects such as the monopole operators vital for the supersymmetry enhancement, but allows us to conveniently describe the operators we are interested in. We will use this simplified picture in chapter3 and in parts of chapter5. In this description, the scalar X and fermion λ transform in the 8v and 8c representations of SO(8), respectively. The 8v scalars
X may be arranged into complex combinations, each of which has charge 1 under precisely one ± of the SO(8) Cartan generators. The 8c fermions λ are each charged under all four generators, with charges ( 1 , 1 , 1 , 1 ) + permutations. Turning on the chemical potential for each Cartan ± − 2 2 2 2 generator will thus affect only two bosons, but all eight fermions.
In the X, λ notation, the chiral primary operators we will be interested in have the form TrXk, k = 2, 3,..., with dimensions ∆ = k/2. The lowest-dimension chiral primary Tr X2 transforms in the 35v of SO(8), and its first descendent is the lowest-dimension gauge-invariant fermionic operator
TrXλ, which has ∆ = 3/2 and sits in the 56s. This operator will be studied in several of the coming chapters. We will also have cause to mention the second descendant, the bosonic operator Tr λ2 with ∆ = 2 in the 35c. A table of operators for this theory using this notation may be found in
[34]. Chapter 3
Fermi Surface Behavior in the ABJM M2-brane Theory
This chapter is an edited version of [17], written in collaboration with Oliver DeWolfe and
Christopher Rosen.
3.1 Introduction and Summary
3.1.1 Holographic Realizations of Non-Fermi Liquids
Many systems of interacting fermions, including most metals, behave as Landau-Fermi liquids, where the interactions dress the fermions into quasiparticles whose fluctuations around a Fermi surface are asymptotically stable at low energies. However, a number of interesting strongly coupled systems — notably cuprate superconductors [35, 36] and heavy fermion systems [37] — display
“strange metal” behavior which deviates from the Fermi liquid paradigm. In such systems, a Fermi surface is evident, but the fluctuations are not stable, and transport properties are correspondingly different. It is of interest to develeop theoretical mechanisms to study such “non-Fermi liquids”.
The gauge-gravity correspondence, or AdS/CFT correspondence [3,4,5], has become a valu- able tool for exploring strongly coupled systems that lack a straightforward quasiparticle descrip- tion. Systems at zero temperature and finite density are described holographically by charged, extremal, asymptotically anti-de Sitter black hole geometries living in one higher dimension [38].
Normal modes of fermions in such backgrounds compute fermionic Green’s functions, whose zero energy, finite momentum poles may be interpreted as Fermi surface singularities, with near-pole behavior determining the dispersion of nearby excitations. 30
Such systems were considered first from a “bottom-up” perspective, where simple Dirac equations were postulated and studied in Reissner-Nordstr¨omblack brane backgrounds [39, 40,
41, 10]. These studies showed that holographic Fermi surfaces could indeed exist, and depending on the charge and mass parameters of the fermion, could manifest either Fermi liquid behavior, with asymptotically stable quasiparticles, or non-Fermi liquid behavior, where the decay width of excitations typically remains of the same size as the energy. Thus gravity duals to systems having non-Fermi liquid behavior were shown to be possible, albeit in systems whose precise field theory dual is not known. Initial studies included (constant) masses and gauge couplings; Pauli couplings were added in [42, 43].
A natural next step is to study “top-down” constructions, where the black brane backgrounds and fluctuating fermions are part of a known supergravity theory descending from string theory, and hence have a precisely known field theory dual. Natural candidate theories for such a study are
= 4 Super-Yang-Mills (SYM) theory in four spatial dimensions, and the = 8 supersymmetric N N Aharony-Bergman-Jafferis-Maldacena (ABJM) theory in three dimensions; these theories are max- imally superconformal and are the most symmetric avatars of four-dimensional non-Abelian gauge theory and three-dimensional Chern-Simons-matter theory, describing the dynamics of stacks of
D3-branes and M2-branes, respectively. The finite-density behavior of these theories is interesting in its own right, and the gauge-gravity correspondence provides an opportunity to study them at strong coupling and large N.
Fluctuations of the gravitino field in supergravity were studied in [44, 45, 46], but no Fermi surface singularities were found. The first Fermi surfaces were identified in [47], where one fermion in one particular background for each of = 4 SYM and ABJM theories was shown to have a N Fermi surface with non-Fermi liquid behavior. A systematic study of the = 4 Super-Yang-Mills N case was carried out in [13], where the Dirac equation of every spin-1/2 fermion not mixing with the gravitino was solved across a one-dimensional slice of the two-dimensional parameter space defined by ratios of the three SO(6) chemical potentials. The Dirac equations were more complicated than the typical bottom-up examples, featuring mass and Pauli terms that depend on scalar fields that 31 generically vary in the background. Every value of the chemical potentials showed at least one fermion with a Fermi surface, and in all cases, the excitations near the Fermi surfaces displayed non-Fermi liquid behavior. As the chemical potentials varied, in general Fermi momenta vary but the existence of a Fermi surface persists, except when the Fermi momentum enters a so-called oscillatory region, where the Green’s function displays log oscillatory behavior and the Fermi surface singularity cannot exist. One class of fermion asymptoted to the case separating Fermi and non-
Fermi liquid behaviors, the marginal Fermi liquid (MFL) which was proposed as a description of the optimally doped cuprates [35], as it approached the edge of the parameter space.
For generic values of the chemical potentials, the extremal black brane backgrounds possess a regular event horizon, which implies a nonzero entropy at zero temperature. Such a feature is shared by the Reissner-Nordstr¨ombackgrounds studied in many bottom-up models, but is somewhat unusual from the field theory point of view. In [14], it was suggested that such states should be understood not as the true ground state of the dual gauge theory, but instead as states in a semi-local quantum liquid (SLQL) phase characterized by scaling at intermediate energies, before a true ground state phase emerges due to the condensation of instabilities or the manifestation of subleading N effects. There are exceptions to this behavior, however, at the edges of the = 4 N SYM chemical potential parameter space. When two of the three charges are set to zero, the
“extremal” geometry loses its horizon, becoming a non-thermodynamic renormalization group (RG)
flow geometry previously studied in [48, 49, 50, 51, 52]. Perhaps more interestingly, when one of the three charges is set to zero, the geometry becomes singular at the horizon, and the entropy at zero-temperature correspondingly vanishes [53, 54]. This case was studied in detail in [55], where it was shown how a lift to six dimensions resolves the singularity as well as providing constraints on consistent parameters for fermion fields. It was found that there is a region in energy around the
Fermi surface where the fermionic fluctuations are perfectly stable, before returning to non-Fermi liquid behavior outside. An interpretation of this is a gap developing in another sector, removing a large number of degrees of freedom and depriving the fermions of the catalyst for their decay. It also shares features with the semi-local quantum liquid resolutions described in [14], as the non-Fermi 32 liquid behavior exists at intermediate energies, while the true ground state is controlled by a Fermi surface with Fermi liquid-like excitations and vanishing entropy.
Hence it has been demonstrated that non-Fermi liquid behavior exists in nonzero-density gauge theories at strong coupling, and studied in great detail for four-dimensional = 4 Super- N Yang-Mills. Given the associations to cuprate superconductors and other strongly correlated sys- tems in two spatial dimensions, it is natural to extend the thorough, systematic study of [47, 55] to the case of the maximally supersymmetric ABJM M2-brane theory, both for its potential ap- plication to realistic systems and for its inherent interest as one of the maximally superconformal theories. This is the goal of the present chapter.
3.1.2 Fermionic Response in the M2-brane Theory
The M2-brane theory has an SO(8) R-symmetry, and hence 4 distinct chemical potentials.
7 The dual description is M-theory on AdS4 S , which reduces to four-dimensional = 8 gauged × N supergravity. Finite density black brane solutions corresponding to rotating M2-brane systems are known in a truncated theory of the metric, gauge fields and three scalars, but no fermions [56].
We use the known embedding [57] of this truncated theory to lift the solutions to the full = 8 N gauged supergravity, and we use these backgrounds to derive the corresponding Dirac equations for all spin-1/2 fluctuations with quantum numbers forbidding mixing with the gravitino. We then solve these Dirac equations in the black brane backgrounds with the infalling boundary conditions at the horizon that calculate retarded Green’s functions in the dual field theory. Because the mass of the fermions approaches zero at the boundary, there is an ambiguity between which terms in the near-boundary expansion to identify as the source, and which as the response; the mass functions are nonzero away from the boundary, however, so the choice has physical content. We demonstrate how to use supersymmetry to resolve this ambiguity, producing a unique prescription for the dual
Green’s functions.
We calculate Green’s functions over two one-dimensional cuts through the three-dimensional space of chemical potential ratios, one where three charges are set equal, and one where the charges 33 are set equal in pairs; these cuts meet at the point where all four charges are equal. Results over this parameter space are strongly in accord with the = 4 SYM case. In particular, Fermi N surface singularities are common and are in all cases associated to non-Fermi liquid behavior. In particular, one class of excitations, the net-charged fermions, are qualitatively identical to the higher-dimensional case; Fermi surfaces persist as the chemical potentials are varied unless the
Fermi momentum falls into an oscillatory region. One such fermion again approaches marginal
Fermi liquid behavior at a limit of the parameter space. The other class of excitations, so-called net-neutral fermions, shows novel behavior: while all Fermi surface singularities still show non-
Fermi liquid behavior, there are no oscillatory regions, and yet a Fermi surface can discontinuously appear or disappear at a nonzero value of the Fermi momentum as one tunes the chemical potentials past the four-charge black hole. This abrupt change in the spectrum at zero temperature as a dimensionless parameter is varied is reminiscent in aspects of a quantum phase transition; however, no singularities in the susceptibilities are visible in the thermodynamics.
Some interest has appeared recently in identifying zeros of a fermionic Green’s function as a sign of Mott insulator behavior, and a duality between zeros and poles in certain bottom-up models has been noted [58, 59]. We also obtain the zeros of the Green’s function, which are also of interest as in the alternate quantization of the fermions — which would correspond to an alternate theory breaking supersymmetry — they exchange roles with the poles. In this alternate quantization, ordinary Fermi liquid behavior would appear for certain excitations, while the true ABJM theory has only non-Fermi liquid excitations. We note that the zero/pole duality of [58, 59] is a consequence of the symmetry of the Dirac equation under a flip of chirality, and does not obtain for our models where the mass and Pauli couplings are nonzero.
As for = 4 SYM, the ABJM theory again has exceptional cases at the limits of parameter N space. When one of the four charges is set to zero, we encounter again a naively singular geometry, with vanishing entropy at zero temperature. An analysis closely following [55] holds, again revealing a region in energy around the Fermi surface where the fermionic fluctuations are perfectly stable.
As in [55], there is a lift to a higher dimension resolving the singularity, which also results in a 34 constraint between the mass, charge and Pauli couplings of consistent fermions, which are obeyed by all the cases in the maximal gauged supergravity. When two or three charges are set to zero, we find renormalization group flow solutions, with only a running scalar modifying the geometry.
While these backgrounds are non-thermodynamic, they may be of interest both in their relation to the nonzero temperature backgrounds with the same charge, and as RG flow geometries in their own right. In these cases we are able to solve for the fluctuations of fermions, and find the corresponding Green’s functions, exactly.
Overall a similar picture has emerged for the ABJM case as for the = 4 case: the bulk of the N parameter space, with all charges nonzero, leads to regular black holes dual to zero temperature states with nonzero entropy showing non-Fermi liquid behavior. Limits of the parameter space either lack horizons, or are naively singular, resulting in zero entropy states with an energy gap around the Fermi surface where the fermionic fluctuations are stable. For the ABJM case, moreover,
Fermi surfaces appear and disappear discontinuously around the most symmetric point in the parameter space, suggestive of a quantum phase transition.
In section 3.2, we describe the reduction of four-dimensional maximally supersymmetric gauged supergravity to the truncated model, present the general black brane solution with four charges, and derive the Dirac equations of the theory’s fermions in these backgrounds. In sec- tion 3.3, we review methods for solving these Dirac equations and generating Green’s functions, presenting discrete symmetries of the equations and using supersymmetry to resolve the apparent ambiguity in quantization for the fermions in asymptotically anti-de Sitter space. In section 3.4, we review the properties of such equations in the background of regular extremal black holes, present the solutions with three charges set equal and with charges set equal in pairs, and numerically obtain Fermi surface singularities and their corresponding Fermi momenta, as well as oscillatory regions and the locations of zeros of the Green’s function, for each fermion. In section 3.5, we con- sider the special case with three charges equal and one charge zero, and demonstrate the existence of an energy gap wherein the fluctuations are exactly stable, and solve for the dispersion relations for each fermion throughout this region. We match these results on to the limit of the regular black 35 holes. In section 3.6, we exactly solve the Dirac equations in the backgrounds where two and three charges are zero, with the rest set equal, and again match the results onto the limit of the regular sequence. Certain details of the lift of the three-charge geometry to five dimensions are presented in appendixA.
3.2 Black Branes and Dirac Equations in Maximal Gauged Supergravity
We begin this section by showing how four-dimensional maximally supersymmetric gauged supergravity reduces to a truncated bosonic theory and present its black brane solutions. Then we derive the Dirac fluctuation equations for the set of spin-1/2 fields not mixing with the gravitino in these backgrounds.
3.2.1 Black Brane Solutions
Following Duff and Liu [56], we can reduce to a truncated theory including only the metric, the four Cartan gauge fields, and three scalars φA using the ansatz
1 12 34 13 24 14 23 φIJKL = [φ1( + )IJKL + φ2( + )IJKL + φ3( + )IJKL] . (3.1) √2
αβ Here the special Levi-Civita symbols IJKL are non-zero only when the indices I, J, K, L take values within the index pairs specified by the superscripts, where α = 1, ..., 4 runs over the SO(8) index pairs 12, 34, 56, 78 . For example, 13 = 1( 1) when I, J, K, L is an even (odd) permutation of { } IJKL − 1, 2, 5, 6.
By plugging (3.1) into (2.27) one may now calculate the u and v tensors in terms of this
a b scalar ansatz, and from them the T -, A- and S-tensors. We also define the gauge fields Aµ, Aµ,
c d IJ Aµ, Aµ in terms of the Cartan generators Aµ as 12 a Aµ 1 1 1 1 Aµ 34 b Aµ 1 1 1 1 1 Aµ − − , (3.2) ≡ 2√2 A56 1 1 1 1 Ac µ − − µ A78 1 1 1 1 Ad µ − − µ 36 where the matrix may be thought of as an SO(8) triality rotation [56], which diagonalizes the couplings to the scalars. The Lagrangian for this restricted set of fields is then
−1 1 2 2 1 X −λi 2 2e = R (∂φ~) + (cosh φ1 + cosh φ2 + cosh φ3) e F , (3.3) L − 2 L2 − 4 i i=a,b,c,d where
λa φ1 φ2 φ3 , λb φ1 + φ2 + φ3 , λc φ1 φ2 + φ3 , λd φ1 + φ2 φ3 , (3.4) ≡ − − − ≡ − ≡ − ≡ − and where we have defined1 L, 1 g = . (3.5) √2L Families of black brane solutions are known in this truncated theory [56, 57]. The black branes asymptote to the Poincar´epatch of four-dimensional anti-de Sitter space. In general the three scalars of the truncated theory run with the radial coordinate, and the electric potentials of the four gauge fields are turned on as well, which will be associated with the nonzero chemical potentials.
The solutions are of the form [57],
2B(r) 2 2A(r) 2 2 e 2 ds = e ( h(r)dt + d~x ) + dr ,Ai = Φi(r)dt , φA = φA(r) . (3.6) 4 − 2 h(r)
They are characterized by four charges Qi and a mass parameter, the latter of which we may trade for a horizon radius rH . It is convenient for us to take Qi > 0, and separate out the signs of the gauge fields ηi 1. Then in terms of the functions ≡ ±
Qi Hi = 1 + , (3.7) r the solutions are r 1 X A(r) = B(r) = log + log Hi , (3.8) − L 4 i
r(r + Qa)(r + Q )(r + Qc)(r + Q ) h(r) = 1 H H b H H d , (3.9) − rH (r + Qa)(r + Qb)(r + Qc)(r + Qd) 1 HaHb 1 HaHc 1 HaHd φ1 = log , φ2 = log , φ3 = log , (3.10) 2 HcHd 2 HbHd 2 HbHc √ 1 Our normalization of g is from [25] and matches [56]; [57] uses a g smaller by 1/ 2. 37 r p ηi Qi (rH + Qa)(rH + Qb)(rH + Qc)(rH + Qd) rH + Qi Φi = 1 . (3.11) L rH rH + Qi − r + Qi
The horizon r = rH is the largest zero of the horizon function h(r). These solutions are asymptot- ically anti-de Sitter at large r,
r A(r ) = B(r ) log , → ∞ − → ∞ → L
h(r ) 1 , φA(r ) 0 , Φi(r ) const , (3.12) → ∞ → → ∞ → → ∞ → with AdS radius L. These black brane solutions, when lifted to 11D, have the interpretation as rotating M2-brane configurations, with the conserved charges corresponding to conserved angular momenta in the eight directions transverse to the branes; this is analogous to the five-dimensional solutions studied in [13], corresponding to rotating D3-branes.
The thermodynamics may be calculated from standard formulas, with the temperature T and entropy density s determined by the metric,
1 0 A(rH )−B(rH ) 1 2A(rH ) T = h (rH )e , s = e , (3.13) 4π 4G and the chemical potentials µi and charge densities ρi for the conserved charges from the near- boundary expansion of the gauge fields,
8πGLρ Φi(r ) µL + ... (3.14) → ∞ → − r
The results are p (rH + Qa)(rH + Qb)(rH + Qc)(rH + Qd) 1 1 1 1 1 T = 2 + + + + 4πL −rH rH + Qa rH + Qb rH + Qc rH + Qd (3.15) 1 p s = (rH + Qa)(rH + Q )(rH + Qc)(rH + Q ) , (3.16) 4GL2 b d r p ηi Qi (rH + Q1)(rH + Q2)(rH + Q3)(rH + Q4) µi = 2 . (3.17) L rH rH + Qi r ηi Qi ρi = s . (3.18) 2π rH
Extremal black holes have T = 0, and generically display a double pole in h(r rH ). It will be → extremal solutions that we will focus on. 38
The simplest special case is the so-called four-charge black hole (4QBH) where Qa = Qb =
Qc = Qd; here the scalars all vanish and we are left with a Reissner-Nordstr¨omblack brane. If
Aa = Ab = Ac = Ad Φ4(r)dt, then the 4QBH solution is ≡ 4 r Q4 r(rH + Q4) A(r) = B(r) = log + log 1 + , h(r) = 1 4 , (3.19) − L r − rH (r + Q4) r η4 Q4 rH + Q4 Φ4(r) = (rH + Q4) 1 . (3.20) L rH − r + Q4
Other simplifications can be chosen where two or three charges are set equal, which we will discuss in later sections. Interesting special cases arise when one or more charges Qi vanishes, which we will explore in turn. For now, we turn to the fermionic Lagrangian. In what follows, we take all the signs of the charges to be positive, ηi = +1.
3.2.2 Linearized Dirac Equations
We are interested in the quadratic action for spin-1/2 fields. As mentioned in chapter2, these may general mix with the spin-3/2 gravitini. The 56s representation consists of 32 unique weight vectors, along with three copies of the weights of the 8s. Because the bosonic fields turned on in the background are all neutral under the Cartan gauge fields U(1)a U(1)b U(1)c U(1)d, × × × and the action must respect this gauge symmetry, fermi fields can only mix in the quadratic action if they have the same weight vector. Thus the 32 spin-1/2 fields that have unique weight vectors cannot mix with the gravitini or each other. We will therefore consider these fields, and drop the couplings to the gravitini.
Since the scalar ansatz and the tensors derived from it furthermore are real, we may then start from (2.42), which we reproduce here for convenience:
−1 i ijk µ ijk i l ijk µ ljk e χχ¯ = χ¯ Γ µχ χ¯ Γ χ L 12 M ∇ M − 8Bµ i M M √2 IJ,KL −1 ij klm µν npq FµνIJ S (u ) ijklmnpqχ¯ Γ χ (3.21) − 576 KL M M √2 + g ijklmnpqA lmnχ¯ijkχpqr 144 2r M M 39
Thinking of the χijk as a 56-component vector ~χ, this Lagrangian can be written in the form
−1 1 µ e χχ¯ = χ~¯(iΓ µ1 + Q + P + M)~χ (3.22) L 2 ∇ where 1, Q, P and M are 56 56 matrices for the kinetic, gauge, Pauli and mass-type terms, × respectively. We evaluate these matrices, using that in this truncation
l = 2gA l (3.23) Bµ i − µ i and redefining the gauge fields as in (3.2). The matrices can then be diagonalized in order to
find eigenvectors and eigenvalues. Diagonalizing first the gauge term, we find 32 eigenvectors with distinct, non-degenerate eigenvalues, and 24 eigenvectors that are degenerate in groups of three, as expected. The latter contain some additional mixing to the gravitini which we have ignored, and therefore we set them aside. The remaining 32 cannot mix thanks to gauge invariance, and are therefore also eigenvectors of the mass and Pauli terms.
In general the eigenvectors are complex linear combinations of the form χ = χ1 + iχ2 where
χ1 and χ2 are two of the χijk, and are hence Dirac spinors; 16 are then conjugates of the other 16.
The Dirac equation for these eigenvectors takes the form
h i µ 1 X λi/2 1 µ X i i µν X −λi/2 i iΓ µ + mie + Γ qiA + Γ pie F χ = 0 . (3.24) ∇ 4L 4L µ 8 µν i=a,b,c,d i=a,b,c,d i=a,b,c,d
Here, mi, qi, and pi, i = a, b, c, d are integer numbers characterizing each fermion. The λi are combinations of the scalars given in (3.4). Table 3.1 shows the 16 independent eigenvectors. The
16 conjugate fermions simply have (mi, qi, pi) (mi, qi, pi). The qi in the table are proportional → − − to the 32 weight vectors of the 56s representation of SO(8) with norm √3, as expected, and can be characterized as follows: one of the four qi is qi = 3. Of the remaining three qj, an odd number
(either one or all three) are 1, with the remaining charges, if any, equal to +1. There are 16 − such combinations. Using the simplified description of ABJM theory from section 2.2.1, we identify the dual operators in the table, where the complex scalars Zj X2j−1 + iX2j, j = 1, 2, 3, 4 have ≡ weight vectors proportional to (+1, 0, 0, 0) and permutations, and Λj λ2j−1 + iλ2j, j = 1, 2, 3, 4 ≡ 40 are complex combinations of the eight spinors with weight vectors proportional to ( 1 , 1 , 1 , 1 ) and − 2 2 2 2 permutations.
Table 3.1: The 16 independent fermion eigenvectors that do not mix with the gravitini.
(qa,q ,qc,q ) χ b d Operator ma mb mc md qa qb qc qd pa pb pc pd (+3,−1,+1,+1) χ Tr Z1Λ2 3 +1 +1 +1 +3 1 +1 +1 1 1 +1 +1 (+3,+1,−1,+1) − − − − χ Tr Z1Λ3 3 +1 +1 +1 +3 +1 1 +1 1 +1 1 +1 (+3,+1,+1,−1) − − − − χ Tr Z1Λ4 3 +1 +1 +1 +3 +1 +1 1 1 +1 +1 1 (−1,+3,+1,+1) − − − − χ Tr Z2Λ1 +1 3 +1 +1 1 +3 +1 +1 1 1 +1 +1 (+1,+3,−1,+1) − − − − χ Tr Z2Λ3 +1 3 +1 +1 +1 +3 1 +1 +1 1 1 +1 (+1,+3,+1,−1) − − − − χ Tr Z2Λ4 +1 3 +1 +1 +1 +3 +1 1 +1 1 +1 1 (−1,+1,+3,+1) − − − − χ Tr Z3Λ1 +1 +1 3 +1 1 +1 +3 +1 1 +1 1 +1 (+1,−1,+3,+1) − − − − χ Tr Z3Λ2 +1 +1 3 +1 +1 1 +3 +1 +1 1 1 +1 (+1,+1,+3,−1) − − − − χ Tr Z3Λ4 +1 +1 3 +1 +1 +1 +3 1 +1 +1 1 1 (−1,+1,+1,+3) − − − − χ Tr Z4Λ1 +1 +1 +1 3 1 +1 +1 +3 1 +1 +1 1 (+1,−1,+1,+3) − − − − χ Tr Z4Λ2 +1 +1 +1 3 +1 1 +1 +3 +1 1 +1 1 (+1,+1,−1,+3) − − − − χ Tr Z4Λ3 +1 +1 +1 3 +1 +1 1 +3 +1 +1 1 1 (+3,−1,−1,−1) − − − − χ Tr Z1Λ¯ 1 3 +1 +1 +1 +3 1 1 1 1 1 1 1 (−1,+3,−1,−1) − − − − − − − − χ Tr Z2Λ¯ 2 +1 3 +1 +1 1 +3 1 1 1 1 1 1 (−1,−1,+3,−1) − − − − − − − − χ Tr Z3Λ¯ 3 +1 +1 3 +1 1 1 +3 1 1 1 1 1 (−1,−1,−1,+3) − − − − − − − − χ Tr Z4Λ¯ 4 +1 +1 +1 3 1 1 1 +3 1 1 1 1 − − − − − − − −
The mi are then determined by the qi: if qi = 3, mi = 3, while if qi = 1, mi = 1. Finally | | − | | the pi are all 1, and are simply the ratios ±
mi pi = , (3.25) qi for each i. Thus the four charges completely characterize the Dirac equation. We note that for each fermion the mi satisfy
ma + mb + mc + md = 0 . (3.26)
We find it useful to sort these fermions into two categories: the first 12 are net-charged fermions, P P for which only one qi is 1, and for which qi = +4 and pi = 0, while the final four are the − i i P P net-neutral fermions, for which three qi are 1, and for which qi = 0 and pi = 4. − i i − 41
3.3 Fermionic Green’s Functions
In this section, we discuss how to solve the Dirac equation obtained in the previous section, and review how the retarded Green’s function may be obtained from such a solution. There is an apparent ambiguity in how to treat the quantization of this fermionic fluctuation, and we discuss how this ambiguity is resolved by supersymmetry.
3.3.1 Solving the Dirac Equation
Solutions to Dirac equations of the form 3.24 were discussed in [10] for constant mass and gauge couplings, and Pauli couplings were added in [42, 43]. Further development, including cases with scalar-dependent couplings, was carried out in [13, 60, 55]. We begin by Fourier transforming the t, ~x directions and rescaling the spinor χ,
χ (e6Ah)−1/4e−iωt+ikxψ , (3.27) ≡ where ω is the frequency and k is the spatial momentum (chosen to lie in the x-direction) of the fermion mode. The factor of (e6Ah)−1/4 is chosen so as to exactly cancel the spin connection term coming from µ in the Dirac equations above. Next we choose a Clifford basis where the relevant ∇ matrices are block diagonal, iσ3 0 σ1 0 iσ2 0 rˆ tˆ ˆi Γ = , Γ = , Γ = . (3.28) 0 iσ3 0 σ1 0 iσ2 − We can characterize the four components of the spinor as
ψα± ΠαP±ψ . (3.29) ≡ with α = 1, 2, in terms of the projectors
1 α rˆ tˆ ˆi 1 rˆ Πα 1 ( 1) iΓ Γ Γ ,P± 1 iΓ . (3.30) ≡ 2 − − ≡ 2 ± 42
The two-component objects ψ+ and ψ− (each with both values of α) transform as three-dimensional
Dirac spinors. However, it is in terms of the two-component objects ψα, ψα− ψα = (3.31) ψα+
(which are not lower-dimensional spinors) that the Dirac equation decomposes into two decoupled pairs of equations:
(∂r + Xσ3 + Y iσ2 + Zσ1)ψα = 0 , (3.32) where
B B−A B−A e X λi/2 e e α X = mie ,Y = u , Z = ( 1) k v , (3.33) − √ − √ − √ − − 4L h i h h with −B 1 1 X e X −λi/2 u = ω + qiΦi , v = pie ∂rΦi . (3.34) √ 4L 4 h i i
We note that the solutions for ψα=1 and ψα=2 are related to each other simply by k k. → − We may turn the coupled first-order equations 3.32 into decoupled second-order equations for each component,
00 0 0 2 2 2 ψ F±ψ + X X + Y Z XF± ψα± = 0 , (3.35) α± − α± ∓ − − ± with F± ∂r log ( Y + Z), where we keep in mind that 3.32 keeps the solutions for different ≡ ∓ components from being independent. The form 3.35 is convenient for an analysis at r , but → ∞ for r rH it is convenient to define the combinations [60], →
U± ψ− iψ+ (3.36) ≡ ± in terms of which the Dirac equations become
0 0 U + iY U− = ( X + iZ)U+ U iY U+ = ( X iZ)U− . (3.37) − − + − − −
From here one can derive the uncoupled second-order equations
00 0 0 2 2 2 U + pU + (iY X + Y Z + iY p)U− = 0 − − − − 00 0 0 2 2 2 U +pU ¯ + ( iY X + Y Z iY p¯)U+ = 0 . (3.38) + + − − − − 43 with p ∂r log( X + iZ). ≡ − − We note there are two independent discrete transformations acting on the Dirac equation.
Conjugation is implemented by showing that if χ satisfies the Dirac equation with parameters
rˆ ∗ mi, qi, pi , then Γ χ satisfies it with parameters mi, qi, pi . Conjugation of 3.27 also ex- { } { − − } changes the signs of k and ω, so the net transformation is
Conjugation : q q , p p , ω ω , k k , (3.39) → − → − → − → − which is equivalent to Y Y , Z Z; one can see the ψ± second-order equations respect this → − → − symmetry, while it exchanges the equations for U+ and U−. Meanwhile one can also show that if
χ satisfies the Dirac equation with parameters mi, qi, pi , then Γ5χ satisfies it with parameters { }
mi, qi, pi . The chirality matrix exchanges both ψ+ and ψ− and the two values of α; since the {− − } latter is equivalent to flipping the sign of k, we have
Chirality flip : m m , p p , k k , ψ+ ψ− , (3.40) → − → − → − ↔ which is X X, Z Z; while this exchanges ψ+ ψ−, it is a symmetry of the U± equations. → − → − ↔
3.3.2 Quantization of Fermi Fields and Green’s Functions
To define any field in an asymptotically anti-de Sitter space, one must impose appropriate boundary conditions at r . The functions appearing in the Dirac equation have the asymptotic → ∞ behavior 2 2 m0L ωL˜ kL X ,Y ,Z , (3.41) → r → − r2 → − r2 with m0 the value of m(φ) at infinity, and
ω˜ ω + qA0(r ) . (3.42) ≡ → ∞
We discuss this first for the case of general m0, discussed in [61], and then specialize to our case, where m0 = 0. The behavior 3.41 leads to the near-boundary second-order equation,
2 2 00 2 0 m0L m0L ψ + ψ ± ψα± , (3.43) α± r α± − r2 44
The asymptotic solutions are then,
m0L −m0L−1 −m0L m0L−1 ψα+ Aα+(ω, k)r + Bα+(ω, k)r , ψα− Aα−(ω, k)r + Bα−(ω, k)r , ∼ ∼ (3.44) or in terms of the original spinor,
−d/2+m0L −d/2−m0L−1 χα+ Aα+(ω, k)r + Bα+(ω, k)r , ∼
−d/2−m0L −d/2+m0L−1 χα− Aα−(ω, k)r + Bα−(ω, k)r . (3.45) ∼
Using the full Dirac equation on the asymptotic solution 3.45, one finds that the B± are not independent of the A∓, but rather are derivatives of them:
L2(˜ω ( 1)αk) Bα∓ = ± − Aα± , (3.46) 2m0L 1 ∓
One must choose whether A+ or A− is the mode that one imposes boundary conditions on; this is only allowed when the mode is normalizable, which depends on the value of m0. The chosen mode is then interpreted as the response (vev) of the dual operator, while the other mode is interpreted as the source. The retarded Greens function is then given by the ratio of the response over the source, for a solution of the Dirac equation for which infalling boundary conditions have been imposed at the black hole horizon.
The A− quantization is allowed for m0L > 1/2 and corresponds to a dual operator with −
∆ = d/2 + m0L, with Green’s function
Aα− GR,α = , (A− quantization) (3.47) Aα+ while the A+ quantization is allowed for m0L < 1/2 and corresponds to a dual operator with
∆ = d/2 m0L, with Green’s function −
Aα+ GR,α = . (A+ quantization) (3.48) Aα−
For the range 1/2 < m0L < 1/2, both quantizations are possible. Note that the Green’s function − is diagonal on the space of two-component spinors α = 1, 2; in what follows we will pick a single component α = 2 for convenience, knowing that GR,1(k) = GR,2( k). − 45
Now for our special case m0 = 0, we find ψ+ and ψ− have the same scaling in r:
Bα+(ω, k) Bα−(ω, k) ψα+ Aα+(ω, k) + , ψα− Aα−(ω, k) + . (3.49) ∼ r ∼ r
The leading term in X is now
m1 X = + ..., (3.50) r2 and the relations between the B∓ and A± from the first-order equations are modified to
2 α Bα∓ = L (˜ω ( 1) k)Aα± m1Aα∓ . (3.51) ∓ ± − ±
There is now an ambiguity in the identification of the source and the response: both A+ and
A− appear in symmetric fashion with the same scaling in r, appropriate to the situation where
∆ = 3/2 = d/2, and the operator and its source have the same conformal dimension. In the case of scalar fluctuations in AdS/CFT, the ∆ = d/2 case involves a term of the form r−d/2 and a term of the form rd/2 log r, and there is only one conformally invariant choice of quantization [62]. For spinors, however, there is no log and two choices of quantization are possible.
In the simple case of a fermion with m = p = 0 exactly, the chirality flip 3.40 implies that the two quantizations are equivalent up to k k, so there is no loss of generality to simply picking → − one; this is the case usually discussed in the literature, for example [10]. However in our case, both p and m are nonzero (though m is asymptotically zero) and depend on r. In this more general situation, the two different choices of quantization lead to distinct physics; in particular, for us, they will exchange poles of the fermionic Green’s function with zeros. This exchange in the m = 0 case was noted in [58]. Thus we must find a way to resolve the ambiguity to correctly identify the fermionic response.
To resolve the issue, we will use supersymmetry. The 70 scalars of maximal gauged super- gravity are divided into 35v parity-even scalars with ∆ = 1, and 35c pseudoscalars with ∆ = 2.
All the scalar modes, however, asymptotically have m2L2 = 2. The well-known analog of 3.45 for − scalars is
r 2 −∆− −∆+ d d 2 2 φ = A−(~x,t) r + ... + A+(~x,t) r + ..., ∆± = + m L . (3.52) 2 ± 4 46 which for the case at hand gives ∆− = 1, ∆+ = 2. Again there is a choice of quantization [62], and to match the dual field theory, we must place the 35v scalars in the alternate quantization to get
∆ = 1, and the 35c pseudoscalars in the regular quantization to obtain ∆ = 2. We now show how supersymmetry relates this choice of scalar quantization to a definite choice of spinor quantization.
These results were discussed in pre-AdS/CFT language in [63, 64]; for a related discussion see [65].
It is sufficient to consider a single = 1 supersymmetry, under which a scalar φ, a pseu- N doscalar and a Majorana spinor χ assemble into a single chiral multiplet. The action for such a P multiplet is
Z 1 4 µν µν µ 2 2 2 2 S = d x√ g g ∂µφ∂νφ g ∂µ ∂ν + iχ¯Γ µχ m φ m mχχ¯ , (3.53) 2 − − − P P ∇ − φ − P P − where the scalars have masses
m 2 m 2 m2 m2 , m2 m2 + . (3.54) φ ≡ − L − L2 P ≡ L − L2
Being in anti-de Sitter space has split the three masses of the multiplet, but all masses are deter- mined by the single fermion mass parameter m. It is straightforward to see that as m varies from m = to m = , the scalar mass-squareds go down from infinity, reach a minimum at the −∞ ∞ Breitenlohner-Freedman bound m2 L2 = 9 , and go back to infinity. The action 3.53 is invariant BF − 4 under the transformations
µ 1 δφ =εχ ¯ , δ = iε¯Γ5χ , δχ = iΓ ∂µ(φ + iΓ5 ) + (φ iΓ5 ) + m(φ + iΓ5 ) ε . P − P L − P P (3.55)
Our strategy is to use a Killing spinor of the AdS background to generate a near-boundary solution for the scalars from a near-boundary solution of the spinor; this will match the fluctuations on which boundary conditions are imposed between the scalar and spinor sectors, which will allow us to choose our spinor quantization. A Killing spinor is obtained by requiring that the gravitino supersymmetry variation [25],
i i ji δψ = 2 µε i√2gA Γµεj + ..., (3.56) µ ∇ − 1 47
ij ij vanishes. In AdS space where A1 = δ this becomes
i i 1 i δψ = 2 µε Γµε = 0 . (3.57) µ ∇ − L
The r-dependent Killing spinor solution for any i is:
1/2 (0) ε(r) = r ε+ , (3.58)
rˆ with Γ -chirality ε+ P+ε+ as in 3.30. The supersymmetry variations of the scalars with this ≡
Killing spinor as supersymmetry parameter then each involve only one of χ±,
δφ =εχ ¯ − , δ = iε¯Γ5χ+ . (3.59) P
Consider the A− quantization of χ; this is permitted for mL 1/2, and has ≥ − 3 ∆χ = + mL . (3.60) 2
We consider a fluctuation of χ with no “source” term; thus only A− and B+ are turned on:
−5/2−mL −3/2−mL χ+ = B+r , χ− = A−r . (3.61)
We then find the corresponding scalar fluctuations, −mL−1 1 (0) −mL−2 i (0) δφ = r ε¯+ A− , δ = r ε¯+ Γ5B+ . (3.62) √2 P √2
Thus supersymmetry requires we pick the quantizations of the scalars giving the operator dimen- sions
∆φ = 1 + mL , ∆P = 2 + mL . (3.63)
Analogously, the A+ quantization would lead to ∆φ = 2 mL, ∆P = 1 mL. For us, we require − −
∆φ = 1, ∆P = 2, which obtains for m = 0 in the A− quantization, with the response in χ− and the source in χ+. Thus we have resolved the ambiguity, and 3.47 will be our expression for the fermionic Green’s function.
The A+ quantization would place the scalars in the regular quantization and the pseudoscalars in the alternate quantization, contrary to maximal gauged supergravity. In principle this repre- sents some other non-supersymmetric AdS/CFT dual pair. Since the Green’s functions for the 48 two quantizations 3.47 and 3.48 are reciprocals, the poles and zeros of the Green’s function are exchanged between the two. We will indicate the zeros of the Green’s function in many of our backgrounds; one may give them the alternate interpretation as Fermi surface singularities for the non-supersymmetric theory of the other quantization.
3.4 Regular Black Holes and Non-Fermi Liquids
We turn now to solving the Dirac equation to obtain retarded Green’s functions for different fermions at zero temperature and various values of the chemical potentials, obtaining information about the fermionic response over the parameter space of the ABJM theory.
In principle, one could study the entire black hole parameter space of four independent charges of black holes. However, dealing with four charges can be somewhat tedious. To simplify matters, we will consider a truncated parameter space, examining two classes of simplified black holes: one class with three charges set equal, and the other distinct (the “3+1-charge black hole”) and one with the four charges set to two values in pairs (the “2+2-charge black hole”). As we will see, each class simplifies the solutions to consist of two gauge fields and a single scalar. Since only the ratio of charges matters, the parameter space consists of two one-dimensional segments that intersect at the point where all four charges are equal, the four-charge black hole, which has vanishing scalars and is simply a Reissner-Nordstr¨omblack brane. A cartoon of the parameter space is displayed in
Figure 3.1.
Generic black branes with all four charges nonzero are “regular”, with a regular horizon, and display qualitatively similar behavior; these will be explored in this section. Novel phenomena occur when one or more charges vanish. These interesting special cases occur at the boundaries of our parameter space, and we will investigate them in more detail in future sections. While we do not cover the entire parameter space of four charges, we expect that the unexplored areas are qualitatively similar to corresponding regions in our explored space with the same number of nonzero charges. 49
2QBH
3+1QBH
1QBH 4QBH 3QBH
2+2QBH
2QBH
Figure 3.1: A cartoon of the parameter space of black holes we consider. 50
3.4.1 Regular Black Holes and Non-Fermi Liquids
Regular black holes are characterized by a regular horizon; for the extremal case there is a
2 double pole in the horizon function, h(r) (r rH ) , but the horizon remains of nonzero size. ∼ − Since the entropy density of the dual field theory is simply proportional to the area of the horizon, these systems have a nonzero entropy density even as the temperature goes to zero. Zero entropy at zero temperature will require a singular event horizon, as we review in a later section.
The fermionic response of regular black holes was considered in [10] for fermions with constant masses, and Pauli couplings were added in [43, 42]. In [13], it was shown that top-down super- gravities in five dimensions generically are of this type, albeit with masses and Pauli couplings depending on the radial coordinate; this did not change the overall structure.
Let us review how the Green’s function may be calculated in this case. One must solve the
Dirac equation for a fermionic fluctuation with infalling boundary conditions at the horizon, and then calculate the ratio 3.47 of the components near the boundary. For the general case of ω = 0, 6 this can be done straightforwardly. Near ω = 0 there is a subtlety [10]. For Dirac equations in the background of regular extremal black holes, the near-horizon (r rH ) limit has the structure → 4 2 0 2 2 00 1 0 #L ω # L ω ν U + + ... U + 4 + 3 2 + ... U = 0 , (3.64) r rH (r rH ) (r rH ) − (r rH ) − − − − where # and #0 are constants we are not interested in and ν2 is a constant we are interested in, and we have neglected both higher-order terms in 1/(r rH ) and in ω. Because the near-horizon and − small-frequency limits do not commute, to study dynamics at low energy one must define an inner region with ω 0, r rH , ω/(r rH ) fixed, where the infalling boundary condition is imposed; → → − this is then matched to an outer region with ω = 0 strictly, and the result may be extended to
d−1 small ω. The inner (IR) region for black branes in AdSd+1 has the geometry AdS2 R , and × this region governs the low energy properties of the dual gauge theory. An infalling solution at the horizon translates to a solution bridging the gap between inner and outer regions with the form
− 1 +ν − 1 −ν U (r rH ) 2 + (ω)(r rH ) 2 , (3.65) ∼ − G − 51 where the relative weighting (ω) between the two solutions depends on k and the other parameters G as well. Thinking of the two terms in 3.65 as the source and response in the near-boundary region of an AdS2 fluctuation, we may interpret (ω) as an AdS2 Green’s function. G The exponent ν takes the form
ν2 = ν2 + ν2 ν2 , (3.66) m k − q where νm depends on the mass parameters mi, νk depends on the momentum k and the Pauli couplings pi, and νq depends on the charges qi; all three depend on the ratios of chemical potentials
2 encoding where we are in the parameter space. The term νk depends on k and the pi only in the combination k˜2, where X k˜ k + αi pi µi , (3.67) ≡ i where αi are some constants and µi is the corresponding chemical potential, so the effect of the
Pauli couplings is to shift the momentum. In [14], the combination of the terms ν2 ν2 was m − q identified as being proportional to the inverse correlation length squared,
r 1 ν k˜2 + , (3.68) ∼ ξ2 where for our more general case we have replaced k2 with k˜2.
For regions where the contribution of the charge to 3.66 is not too strong, ν2 is positive and one may find Fermi surface singularities where the retarded Green’s function GR diverges at ω = 0 for some k = kF , corresponding to the vanishing of the source term A−. Negative values of kF correspond to Fermi surfaces for the antiparticles associated to our (Dirac) fermionic operators.
One may then determine the properties of excitations near the Fermi surface using (ω). G The full form of (ω) is recorded in [10]; for small ω it scales as a power law, G
(ω) = c(k) eiγk (2ω)2ν , (3.69) G | | with real quantities c(k) and γk. The phase γk can be written as | |
−2πiν −2πνq γk arg Γ( 2ν) e e . (3.70) ≡ − − 52
The retarded Green’s function near the Fermi surface for small ω takes the form
h1 GR(k, ω) , (3.71) 1 iγk 2νk ∼ k⊥ ω h2e F (2ω) F − vF − with h1, h2 positive constants and k⊥ k kF . ≡ −
While h1 and h2 depend on the details of the UV physics, certain properties are determined solely by the IR AdS2 region [10]. The denominator of 3.71 determines the dispersion relation of
fluctuations near the Fermi surface.
The nature of the dispersion relation depends crucially on νkF . For νkF > 1/2, the leading imaginary part comes from (ω), but the leading real part comes from the generic (ω) corrections G O given by the 1/vF term. In this case the ratio of excitation width Γ to excitation energy ω∗ goes to zero as one approaches the Fermi surface; the excitations are true quasiparticles and the system behaves as a Fermi liquid. The leading dispersion relation is ω∗ vF k⊥, and the residue Z ∼ quantifying the overlap between the state created by the fermionic operator and the quasiparticle excitation approaches a nonzero constant proportional to vF .
On the other hand, if νkF < 1/2, both the leading real and imaginary parts of the dispersion relation come from (ω), and they are of the same order; we can ignore the Fermi velocity vF term G as subleading. The ratio of the excitation width to its energy then approaches a constant, given by
Γ γkF = tan , k⊥ > 0 , ω∗ 2νkF γkF = tan πz , k⊥ < 0 , (3.72) 2νkF − where the exponent is 1 z . (3.73) ≡ 2νkF In this case the excitations remain unstable as one approaches the Fermi surface; this behavior is similar to what one expects in a non-Fermi liquid. The dispersion relation between the excitation energy ω∗ and the momentum k⊥ is then
z ω∗ (k⊥) . (3.74) ∼ 53
Furthermore the residue Z vanishes at the Fermi surface like
z−1 Z (k⊥) , (3.75) ∼
another property characteristic of a non-Fermi liquid. The intermediate case of νkF = 1/2 is the so-called marginal Fermi liquid, where the ratio Γ/ω∗ and the residue Z vanish logarithmically in
ω as the Fermi surface is approached.
If the charge contribution to 3.66 is sufficiently strong, ν will become imaginary. This has been interpreted as the AdS2 region developing an instability to pair creation of charged excitations
[66]. The range of k for which this is the case is called an oscillatory region, as the retarded Green’s function displays periodic behavior in log ω [40, 10]. In this case the boundary condition 3.65
−1 −1 acquires a complex exponent, and in general one cannot have Im GR = 0 even when Re GR = 0; thus there are no Fermi surface singularities, as the width of would-be excitations persists even as the energy goes to zero, washing out the Fermi surface. We will find lines of Fermi surfaces as we vary the chemical potentials that terminate at an oscillatory region.
While bottom-up models can easily show both Fermi and non-Fermi liquid behavior, = 4 N Super-Yang-Mills at finite density was found at strong coupling to exclusively have excitations behaving as a non-Fermi liquid [13]. A primary result of the work in this chapter is that the ABJM theory is the same: only non-Fermi liquid behavior is found. It is interesting to note that for the alternate non-supersymmetric quantization this is no longer the case.
3.4.2 The 3+1-Charge Black Hole
The 3 + 1-charge black hole solutions (3+1QBH) are defined by setting three of the charges equal, while allowing the fourth to vary independently:
Q1 Qa ,Q3 Qb = Qc = Qd . (3.76) ≡ ≡
The corresponding gauge fields turned on in the bulk are
a Aa Φ1(r)dt , A Ab = Ac = Ad Φ3(r)dt , (3.77) ≡ ≡ ≡ ≡ 54 with field strengths f da, and F dA. This simplification also relates the three active scalars ≡ ≡ to one another,
φ φ1 = φ2 = φ3 , (3.78) ≡ − − − where the minus sign is for later convenience. The simplified Lagrangian then becomes
3 6 3 1 2e−1 = R (∂φ)2 + cosh φ eφF 2 e−3φf 2 . (3.79) L − 2 L2 − 4 − 4
The (3+1)QBH solutions are r 1 Q1 3 Q3 A(r) = B(r) = log + log 1 + + log 1 + − L 4 r 4 r 3 r(rH + Q1)(rH + Q3) 1 Q3 1 Q1 h(r) = 1 3 , φ = log 1 + log 1 + − rH (r + Q1)(r + Q3) 2 r − 2 r r 3/2 η1 Q1 (rH + Q3) rH + Q1 Φ1(r) = 1/2 1 L rH (rH + Q1) − r + Q1 r η3 Q3 rH + Q3 Φ3(r) = (rH + Q3)(rH + Q1) 1 , (3.80) L rH − r + Q3 with temperature and entropy density
2 s 3rH + 2Q1rH Q1Q3 rH + Q3 1 3/2 1/2 T = 2 − , s = 2 (rH + Q3) (rH + Q1) , (3.81) 4πL rH rH + Q1 4GL and the chemical potentials and charge densities
r 3/2 r η1 Q1 (rH + Q3) η3 3 Q3 µ1 = 2 1/2 , µ3 = 2 (rH + Q3)(rH + Q1) , (3.82) L rH (rH + Q1) L rH r r η1 Q1 η3 3Q3 ρ1 = s , ρ3 = s , (3.83) 2π rH 2π rH where the factor of √3 comes from defining µ3 and ρ relative to a canonically normalized gauge
field √3A.
We will be interested in extremal black holes, which satisfy
2 3r + 2Q1rH Q1Q3 = 0 (extremal (3 + 1)QBH) . (3.84) H −
To solve 3.84 it is generally most convenient to eliminate Q1 in favor of Q3 and rH ,
2 3rH Q1 = (extremal (3 + 1)QBH) , (3.85) Q3 2rH − 55
1.0 4QBH
0.5 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ k æ æ æ æ æ 0.0 Μ3
ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç -0.5
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
ΜR
Figure 3.2: Class 1 fermions for the (3+1)QBH. Fermi surface singularities are shown as blue dots, while zeroes are marked by empty circles. The green hatched region is the “oscillatory region” characteristic of an infrared instability towards pair production in the bulk. The solid blue contours bound the region of Fermi surfaces with non-Fermi liquid-like excitations. 56 indicating Q3 2rH for extremal solutions (recall we have taken the Qi positive). These black holes ≥ all have a nonsingular event horizon at r = rH , and are thus regular. Correspondingly, the entropy density (which is just the area density of the event horizon) is nonzero, even at zero temperature.
The parameter space is naively two-dimensional, but since the underlying field theory is conformal, only the ratio of dimensionful quantities matters; hence there is a one-parameter space of extremal solutions, given by rH /Q3, or equivalently by the ratio of the chemical potentials:
r µ1 2rH µR = 1 (extremal (3 + 1)QBH) . (3.86) ≡ µ3 − Q3
This runs over values 0 µR 1. The endpoints of the parameter range are not regular black ≤ ≤ holes: the limit µR 1 connects to the three-charge black hole, to be discussed in section 3.5, → while the opposite limit µR 0 connects to the one-charge black hole, discussed in section 3.6. At →
µR = 1/√3, we obtain the Reissner-Nordstr¨omfour-charge black hole 3.19.
The Dirac equation 3.24 in the (3+1)QBH backgrounds is
h µ m −φ/2 3φ/2 q1 µ q3 µ i µν −3φ/2 φ/2 i iΓ µ + (e e )+ Γ aµ + Γ Aµ + Γ p1e fµν +p3e Fµν χ = 0 . (3.87) ∇ 4L − 4L 4L 8
The quantities m, q1, q3, p1, p3 are combinations of the mi, qi, pi characterizing each fermion:
m ma = mb + mc + md , ≡ −
q1 qa , q3 qb + qc + qd , p1 pa , p3 pb + pc + pd . (3.88) ≡ ≡ ≡ ≡
In these backgrounds the functions X, u and v are
3φ/2 −φ/2 B −B m (e e ) e 1 q1 q3 e −3φ/2 0 φ/2 0 X = − , u = ω + Φ1 + Φ3 , v = p1e Φ1 + p3e Φ3 . 4L√h √h 4L 4L 4 (3.89)
Several fermions that have distinct charges in general backgrounds satisfy the same Dirac equation when restricted to the (3+1)QBH backgrounds. We find that the 16 fermions given in the previous section organize into five distinct (3+1)QBH equations, which we label as classes 1-5: 57
1.0 4QBH
æ æ æ æ æ 0.5 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ k æ æ æ 0.0 Μ3 ç ç æ æ æ ç æ æ æ ç ç ç ç ç ç ç ç -0.5 ç ç ç ç ç ç ç ç ç ç ç
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
ΜR
Figure 3.3: Class 2 fermions for the (3+1)QBH. These modes are unique in that they exhibit multiple Fermi surfaces for small µR. 58
(qa,q ,qc,q ) Class χ b d m q3 q1 p3 p1
χ(+1,+3,−1,+1), χ(+1,−1,+3,+1), χ(+1,+1,+3,−1), 1 1 3 1 1 1 − − χ(+1,+3,+1,−1), χ(+1,+1,−1,+3), χ(+1,−1,+1,+3)
2 χ(−1,+1,+1,+3), χ(−1,+3,+1,+1), χ(−1,+1,+3,+1) 1 5 1 1 1 − − − 3 χ(+3,−1,+1,+1), χ(+3,+1,−1,+1), χ(+3,+1,+1,−1) 3 1 3 1 1 − 4 χ(−1,+3,−1,−1), χ(−1,−1,+3,−1), χ(−1,−1,−1,+3) 1 1 1 3 1 − − − − 5 χ(+3,−1,−1,−1) 3 3 3 3 1 − − −
Classes 1-3 are net-charged fermions, while classes 4 and 5 are net-neutral. We note that at the
4QBH point, the vanishing scalar makes the mass function vanish, while the gauge and scalar P P couplings depend only on q1 + q3 = i qi and p1 + p3 + i pi; thus all net-charged fermions have the same Dirac equation at the 4QBH point, with a gauge coupling only, and all net-neutral fermions have the same Dirac equation at the 4QBH point, with a Pauli coupling only.
The parameter ν 3.66 is given by
2 2 2 2 2 3 2 m (1 3µ ) 2 k˜ (q3(1 µ ) + 2√3q1µ ) ν2 = − R + − R R , (3.90) 48(1 µ4 ) (1 + µ2 ) µ3 − 72(1 µ2 )(1 + µ2 )2 − R R 3 − R R where the shifted momentum 3.67 is
α ( 1) p3 k˜ = k − p1µ1 + µ3 . (3.91) − 4 √3
We numerically obtained ω = 0 Green’s functions as a function of k for all five classes over the range
0 < µR < 1, imposing infalling boundary conditions by requiring U to satisfy 3.65 with ω = 0.
Fermi surface singularities are then identified as momenta k = kF for which the source is zero
A+ = 0; we also identify zeros as momenta k = kL for which the response vanishes A− = 0. These results are plotted in figures 3.2-3.6, with Fermi surface singularities given as blue dots, and zeros as open circles. The plots show the α = 2 component of each spinor; α = 1 modes are obtained simply by exchanging k k. We also indicate oscillatory regions in green crosshatch, with their → − boundary k = kosc determined by νk 0. We additionally plot the lines of k for which νk = 1/2; osc ≡ this describes the boundary between the non-Fermi liquid behavior region (inside) and the Fermi liquid region (outside). 59
1.0 4QBH
0.5 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ k æ æ æ æ 0.0 Μ ç 3 ç ç ç ç ç ç -0.5 ç ç ç ç ç ç ç ç ç ç
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
ΜR
Figure 3.4: Class 3 fermions for the (3+1)QBH. The poles end at the oscillatory region just before µR = 1.
1.0 4QBH
0.5
k 0.0 Μ3
ç ç ç ç ç ç ç ç ç ç ç -0.5 ç ç ç ç ç ç ç ì æ æ æ æ æ æ æ æ æ æ æ æ æ
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
ΜR
Figure 3.5: Class 4 fermions for the (3+1)QBH. For the net-neutral modes, there is a novel transition at the 4QBH state from Fermi surface singularities to zeroes. 60
Examining the results, the five classes fell into two distinct categories: the net-charged and net-neutral fermions behave rather differently. For the net-charged fermions (classes 1, 2 and 3) one always finds an oscillatory region; for the first class this extends across the entire region, while in the other cases it begins on the left or right side respectively, but terminates some distance after crossing the four-charge black hole line. In general lines of poles (or zeros) either persist to the edge of the parameter space, or end on an oscillatory region. Each fermion has at least one Fermi surface singularity for any given value of µR, with the exception of class 3 where the line of poles disappears into the oscillatory region just before µR = 1. Class 2 has two Fermi momenta kF with opposite sign for some small values of µR; similar situations have been interpreted as a thick shell of occupied states between the two values of kF [47]. The three classes match precisely at the | | four-charge point, as they must.
The net-neutral fermions (classes 4 and 5) look rather different. On one side of the four- charge point, there is a line of zeros; on the other side, a line of poles. Precisely at the four-charge point, one line turns into the other. Also unlike the net-charged case, there is no oscillatory region.
However, one can determine that precisely at the four-charge point, there is a single point indicated by a red diamond where kosc = 1/√3 gives νk = 0; this “oscillatory point” is precisely where − osc the line of poles turns into a line of zeros, respecting the pattern that a line of poles or zeros may terminate only at a momentum k = kosc. Precisely at this point — which agrees between the two classes — the Green’s function is a nonzero, finite constant. We will comment more on this point at the end of the next subsection.
In all cases, both net-charged and net-neutral, the Fermi surface singularities stay within the non-Fermi liquid region. Thus continuing the pattern observed in the case of = 4 Super-Yang- N Mills theory, this strongly coupled maximally supersymmetric conformal field theory seems only to show non-Fermi liquid behavior, not Fermi liquid behavior. The same is not true for the zeros; in classes 2 and 3 the line of zeros extends into the ν > 1/2 region. This implies that in the alternate quantization — which is not dual to ABJM theory but in principle defines a dual CFT, as much as any bottom-up construction — Fermi liquids would be present. It is very interesting that the 61
1.0 4QBH
0.5
k 0.0 Μ3
æ æ æ æ æ æ æ æ æ æ -0.5 æ æ æ æ æ æ æ æì ç ç ç ç ç ç ç ç ç ç ç ç ç
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
ΜR
Figure 3.6: Class 5 fermions for the (3+1)QBH. Unlike their net-charged brethren, there exists no oscillatory region for the net neutral modes, but a single “oscillatory point” at the pole/zero transition. 62 top-down theories seem to avoid Fermi liquid behavior, when this is easy to obtain in a bottom-up construction; these results make this distinction sharper still.
3.4.3 The 2+2-Charge Black Hole
Another interesting sector of the supergravity theory is made of the 2 + 2-charge black holes
(2+2QBH), which are defined by setting the charges equal in pairs,
Q2 Qa = Qb , Q˜2 Qc = Qd , (3.92) ≡ ≡ corresponding to turning on the gauge fields
B Aa = Ab Φ2(r)dt , B˜ Ac = Ad Φ˜ 2(r)dt , (3.93) ≡ ≡ ≡ ≡ with field strengths G dB, and G˜ dB˜. ≡ ≡ In addition to simplifying the gauge sector, this also sets two of the three scalars to zero, and we define
γ φ1, φ2 = φ3 = 0 . (3.94) ≡
The Lagrangian then becomes
1 4 2 1 1 2e−1 = R (∂γ)2 + + cosh γ eγG2 e−γG˜2 , (3.95) L − 2 L2 L2 − 2 − 2 and the black hole backgrounds are
" !# r 1 Q2 Q˜2 A(r) = B(r) = log + log 1 + + log 1 + , − L 2 r r 2 2 ! r(r + Q2) (r + Q˜2) Q2 Q˜2 h(r) = 1 H H , γ = log 1 + log 1 + , 2 2 − rH (r + Q2) (r + Q˜2) r − r s r ! η2 Q2 rH + Q2 η˜2 Q˜2 rH + Q˜2 Φ2(r) = (rH + Q˜2) 1 , Φ˜ 2(r) = (rH + Q2) 1 (3.96). L rH − r + Q2 L rH − r + Q˜2
The thermodynamic properties are
2 ˜ ˜ 3rH + (Q2 + Q2)rH Q2Q2 1 ˜ T = 2 − , s = 2 (rH + Q2)(rH + Q2) . (3.97) 4πL rH 4GL 63
s √ r √ ˜ 2 η2 Q2 ˜ 2η ˜2 Q2 µ2 = 2 (rH + Q2) , µ˜2 = 2 (rH + Q2) . (3.98) L rH L rH r s η2 2Q2 η˜2 2Q˜2 ρ2 = s , ρ˜2 = s . (3.99) 2π rH 2π rH where again the chemical potentials are defined with respect to canonically normalized gauge fields.
Extremality occurs for
2 3r + (Q2 + Q˜2)rH Q2Q˜2 = 0 (extremal (2 + 2)QBH) . (3.100) H −
Solving for Q˜2, we have
2 Q2rH + 3rH Q˜2 = (extremal (2 + 2)QBH) , (3.101) Q2 rH − indicating rH Q2. These extremal (2+2) charge solutions are again regular black holes. There is ≤ a one-parameter space of such solutions, s µ2 Q2rH µ˜R = 2 2 2 (extremal (2 + 2)QBH) . (3.102) ≡ µ˜2 Q + 2Q2rH 3r 2 − H
Hereµ ˜R can take all positive values 0 µR . The limitµ ˜R 0 corresponds to the “extremal” ≤ ≤ ∞ →
2QBH, described in section 3.6, and the pointµ ˜R = 1 is the 4QBH, connecting to the parameter space of the (3+1)QBHs.
Since the original charges are all equivalent, gauge invariance mandates a symmetry exchang- ing Q2 and Q˜2, which exchanges the corresponding gauge fields and changes the sign of the scalar
γ:
Q2 Q˜2 , γ γ , Φ2 Φ˜ 2 , (3.103) ↔ → − ↔ which also exchanges (µ2, ρ2) and (˜µ2, ρ˜2) and hence sendsµ ˜R 1/µ˜R. Thus the region 1 µ˜R → ≤ ≤
is equivalent to 0 µ˜R 1. ∞ ≤ ≤ The Dirac equation in the (2+2)QBH background is
h µ m˜ −γ/2 γ/2 q2 µ q˜2 µ i µν γ/2 −γ/2 i iΓ µ + (e e )+ Γ Bµ + Γ B˜µ + Γ p2e Gµν +p ˜2e G˜µν χ = 0 , (3.104) ∇ 4L − 4L 4L 8 64
1.0 4QBH 4QBH 1.0
0.5 æ æ æ 0.5 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ k æ æ æ k æ æ æ 0.0 æ æ æ æ 0.0 ç ç Μ ç Μ 2 ç 2 ç ç ç ç ç ç ç ç ç ç -0.5 ç ç ç -0.5 ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç
-1.0 -1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 ΜR ΜR
Figure 3.7: Poles and zeros of the retarded Green’s function for the class I fermion from 0 < µ˜R < 1 (left) and for the class II fermion from 1 < µ˜R < 0 (right). Viewed together the two plots depict the entire range 0 < µ˜R < for class I or > µ˜R > 0 for class II. ∞ ∞
with parameters
m˜ ma + mb = (mc + md) , ≡ −
q2 qa + qb , q˜2 qc + qd , p2 pa + pb , p˜2 pc + pd . (3.105) ≡ ≡ ≡ ≡
In this case, there are six distinct non-degenerate fermion eigenvectors, which we sort into classes
I-VI:
(qa,q ,qc,q ) Class χ b d m˜ q2 q˜2 p2 p˜2
I χ(−1,+1,+1,+3), χ(+1,−1,+3,+1), χ(−1,+1,+3,+1), χ(+1,−1,+1,+3) 2 0 4 0 0
II χ(+1,+3,−1,+1), χ(+3,+1,+1,−1), χ(+1,+3,+1,−1), χ(+3,+1,−1,+1) 2 4 0 0 0 − III χ(+1,+1,+3,−1), χ(+1,+1,−1,+3) 2 2 2 2 2 − IV χ(−1,+3,+1,+1), χ(+3,−1,+1,+1) 2 2 2 2 2 − − V χ(−1,−1,+3,−1), χ(−1,−1,−1,+3) 2 2 2 2 2 − − − VI χ(−1,+3,−1,−1), χ(+3,−1,−1,−1) 2 2 2 2 2 − − − −
The first four classes are net-charged, and the last two net-neutral.
Gauge invariance requires that under the exchange 3.103 the total physics of all fluctuations is invariant. For this to be true, the form of the Dirac equation 3.104 demands that the spectrum 65
Figure 3.8: Poles and zeros of the retarded Green’s function for the class III fermion from 0 < µ˜R < 1 (left) and for the class IV fermion from 1 < µ˜R < 0 (right), or the entire range 0 < µ˜R < for ∞ class III or > µ˜R > 0 for class IV. ∞
of fermions be carried into itself under
m˜ m˜ , q2 q˜2 , p2 p˜2 . (3.106) → − ↔ ↔
Glancing at the table we see this is indeed the case; the six classes form three partner pairs (I, II),
(III, IV) and (V, VI) that are exchanged.
The X, u and v functions appearing in the Dirac equation are now
m˜ (eγ/2 e−γ/2) eB me˜ B sinh γ/2 X = − = 4L√h 2L√h −B 1 q2 q˜2 ˜ e γ/2 0 −γ/2 ˜ 0 u = ω + Φ2 + Φ2 , v = p2 e Φ2 +p ˜2 e Φ2 , (3.107) √h 4L 4L 4 and the ν parameter 3.66 is
2 2 3/2 2 3/2 q 2 1 µ2 + 2 2 k˜2 q2µR 2 + µR + 2 +q ˜2 1 + µR 1 + µR + ν2 = − − R S m˜ 2+ − S − S − S , 4(1 + µ2 ) 1 + µ2 µ˜2 − 48(1 + µ2 )2( 2 + µ2 + 2 )2 R R 2 R − R S (3.108) where we defined q 1 µ2 + µ4 , (3.109) S ≡ − R R and the shifted momentum 3.67 is
( 1)α k˜ k − (p2µ2 +p ˜2µ˜2) . (3.110) ≡ − 4√2 66
1.0 1.0 4QBH 4QBH
0.5 0.5 k k 0.0 0.0 Μ2 Μ2
ìç ç ç æ æ æì ç ç ç æ æ æ ç ç ç ç æ æ æ æ ç ç ç ç æ æ æ æ -0.5 ç ç ç ç æ æ æ æ -0.5 ç ç ç ç æ æ æ æ ç ç ç æ æ æ ç ç ç æ æ æ ç ç çì ìæ æ æ
-1.0 -1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 ΜR ΜR
Figure 3.9: Poles and zeros of the retarded Green’s function for the class V fermion from 0 < µ˜R < 1 (left) and for the class VI fermion from 1 < µ˜R < 0 (right), or the entire range 0 < µ˜R < for ∞ class V or > µ˜R > 0 for class VI. ∞
Due to the equivalence 3.103, 3.106, it is not necessary to study all six classes of fermions over the entire range 0 < µ˜R < ; one fermion for 0 < µ˜R 1 is equivalent to its partner fermion over ∞ ≤
1 < µ˜R < . Thus we may either study three classes of fermions over the entire parameter space, ∞ or all six over half the parameter space. In practice we studied all six fermions over the range
0 < µ˜R 1, to avoid the complications ofµ ˜R extending over an infinite range. However, we find it ≤ convenient to show the plots of classes II, IV, and VI with the horizontal axis reversed, placed next to their partner classes I, III, and V, respectively. Then one can alternately interpret each pair of
figures as describing the behavior of a single fermion over the entire range 0 < µ˜R < ; to do this ∞ one simply interprets one of the two plots as being the other fermion in the pair, withµ ˜R 1/µ˜R → and k/µ2 instead of k/µ˜2. Thus we can visualize the entire range ofµ ˜R for all distinct fermions in a compact way.
The (2+2)QBH results are qualitatively very similar to the (3+1)QBH results. Again the net-charged fermions have an oscillatory region as well as lines of poles and zeros that begin either
at the ends of the parameter space or at oscillatory regions. Again the poles stay within νkF < 1/2, indicating non-Fermi liquid behavior exclusively. The net-neutral fermions again have a line of poles turning into a line of zeros at a special “oscillatory point” where k = kosc, that is, where
νk = 0. The (2+2)QBH results for the net-charged and net-neutral classes must match on to the 67
(3+1)QBH results at the 4QBH, and indeed they do.
It is evident that the results shown in figures 3.8 and 3.9 have rotational and reflectional symmetries, respectively, when combined with an exchange of zeros and poles. One can show using the inversion equivalence 3.103, 3.106 in conjunction with charge conjugation 3.39 and the chirality
flip 3.40 that a 180-degree rotation on the combined figures along with an exchange of poles and zeros is equivalent to (q2, p2) (˜q2, p˜2) on the fermions, while left-right parity along with an ↔ − exchange of poles and zeros is equivalent to (q2, p2) ( q˜2, p˜2); the invariance of classes (III, IV) ↔ − under the former operation and of classes (V, VI) under the latter explains the symmetries of their respective figures. Note that one can argue that any fermion with (q2, p2) = ( q˜2, p˜2) such as our − class V and VI net-neutral fermions must be invariant under the inversion of the Green’s function at the 4QBH, and thus cannot have a pole or zero there, but can have the transition between a pole and zero that we observe.
One fermion of particular interest is class II, for which the line of Fermi surfaces has k/µ˜2 0 → asµ ˜R 0. This is the one case neutral under Q˜2; as a result, one can renormalize to k/µ2 and → one discovers k/µ2 0.354 1/√8 which gives → ≈ 1 νk ,II(˜µR 0) . (3.111) F → → 2
Hence while it never reaches the Fermi liquid region of νkF > 1/2 it asymptotically approaches a marginal Fermi liquid in the limit. An analogue of this MFL fermion exists in the five-dimensional case as well [13]. We comment more on this case, and show the plot of kF /µ2, in section 3.6.
The abrupt variation in the spectrum at the 4QBH, where some net-neutral Fermi surfaces disappear and others appear, is reminiscent of a phase transition; since this occurs while varying parameters at zero temperature, it would constitute a quantum critical point. This interpretation is in accord with the interpretation [14] of the inverse correlation length squared as the sum of mass and charge contributions to the ν2 parameter 3.68; indeed at the four-charge black hole the masses of all fermions go to zero, while for the net-neutral fermions the charge contribution vanishes as well. (The lack of massless, chargeless fermions explains the failure of such an apparent critical 68 point to appear when all charges were equal in the = 4 SYM case.) The presence of the Pauli N couplings shifting the momentum allows the pole-to-zero transition to occur at a nonzero kF . We note that no sign of such a critical point is visible in the susceptibilities coming from the black hole thermodynamics 3.15-3.18; it has been argued in [54] that such a discrepancy is a result of the large-N limit.
While we have restricted ourselves to considering the (3+1) and (2+2) charge black holes only, given a fermion with a pole on one side of the 4QBH but a zero on the other it is natural to ask what happens to the Green’s function if one circles around the 4QBH in the full parameter space. A natural guess is that the critical point we observe may extend into a critical surface, separating pole and zero regions. Indeed, if one makes a small variation of the chemical potentials near the 4QBH (which has µa = µb = µc = µd) and asks when a solution to νk = 0 exists, one finds that the fermion χ(+3,−1,−1,−1) still has an oscillatory point at a single value of k if the chemical potentials vary along the codimension-one surface
δµa = 0 , δµb + δµc + δµd = 0 , (3.112) while along other directions there are no oscillatory regions or points. It is natural to surmise that the pole-zero transition again occurs at each oscillatory point, thus separating the parameter space into disjoint regions of poles and of zeros for this fermion. For other net-neutral fermions, the analogous permutation of 3.112 holds; note that for different fermions, the transitions between poles and zeros occur in different places in the general parameter space. 69
3.5 The Extremal Three-Charge Black Hole and the Gap
We now turn to study the 3-charge black hole (3QBH), the special case of the (3+1)QBH where the single charge Q1 is set to zero. The background geometry is given by r 3 Q3 A(r) = B(r) = log + log 1 + , (3.113) − L 4 r 3 r + Q3 1 Q3 h(r) = 1 H , φ = log 1 + , (3.114) − r + Q3 2 r η3 p rH + Q3 Φ3(r) = Q3(rH + Q3) 1 , Φ1(r) = 0 , (3.115) L − r + Q3 and the associated thermodynamic quantities are
3 p √rH 3/2 T = rH (rH + Q3) , s = (rH + Q3) , (3.116) 4πL2 4GL2 r η1 p η3 3Q3 µ3 = 2 3 Q3(rH + Q3) , µ1 = 0 , ρ3 = s , ρ1 = 0 . (3.117) L 2π rH
This background describes a state in the M2-brane theory where three chemical potentials are set equal and the fourth one is set to zero. Zero temperature corresponds to the extremal limit rH 0. Unlike the 1QBH and the 2QBH discussed in the next section, the extremal solution → remains a black hole. The extremal geometry, however, is singular at the horizon rH = 0. This singularity allows the horizon radius to go to zero, so this background has zero entropy density at zero temperature. This situation is very similar to the 5D 2QBH geometry studied in [13], dual to a zero-entropy state in = 4 super-Yang-Mills. One might expect that the fermions in our 3QBH N background will behave similarily to the fermions in the 5D 2QBH background, a suspicion that will be confirmed throughout this section.
In appendixA, we describe how the near-horizon limit of this solution lifts to a five-dimensional
2 geometry of the form AdS3 R , analogous to the six-dimensional lift discussed in [13]; AdS3 regions × in the near-horizon limits of zero-entropy extremal black holes are also discussed in [67, 68, 69, 70].
This lift removes the singularity, showing it is harmless. The inactive gauge field a is identified as the gravitphoton, and consistent reduction of any fermion requires that its parameters obey
m = q1p1 m = q1 , (3.118) − → | | | | 70 where the second equality follows since p1 = 1; due to 3.25, 3.88 this is indeed satisfied by all ± fermions we consider.
The analysis of the 3QBH is done with the same method used in previous sections, analyzing the Dirac equation in the near-horizon limit to identify infalling boundary conditions, then solving the full Dirac equations numerically and identifying poles in the Green’s function. The essential difference from the regular case is an interval of frequencies centered on ω = 0, bounded above and below by an energy scale ∆ which we call the gap, inside which excitations have zero width and are thus stable. Outside this region we again recover the more familiar non-zero excitation widths observed for the regular black holes.
A similar gap was also observed in [55], see section 1.2 and Fig. 1 therein. In that paper, a possible interpretation of the gap region was given as follows: One postulates a sector of the theory, additional to the fermion sector studied, that for generic cases has no mass gap, and thus is responsible for the generically non-zero ground state entropy. This sector could also couple to the fermions which are the subject of study, providing a channel for them to decay through. However, at the 3QBH the ground state entropy vanishes, which is interpreted as an indication that this sector becomes gapped. This gapping then also causes fermions with low enough energy to become stable, implying that their decay channel has been removed and that they cannot decay due to self-interactions; this could potentially be a large-N effect.
3.5.1 Near Horizon Analysis of the 3QBH
Taking the extremal limit rH 0 of the 3QBH background, and then expanding near the → singular horizon r 0, we obtain the same equation for both U±, or equivalently for ψ±: → 1 ! 3 L4(ω2 ∆2) ν2 + ( ω ∆) 00 0 16 − 3Q O | | − ψ + + ψ + −3 + 2 + ψ = 0 ; (3.119) 2r ··· 9Q3r r ··· where the dots represent terms of higher order in r or in ( ω ∆), and we have defined the scale O | | − ∆, √3 m Q 1 ∆ | | = m µ , (3.120) ≡ 4L2 4| | 71 along with the parameter
α 2 2 2 ˜ ( 1) m q3 m ν3Q k3Q + − sgn(ωm) + sgn(ωm) , (3.121) ≡ 4 − 48 24√3 which includes the shifted momentum
2 L k α p3 k α p3 k˜3Q + ( 1) = + ( 1) . (3.122) ≡ √3Q − 4√3 µ3 − 4√3 | | 2 2 The quantity ν3Q will play a similar role to ν for the regular black holes; in fact, we will show that
3.90 coincides with 3.121 in the µR 1 limit for the fermions we consider. → Equation (3.119) shows us that there is something special about the energy scale ∆. For the regular extremal black holes, the leading order, no-derivative term was suppressed close to ω = 0, forcing one to consider inner and outer regions there. Here instead, as for the 5D 2QBH [55], that occurs around ω = ∆; in fact, equation (3.119) has the same structure as Eq. (57) of [55] with the
2 correspondence r4D r , and thus the same analysis can be used. ↔ 5D When ω is not near ∆, the 1/r2 term in 3.119 can be neglected. Then for ω > ∆ we have | | | | complex oscillatory solutions ! i2L2√ω2 ∆2 ψ exp − , (3.123) ∼ ± 3√Q3r clearly representing infalling and outgoing waves. For ω < ∆ on the other hand, the equation is | | purely real and we get growing and dying exponentials, ! 2L2√∆2 ω2 ψ exp − . (3.124) ∼ ± 3√Q3r
When the frequency is close to the gap energy, the two no-derivative terms in the second parenthesis of 3.119 can be of similar magnitude. Therefore, we will divide the problem into an outer region where r is large enough to neglect the r−3-term, and an inner region where we must take both terms into account. The outer region admits power law solutions,
− 1 ±ν ψ r 4 3Q . (3.125) ∼
The inner region equation can be solved by r−1/4 times Bessel functions or modified Bessel functions for ω > ∆ and ω < ∆, respectively. After imposing infalling boundary conditions on the inner | | | | 72 region solutions, we can study their near-boundary (r ) behavior, allowing us to determine → ∞ the “IR Green function”, (ω)3Q. This plays the same role near ω = ∆ as its cousin (ω) does G | | G near ω = 0 for the regular black hole solutions. Near ω ∆ we have | | ≈
2ν3Q (ω)3Q ( ω ∆) . (3.126) G ∼ | | −
Using 3.126 we can derive expressions for the fluctuations near ω = ∆ analogous to 3.71. Let k∆ | | be the momentum leading to a pole at ω = ∆. Then for ω > ∆, we have a formula similar to 3.71, | |
h1 GR , (3.127) |ω|−∆ −2πiν 2ν ∼ (k k∆) + h2e ∆ ( ω ∆) ∆ − − vF · · · − | | − where ν∆ ν3Q(k∆). Similar to the regular case, there is an imaginary part which controls the ≡ width of the fluctuation, and if the value of ν3Q at the pole is greater than 1/2 the excitations behave like those of a Fermi liquid, while if ν < 1/2 their behavior is non-Fermi liquid type. For the case ω < ∆, we instead obtain | |
h1 GR , (3.128) |ω|−∆ 2ν ∼ (k k∆) + h2(∆ ω ) ∆ − − vF · · · − − | | which is the analytic continuation of the previous equation to negative ω ∆ (and similarly | | − the Bessel function solution regular at the horizon for ω < ∆ is the continuation of the infalling | | solution at ω > ∆). Importantly, the phase has disappeared and the Green’s function is manifestly | | real. This implies that the width of the fluctuations it describes are zero and they are thus stable.
From the point of view of the five-dimensional lift, the relation 3.118 allows one to interpret the gap 3.120 as the momentum of the fermion in the fifth dimension. The appearance of ω2 ∆2 − in the Dirac equation can then be understood as the five-dimensional momentum-squared. Thus excitations inside the gap are spacelike from the higher-dimensional point of view, and beyond the gap they become timelike. Analogous behavior was seen in [55].
3.5.2 Connection with Extremal (3+1)-Charge Black Holes
The sequence of extremal (3+1)QBHs parameterized by µR µ1/µ3 approaches the 3QBH ≡ extremal solution as µR 1. This limit is somewhat counterintuitive, as µ1 = 0 for the strict → 73
3QBH; there is an associated discontinuity in µ1/µ3 as a function of Q1/Q3 (or equivalently ρ1/ρ3, as discussed for the analogous five-dimensional case in [55]; see in particular figure 4). Associated to this subtlety is the failure to commute of the operations of going to the 3QBH and taking T 0; → going to the 3QBH first leads to the solution presented in 3.113-3.115 with Φ1 = 0, while taking the extremal limit of (3+1)QBHs first shifts the potential by a constant:
√3Q3 Φ1 . (3.129) → L
Due to the structure of 3.34, the only effect in the Dirac equation is to shift the zero point of the energy between the two descriptions:
q1 √3Q3 1 ω3Q = ω + = ω + q1µ3 , (3.130) (3+1)Q 4L L (3+1)Q 4
Using the relation 3.118 between m and q1 and the definition 3.120 of the gap ∆, we can further | | | | write
ω3Q = ω(3+1)Q + sgn(q1)∆ , (3.131) indicating that the limit of a sequence of Fermi surface singularities at ω3+1 = 0 as µR 1 in the → (3+1)QBHs will manifest itself as a singularity in the corresponding 3QBH at ω = ∆. This is ± reasonable, since it is above ω = ∆ that the 3QBH has decaying fluctuations matching those at | | the Fermi surface in the regular case. In the next subsection we will see that the figures from the
(3+1)QBH section do indeed match to the 3QBH in each case on the appropriate side of the gap region.
These fluctuations are controlled by ν in the (3+1)QBH case, and ν3Q in the 3QBH case; we would thus expect the two quantities to agree in the limit. Expanding 3.90 around µR = 1 we find
2 2 2 2 2 2 m q1 1 α 3m 2q1q3 7q1 ν = − + k˜3Q ( 1) p1 + + (µR 1) . (3.132) −48(µR 1) 16 − − − 2 − 3√3 6 O − −
This would diverge in the limit µR 1, but the relation 3.118 sets the would-be diverging term → to zero for all physical fermions. Moreover, the finite part can be shown to agree exactly with
3.121, using the relationship 3.118 together with sgn(q1) = sgn(ω), which follows from 3.131. It 74
0.6
æ æææææ ææ ææ ææ ææ ææ 0.4 ææ ææ ææ ææ ææ ææ ææ ææ ææ ææ ææ æ ææ æ ææ æ ææ 0.2 ææ æ ææ æ ææ ææ æ ææ æ ææ æ ææ ææ æ ææ æ k ææ ææ æ ææ æ ææ æ 0.0 ææ æ ææ æ ææ æ Μ ææ æ 3 ææ æ ææ ææ ææ æ æ æ -0.2 æ æ æ æ
-0.4
-0.6
-0.2 -0.1 0.0 0.1 0.2 Ω
Μ3
Figure 3.10: Class 1 fermions for the 3QBH. There is both a line of poles throughout the stable region ∆ < ω < ∆, and a pair of poles nucleating very close to ω = ∆ before ending on the − oscillatory region (green). 75 is interesting that the limits only coincide for fermions that can be lifted to five dimensions; this indicates that not just any fermion quantum numbers lead to consistent behavior throughout the parameter space, reinforcing the importance of a top-down description.
3.5.3 Fermion Fluctuations and Fermi Surfaces
We now describe the results of numerically solving the Dirac equations for fermionic modes in the 3QBH. For the regular black holes, we were chiefly interested in whether a Fermi surface existed at a given k = kF at ω = 0, and we obtained the properties of nearby fluctuations. Here, we do more: we will look for poles in the Green’s function as a function of k for the entire stable region
∆ ω ∆, and plot the results over the ω-k plane in Figs. 3.10-3.14; each of the five classes of − ≤ ≤ fermion for the (3+1)QBH retains a distinct Dirac equation in the 3QBH limit. The location of the gap is marked with a vertical dotted red line. When applicable we also plot the extent of the oscillatory region at the gap in green.
For the fermions in class 1, 2 and 4, there is a Fermi singularity at zero frequency, part of a line of poles that stretches from one end of the stable region to the other. The dispersion near
ω = 0 is approximately linear in each case, which allows us to define a corresponding Fermi velocity vF . Fermions of class 3 and 5, on the other hand, have no Fermi surface singularities near zero energy and hence are truly gapped.
As one moves away from ω = 0 another phenomenon emerges: in some cases a new pair of poles nucleates at some nonzero ω and spreads apart as one approaches the gap. This is observed for fermions in class 1, 3 and 5. Poles that run into an oscillatory region at ω = ∆ cease to exist. ± Note that for class 1 — the only case with both a line of poles over the whole stable region as well as a nucleating pair — it is difficult to see the poles nucleating on the right, since they appear very close to the gap and thus seem flattened along the horizontal direction.
Poles for each fermion in the (3+1)QBH case in the µR 1 limit should match onto either → the left or right side of the gap in the 3QBH. Which side of the gap ought to match is decided by the sign of the fermions q1 eigenvalue, with a positive (negative) q1 meaning matching takes place 76
æ ææ ææ æææ æææ æææ æææ æææ æææ æææ æææ æææ æææ æææ 0.5 æææ æ ææ æææ æææ æææ æææ æææ æææ æææ æææ æææ æææ ææ ææ ææ ææ ææ ææ k æ 0.0 Μ3
-0.5
-0.2 -0.1 0.0 0.1 0.2 Ω
Μ3
Figure 3.11: Class 2 fermions for the 3QBH. Here there is a line of poles only.
0.5
æ æ æ æ æ æ æ æ æ æ æ æ k æ 0.0 æ
æ Μ3 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ -0.5 æ æ æ
-0.5 0.0 0.5 Ω
Μ3
Figure 3.12: Class 3 fermions for the 3QBH. No line of poles through ω = 0 exists, but a pair of poles nucleate near ω = ∆ and end on the oscillatory region. 77
0.5
ææ ææ æææ æææ æææ k æææ æææ æææ æææ 0.0 æææ æææ æææ æææ Μ æææ 3 æææ æææ æææ æææ æææ æææ æææ æææ ææ ææ ææ ææ ææ ææ ææ ææ - ææ 0.5 ææ æ æ æ æ
-0.2 -0.1 0.0 0.1 0.2 Ω
Μ3
Figure 3.13: Class 4 fermions for the 3QBH, with a line of poles only. 78 at ω = ∆( ∆). In all cases the appropriate matching occurs. For class 1, both of the nucleating − poles run into the oscillatory region, but the pole at the right end of the long line matches with the one from Fig. 3.2. For class 2, the pole on the left side of the gap matches with Fig. 3.3. For class
3 both of the nucleating poles run into the oscillatory region, agreeing with Fig. 3.4 where the line of poles also hits the oscillatory region just before the edge. For class 4, the left side of the line of poles matches with the right side of Fig. 3.5. And finally, for class 5 the right side where we do the matching displays no poles, agreeing with Fig. 3.6.
We summarize our results in table 3.2, which lists the five classes of fermions and their q1 values. When there is a pole at ω = 0, we list the corresponding kF /µ3 and vF . Furthermore, we list the values of k±∆/µ3 for poles at ω = ∆; in some cases there is more than one. If a pole ± momentum at the appropriate side of the gap matches with the (3+1)QBH, we underline it and provide the value of ν±∆ at that pole.
Table 3.2: A summary of the results for fermion modes in the 3QBH.
Class q1 kF /µ3 vF k∆/µ3 ν∆ k−∆/µ3 ν−∆ 1 1 0.163 0.769 0.0832; 0.254; 0.499 0.248 0.271 − − × 2 1 0.467 0.786 0.796 0.0931 0.373 − × 3 3 None None 0.578; 0.398 None None − × 4 1 0.143 0.748 0.189 0.672 0.0578 − − × − 5 3 None None None None 0.715; 0.328 − ×
3.6 RG Flow Backgrounds: 2QBH and 1QBH
We finally turn to the remaining boundaries of the parameter space: the two-charge black hole (2QBH), where two charges are set to zero and the other two are set equal, and the one-charge black hole (1QBH), where three charges are chosen to vanish. These cases have the property that it is not possible to take the extremal, zero-temperature limit without also setting the remaining chemical potential(s) to zero. In both cases one can take an rH 0 limit with the appropriate → Q fixed, but one does not get a black hole: this limit removes the horizon, as well as shutting 79
0.5
æ æ æ æ æ æ k æ æ 0.0 æ
æ æ Μ3 æ æ
æ æ æ æ æ æ -0.5 æ æ æ æ æ æ æ
-0.5 0.0 0.5 Ω
Μ3
Figure 3.14: Class 5 fermions for the 3QBH, with a pair of poles nucleating near ω = ∆. − 80 off the remaining gauge field, leaving a background with only the running scalar disturbing the metric, a so-called renormalization group (RG) flow geometry. The remaining parameter Q no longer measures a charge, but instead controls the strength of the scalar perturbation.
The simplifications in the bosonic background render the Dirac equations fully solvable, and in the following subsections we will present the exact Green functions and briefly discuss the spectrum of the fermions. Although the backgrounds are nonthermodynamic, they are still worth commenting on. First, although this is outside the main thrust of our work, they are interesting as RG flow geometries and the fermionic Green’s function reveals whether the spectrum is discrete or continuous, gapped or ungapped. For analogous five-dimensional cases, all fluctuations share the same spectrum due to the large supersymmetry [50]. Second, matching these results to the endpoints of the regular cases can provide a check and analytic values for kF in the limit. Finally, these geometries are limits of nonzero-temperature backgrounds and may provide information about those thermodynamic cases.
3.6.1 The 1-Charge Black Hole
By setting Q3 = 0 in the (3+1)QBH background we obtain the 1QBH:
r 1 Q1 A(r) = B(r) = log + log 1 + , (3.133) − L 4 r 2 rH (rH + Q1) 1 Q1 h(r) = 1 2 , φ = log 1 + , (3.134) − r (r + Q1) −2 r s rH Q1 rH + Q1 Φ3(r) = 0 , Φ1(r) = η1 1 . (3.135) L rH + Q1 − r + Q1
The thermodynamic quantities are now
r 3/2 3rH + 2Q1 rH rH 1/2 T = 2 , s = 2 (rH + Q1) , (3.136) 4πL rH + Q1 4GL s r rH Q1 η1 Q1 µ3 = 0 , µ1 = η1 2 , ρ3 = 0 , ρ1 = . (3.137) L rH + Q1 2π rH
From (3.136) we see that the extremal limit corresponds to taking rH 0. This limit also causes all → other thermodynamic quantities to vanish, since the horizon function h(r) 1, and the remaining → 81 gauge field Φ1(r) 0 as well. As promised we are then left with an RG-flow background consisting → of AdS space deformed by an r-dependent scalar field. Unlike the 3QBH and the 2QBH discussed in the next subsection, there is no order-of-limits issue here, the same background is reached starting from the (3+1)QBH regardless of whether the 1QBH-limit or the extremal limit is taken first.
With the horizon function gone, fluctuations respect three-dimensional Lorentz invariance, and with no gauge fields the qi and pi parameters are irrelevant. Hence the uncoupled second order
2 2 Dirac equation depends only on ω k , the mass parameter m and the parameter Q1. Defining a − new variable v = r/(r + Q1), we obtain an exact solution for the spinor component infalling at the horizon in terms of Hankel functions of the first kind:
2 2 1/4 r 2 2 ! ω k (1) 2 ω k χ+(r) = − H 1−m − . (3.138) v 2 Q1 v
The Green function is obtained, as usual, from the asymptotic behavior of the spinor. The result is √ 2 2 (1) 2L2 ω2−k2 √ω k H −1−m Q1 − 2 G(ω, k) = √ . (3.139) (1) 2L2 ω2−k2 (ω + k)H 1−m Q1 2 The imaginary part of the Green’s function reveals a continuous spectrum; unlike the 2QBH in the next subsection, it does not display a mass gap.
The poles and zeros at ω = 0 for the (3+1)QBHs plotted in figures 3.2-3.6 all approach finite, nonzero values of k/µ3 as µR 0, implying their values of k/µ1 are in all cases driven to zero; → we can ask whether we see the corresponding poles or zeros at ω2 k2 = 0 in the 1QBH Green’s − function. In fact 3.139 has a zero for m = 1 and a pole for m = 3, which matches correctly the − classes 3, 4 and 5 with only a single pole or zero approaching µR 0 limit. For classes 1 and 2 → there are both poles and zeros as µR 0; in both cases it is the zero, which has the larger value → of k, that is reflected in the 1QBH Green’s function. 82
3.6.2 The 2-Charge Black Hole
The 2QBH is the special case where Q˜2 is set to zero in the (2+2)QBH background. The geometry is
r 1 Q2 A(r) = B(r) = log + log 1 + , (3.140) − L 2 r 2 rH (rH + Q2) Q2 h(r) = 1 2 , γ = log 1 + , (3.141) − r(r + Q2) r √Q2rH rH + Q2 Φ2(r) = η1 1 dt , Φ˜ 2(r) = 0 . (3.142) L − r + Q2
The thermodynamics are given by
3rH + Q2 rH T = , s = (rH + Q2) , (3.143) 4πL2 4GL2 r η2 p η2 2Q2 µ2 = 2 2Q2rH , µ˜2 = 0 , ρ2 = s , ρ˜2 = 0 . (3.144) L 2π rH
Looking at (3.143), it is clear that in order to continuously tune the temperature down to zero, both rH and Q2 must be taken to zero, leaving us with nothing but empty AdS space. The closest analogue to an extremal solution is to set rH = 0 with Q2 fixed and non-zero. In the limit
2 rH 0 the temperature approaches the value Q2/(4πL ), but this limiting temperature does not → strictly obtain since the horizon disappears at the endpoint. Similarly to the case for the 1QBH, this removes the last remaining gauge field and thus the chemical potential, leaving an RG flow geometry with a running scalar.
As was the case for the extremal 3QBH, there is an order-of-limits issue when this “extremal”
2QBH is approached from the (2+2)QBH, depending on whether T 0 or Q˜2 0 is imposed → →
first. Taking the extremal limit first and then going to the 2QBH shifts the gauge potential Φ˜ 2 relative to 3.142 by a constant,
Q2 Φ˜ 2 . (3.145) → L
The second-order Dirac equation depends only on ω2 k2, the mass parameterm ˜ and the parameter −
Q2, and as in the 1QBH case, it admits an analytic solution. Again we impose appropriate boundary 83 conditions in the interior of the bulk. The solution for the spinor component χ+ is then
√16L4(ω2−k2)−m˜ 2Q2 −i 2 r 4Q2 χ+(r) = , (3.146) r + Q2 and the resulting Green function is
p 4 2 2 2 2 mQ˜ 2 + i 16L (ω k ) m˜ Q G(ω, k) = − − 2 . (3.147) 4L2(k + ω)
2 The imaginary part of the Green function displays a mass gap of Q2/2L , then a continuum above the gap. This is very similar to the 1QBH in 5 dimensions studied in [13], dual to the Coulomb branch flow of = 4 super-Yang-Mills, which also showed a continuous spectrum above a gap N [49, 50].
Similarly to what we did for the 3QBH, we can match this to theµ ˜2 0 limit of the → (2+2)QBH results, in the process deducing the precise values that the Fermi momenta and the momenta of zeros in the Green function should approach in this limit. Thanks to the shift 3.145, there is a relation between the energies analogous to 3.131,
q˜2Q2 ω2Q = ω + . (3.148) (2+2)Q 4L2
The analytic solution (3.147) tells us that fermions with negative (positive)m ˜ have poles (zeros) at exactly ω2 k2 = 0, meaning for ω = 0 there is a pole (zero) at 2Q − (2+2)Q
q˜2Q2 q˜2µ˜2 kF (L) = . (3.149) 4L2 → 4√2
This indicates that class I should have a zero at k =µ ˜2/√2, class II should have a pole at k = 0, | | and class IV and VI (class III and V) should have a pole (zero) at k =µ ˜2/√8. This agrees very | | well with our numerical results as seen in figures 3.7-3.9.
The fermions in class II have a special behavior in the 2QBH limit since they are not charged under the gauge field Φ˜ 2. We refer to them as marginal Fermi liquid (MFL) fermions since they approach the value νk = 0.5 in this limit. For these fermions, we can normalize the Fermi momentum by µ2 instead ofµ ˜2 and still get a finite result, as is shown in Fig. 3.15. For all the other fermions, 84
1.0 4QBH
0.5 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ k 0.0 ç Μ ç 2 ç ç ç ç ç -0.5
-1.0
0.0 0.2 0.4 0.6 0.8 1.0 ΜR
Figure 3.15: Class II fermions for the (2+2)QBH, with k normalized relative to µ2 instead ofµ ˜2. The Fermi surfaces all lie at kF µ2/√8. ≈ 85 the line of poles or zeros diverges in the µR 0 limit with this normalization. Looking at this → plot, it appears possible that the line of Fermi surfaces all lie at kF = µ2/√8 independent ofµ ˜R, although we will not try to prove that here. This same behavior was observed for the MFL fermions in [13]. Chapter 4
Fermionic Response in Finite-Density ABJM Theory with Broken Symmetry
This chapter is an edited version of [18], written in collaboration with Oliver DeWolfe, Steven
S. Gubser and Christopher Rosen.
4.1 Introduction
In this chapter, we move on from gravitational backgrounds where the U(1) symmetry re- sponsible for the finite density is unbroken, such as those in chapter3, and begin our study of backgrounds that break that symmetry. This includes the holographic superconductors, in which a charged scalar condenses outside the horizon in the gravity background for sufficiently low tem- perature [11, 71, 72, 73]. The zero-temperature limits of such backgrounds are expected to be horizonless domain wall-type geometries, where Lorentz invariance or even conformal invariance may be regained in the infrared [73, 74, 75]. Such backgrounds could be related to the non-Fermi liquids discussed previously, as real-world non-Fermi strange metals in high-temperature super- conductors become hidden behind a superconducting dome below a critical temperature, but the zero-temperature quantum critical point is still thought to control their behavior. Symmetry- breaking systems have been studied using bottom-up generic fermion actions in [15]. In [16], a
Majorana Yukawa coupling was studied where a charged fermion couples to itself (not its conju- gate) as well as a “Cooper pair” scalar with twice the charge. It is interesting to ask whether the
Fermi system becomes gapped in the presence of symmetry breaking; [16] found that the Majorana coupling could lead to the generation of such a gap by a “level crossing” repulsion between two 87 lines of poles in the Green’s function.
Naturally, it is interesting to consider symmetry-breaking systems from a top-down perspec- tive. Holographic superconductors descending from string/M-theory were constructed in [76, 77, 78,
79, 80], and one may ask about the fermionic response of these systems, using the Dirac equations derived from the top-down theory. Here we will focus on a class of symmetry-breaking geometries described in [81]. These backgrounds are solutions to the equations of four-dimensional = 8 N gauged supergravity invariant under a SO(3) SO(3) subgroup of the SO(8) gauge group, and be- × sides the metric involve a gauge field corresponding to the chemical potential and a charged scalar.
These geometries are domain walls, interpolating between the maximally symmetric AdS4 vacuum in the UV and a different AdS4 region corresponding to a nonsupersymmetric critical point of the scalar potential in the IR, and as such are similar to the zero-temperature limit of holographic superconductors. Both the UV and IR fixed points are known to be stable under perturbative scalar fluctuations [82]. The dual descriptions of these solutions are states of three-dimensional superconformal ABJM theory with a chemical potential for monopole charge, and both a source and an expectation value turned on for charged operators dual to the bulk scalar; as a result the system is not precisely a superconductor, since the associated U(1) is broken explicitly as well as spontaneously.
To understand fermion response in these ABJM states, we analyze the spectrum of fermionic
fluctuations in the holographically dual backgrounds. We study spin-1/2 modes whose SO(3) × SO(3) quantum numbers prevent them from mixing with the gravitino fields. While they do not mix with the gravitino, these spin-1/2 modes mix with each other; another system of mixed fermionic excitations in a symmetry-breaking background was studied in [80]. We study a simplified, non- chiral version of the full top-down fermion mixing matrix, where in a more intricate generalization of the Majorana “Cooper pair” coupling of [16], a charged fermion mixes with a neutral fermion via the condensed charged scalar, which in turn has a Majorana-type coupling to its own conjugate.
Thus the analog of a Cooper pair in this system is a condensation of a charged/neutral bound state.
Since the charge is monopole charge, the associated charged operators may be thought of as bound 88 states of fundamental fermions with vortices, so-called composite fermions.
We study two backgrounds, one having a source for a fermion bilinear and an expectation value for a boson bilinear, and the other with the roles reversed. We first elaborate on the analysis of [81] of the conductivities of these backgrounds. In geometries approaching AdS in the infrared, the emergent Lorentz invariance generates a light-cone structure on the energy-momentum, and modes living outside this light-cone have regular (rather than infalling) IR boundary conditions; normal modes in this region are associated to fluctuations in the field theory with zero dispersion.
We determine the locations of such normal modes, and find in both cases two lines of modes, one gapped and the other ungapped. By studying the pole structure of the matrix of Green’s functions, we can see that each line is a mixture of both charged and neutral fermionic excitations. This matrix also provides information about the presence of unstable fluctuations inside the lightcone.
Following the ideas of [16], it is natural to inquire how the charged fermion/neutral fermion coupling in our system of Dirac equations affects the dynamics, and in particular whether it causes a repulsion between lines of poles. We study a modification of the Dirac equations removing the couplings between different fermions, and see that without the charged/neutral coupling there is an ungapped, purely charged band, a purely neutral band that asymptotes to the origin in energy/momentum space, and a new gapped band with the conjugate charge; each pair of bands has a point of intersection. The charged/neutral coupling thus has a number of effects: it pushes the third band outside the region of stable excitations, it repels the crossing between the other two bands while mixing the charged and neutral contributions, and it turns what was the neutral band away from the origin leaving it fully gapped. While modifying the Dirac equations departs from the top-down structure of = 8 supergravity, it allows us to see how the couplings are responsible N for the structure of the quasiparticle excitations. The fully top-down, chiral fermion mixing matrix is studied in chapter5. 89
4.2 The Bosonic Background Geometries
In this section we isolate a sector of maximal gauged supergravity in D = 4 whose bosonic content includes the vierbein, one U(1) gauge field, and one complex scalar. Being a consistent truncation, any solution to the resulting equations of motion can in principle be uplifted to a solution of supergravity in D = 11. This feature is particularly notable in that it will allow us to exploit the holographic duality between solutions of M-theory and states in ABJM theory to study the properties of an explicitly known field theory at strong coupling.
We will be interested in solutions to the equations of motion wherein the scalar interpolates between two fixed points of its potential, and the geometry asymptotes to different AdS4 regions in the IR and UV. Holographically, these solutions are dual to zero-temperature states of ABJM theory in which both the low energy and high energy physics is governed by (distinct) conformal
field theories. In [81] such domain wall solutions were found numerically, and their identification with zero-temperature limits of novel states in the ABJM theory was discussed in some detail. We review these solutions, and provide further commentary on their dual holographic description.
4.2.1 The SO(3) SO(3) truncation ×
For the backgrounds we study we will be interested in a particular truncation of the gauged supergravity described in section 2.1, retaining only modes invariant under an SO(3) SO(3) × subgroup of SO(8). We will choose I = 3, 4, 5 and I = 6, 7, 8 to be the directions in which the two
SO(3) groups act on the 8s; the 1, 2 directions correspond to an additional SO(2) gauge symmetry commuting with SO(3) SO(3). The truncation corresponds to an = 2 gravity multiplet plus × N a hypermultiplet, with the bosonic sector consisting of the vierbein, a single graviphoton gauge
field for the SO(2) in the 12-directions, and two complex scalars charged under it. Moreover, it is consistent with the equations of motion to set one complex scalar to zero, and we will do this in what follows. This SO(3) SO(3)-invariant truncation is characterized by the ansatz [83] ×
λ(x) h + − + − i φIJKL(x) = cos α(x) Y + i Y sin α(x) Z i Z , (4.1) 2√2 IJKL IJKL − IJKL − IJKL 90 where λ and α are four-dimensional scalars, and Y ± and Z± are self dual (+) and anti-self dual
( ) invariant four-forms on the scalar manifold, defined as −
+ 3451 2678 − 3452 1678 YIJKL = 4! δIJKL + δIJKL YIJKL = 4! δIJKL + δIJKL (4.2)
Z− = 4! δ3451 δ2678 Z+ = 4! δ3452 δ1678 . (4.3) IJKL IJKL − IJKL IJKL IJKL − IJKL
The more general case of two complex scalars would involve four independent coefficients for the four tensors.
Evaluating the Lagrangian (2.29) in the SO(3) SO(3) invariant truncation gives × 2 −1 1 1 µν µ sinh (2λ) µ µ e = R FµνF ∂µλ∂ λ (∂µα gAµ)(∂ α gA ) , (4.4) L 2 − 4 − − 4 − − − P
2 where κ has been set to one, the gauge field is A A12, and the potential is ≡ g2 = s4 8s2 12 with s sinh λ. (4.5) P 2 − − ≡
It is easy to see that this potential has critical points at s = 0 and s = 2. In anticipation of the ± domain wall geometry, we will refer to the corresponding values of the scalar λ as
λUV 0 and λIR log(2 + √5) , (4.6) ≡ ≡ ± q corresponding to AdS solutions with AdS radii L = √1 and L = 3 L , respectively. In 4 UV 2g IR 7 UV the next subsection, we discuss solutions to the equations of motion coming from (4.4), which will provide the classical backgrounds we wish to probe.
4.2.2 The Domain Wall Backgrounds
The solutions of interest [81] are given in the ansatz,
dr2 ds2 = G(r)e−χ(r)dt2 + r2d~x2 + ,A = Ψ(r) dt, λ = λ(r), α = 0 . (4.7) − G(r) 91
This ansatz sets g = 1, equivalent to LUV = 1/√2. The equations of motion are
sinh2(2λ)Ψ2 0 = eχ r 2rλ02 χ0 , (4.8) − 2G2 − − 1 sinh2(2λ)Ψ2 G0 eχΨ02 0 = + P + eχ + + λ02 + , (4.9) r2 G 4G2 rG 2G sinh2(2λ)Ψ 2Ψ0 1 0 = + + χ0Ψ0 + Ψ00 , (4.10) − 2G r 2 sinh2(4λ)Ψ2 0 2 G0 1 0 = eχ P + λ0 + χ0 + λ00 . (4.11) 4G2 − 2G r G − 2
Near the boundary, the solution approaches the maximally supersymmetric AdS4 vacuum with
λ = 0, given by
2 2 r GUV = 2r = 2 , χUV = const , ΨUV = const , (4.12) LUV while in the IR region far from the boundary, the scalar approaches the extremal value λIR = log(2 + √5) and the equations are solved by
2 14 2 r GIR = r = 2 , χIR = const, and ΨIR = 0 . (4.13) 3 LIR
By rescaling the time coordinate one sees only χUV χIR is physical, but for convenience in finding − solutions we will allow both to be free. Fixing ΨIR = 0 allows us to identify ΨUV as proportional to the chemical potential µ of the U(1)b conserved current.
A useful invariant of the domain wall solution is the index of refraction n, defined by the ratio of the speed of light in the UV and IR CFTs: r vUV LIR 1 (χ −χ ) 3 1 (χ −χ ) n = e 2 IR UV = e 2 IR UV . (4.14) ≡ vIR LUV 7
This quantity is invariant under coordinate transformations, and characterizes the causal properties of the emergent IR conformal dynamics.
To construct the domain wall geometries, it is convenient to consider IR–irrelevant pertur- bations about the fixed point solution (4.13). These perturbations can be used to numerically integrate away from the IR critical point at r = 0 along the radial direction, tracking the RG
flow “upstream” to the UV fixed point at r = . To identify these perturbations, one performs a ∞ 92 linearized fluctuation analysis by substituting into the equations of motion the following ansatz for the IR form of the bulk fields:
14 G(r) = r2 1 + δG rγ , (4.15) 3 ξ χ(r) = χIR + δχ r , (4.16)
Ψ(r) = δΨ rβ , (4.17) λ(r) = log 2 + √5 + δλ rα . (4.18)
The requirement that the perturbations represent irrelevant deformations to the IR fixed point constrains the various exponents. Specifically, one requires α, β, ξ > 0 and γ > 2. − Substituting the fluctuations (4.15-4.18) into the equations of motion (4.8-4.11) one finds that the linearized equations decouple, can be solved non-trivially by r r 303 3 247 1 δG = δχ = 0, α = and β = , (4.19) 28 − 2 28 − 2 and that the amplitude δΨ can be rescaled to any value under symmetries of the equations of motion. Thus the IR deformations are described by a single free parameter, δλ, which we will tune to produce domain wall solutions with various UV asymptotics.
The asymptotic mass of the scalar field λ near the boundary is
1 ∂2 2 m2 = P = , (4.20) λ 2 ∂λ2 −L2 λ=0 UV which implies that near the UV boundary the scalar behaves like
λ1 λ2 λ(r ) + + ... (4.21) → ∞ ∼ r r2
We will consider two particular cases, solutions in which either λ1 or λ2 vanish. For reasons we will explain in the next subsection, we will refer to these as “Massive Boson” and “Massive
Fermion” backgrounds, respectively. In practice, it is straightforward to produce such solutions by integrating the equations of motion from very near the IR fixed point to the UV boundary.
This involves employing the scaling symmetries of the equations of motion to fix δΨ and χIR, then 93 using the fluctuations defined in (4.15-4.18) to produce IR boundary conditions for the numerical integration of (4.8-4.11) for many choices of δλ. After each successful integration throughout the bulk, one can fit the near boundary behavior of the numerical solution obtained for λ to the form given in (4.21), and subsequently extract the values of λ1 and λ2, χUV and ΨUV characterizing that solution.
“Massive Boson” and “Massive Fermion” solutions constructed from this procedure are shown
1 in figures 4.1 and 4.2. The Massive Boson solution, with nonzero λ2, has interesting similarities to the extremal AdS Reissner-Nordstr¨omsolution, and in some sense is “almost” AdSRN. Extremal
2 AdSRN is characterized by an AdS2 R near horizon geometry, which manifests as a double pole × in the metric function grr. From figure 4.1, a similar feature can be seen around r 0.4 where G ≈ very nearly vanishes quadratically, before reverting to a nonzero value. Moreover, the figure shows that nearly all the scalar hair is bunched behind this “almost” horizon. Perhaps not surprisingly, similar properties have been observed in the extremal limits of various holographic superconductors studied in the literature [75].
The index of refraction in this solution is large, n = 26.900, which implies that the effective speed of light is very slow in the IR CFT compared to the UV theory, in turn suggesting that the
IR dynamics is nearly 0 + 1 dimensional, reminiscent of the Semi-Local Quantum Liquid [84]. This is another sense in which the solution is “almost” extremal AdSRN, since the black hole horizon corresponds to an n limit. We will learn in the next subsection that λ2 is proportional to a → ∞ dimension-2 source for a scalar bilinear; the dimensionless ratio of the source to the U(1) chemical potential can be measured to be
1/2 λ 2 0.0308 , (4.22) ΨUV ≈ indicating that indeed this solution can be thought of as a small perturbation by λ on top of the no-scalar background, which has extremal AdSRN as a solution.
The Massive Fermion background has λ2 = 0, and unlike the previous case, this geometry
1 It is possible that these solutions are not unique, even up to rescalings; there may be additional solutions with nodes in λ. If such additional solutions exist, they are probably unstable toward bosonic perturbations. 94 5 4 4 2 3 G β 0 2 r 2 -2 1 -4 0 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 r r
8 1.5
6 1.0 Ψ λ 4 0.5
2 0.0 0 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r r
Figure 4.1: The Massive Boson background. The dashed lines in the plot of G/r2 are at 14/3 and 2, indicating the values obtained in the IR and UV AdS4 fixed points respectively. The ratio of the speed of light in the UV CFT compared to that of the IR theory is n = 26.900, and the λ1/2 non-vanishing scalar fall-off is 2 0.0308. ΨUV ≈
5 4 4 2 3 G β 0 2 r 2 -2 1 -4 0 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 r r
0.5 1.5 0.4 1.0 0.3 Ψ λ 0.2 0.5
0.1 0.0 0.0 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r r
Figure 4.2: The Massive Fermion background. The dashed lines in the plot of G/r2 are at 14/3 and 2, indicating the values obtained in the IR and UV AdS4 fixed points respectively. This geometry is characterized by n = 1.861 and λ1 1.227. ΨUV ≈ 95 is not “almost” AdSRN in any sense. The function G(r) never comes close to vanishing, so the solution is not “close” to having a horizon. The index of refraction is substantially closer to unity at n = 1.861, so the speed of light does not change that dramatically between the UV and the IR.
In this case λ1 is proportional to a dimension-1 source for a fermion bilinear, and we have
λ1 1.227 , (4.23) ΨUV ≈ so the two massive perturbations to ABJM theory, the source and the chemical potential, are the same to a factor of order unity. We plot the IR light cones for the two solutions next to a UV light cone normalized at right angles in figure 4.3.
These backgrounds interpolate between UV and IR fixed points that are known to be stable in the following sense. The ultraviolet AdS4 is stable on account of supersymmetry; this corresponds simply to the unitarity of ABJM theory. It is shown in [82] that all the scalar fluctuations in the non- supersymmetric IR AdS4 geometry satisfy the Breitenlohner-Freedman bound [63, 64]. However, we do not know of a demonstration of stability of the non-supersymmetric AdS4 solution against perturbations involving non-scalars; also, stability of the anti-de Sitter endpoints of these domain wall solutions does not by itself demonstrate the stability of the whole domain wall. Nonetheless, these domain wall backgrounds are the best candidates available for a stable holographic dual of a
finite-density state in ABJM theory.
4.2.3 Holographic Interpretation
The domain wall backgrounds constructed in the previous subsection are horizonless solu- tions to = 8 gauged supergravity with a non-vanishing electric potential for the gauge field. N Thus, we broadly expect that these bulk solutions provide a holographic description of certain zero temperature states of ABJM theory at finite density. The fact that a charged scalar is turned on in these backgrounds implies that we are studying either a deformation of the ABJM theory by the addition of a dual scalar operator, a state of the ABJM theory with non-vanishing expectation values for this scalar operator, or some combination of these. Since we have a top-down solution, 96
t 1.0
0.5
x -1.0 -0.5 0.5 1.0
-0.5
-1.0
Figure 4.3: In units where the UV light cone is 45o (dotted black), we compare the Massive Boson (solid blue) and Massive Fermion (dashed red) IR light cones. 97 we can determine the nature of the background precisely using the explicit mapping between the bulk fields and various single trace operators of the ABJM theory.
The truncation of the maximal gauged supergravity breaks the SO(8) gauge symmetry to
SO(3) SO(3) SO(2). The surviving bulk fields are all singlets under the SO(3) SO(3), × × × and carry charge only under the remaining SO(2) ∼= U(1). The gauge field associated with this U(1) is A, the active gauge field in the domain wall backgrounds constructed above. The gauge symmetry present in the supergravity theory is holographically dual to the global R-symmetry of the ABJM theory, and thus bulk solutions with non-zero At = Ψ correspond to ABJM theory with a chemical potential µ turned on for the conserved global U(1)b current. The dual U(1)b current counts monopole number, and hence takes the form
µ µνλ J Tr (Fνλ + Fˆνλ) , (4.24) b ∼ where F and Fˆ are the field strengths for U(N) U(N). × Let us now connect the scalar λ in our background to dual ABJM operators. As mentioned previously, the SO(3) SO(3)-invariant sector of = 8 gauged supergravity contains a hypermul- × N tiplet, corresponding to two complex scalars, which can be packaged in various ways. The gauged supergravity naturally gives rise to φ1, φ2 which are a complex SO(2) doublet, the real parts being
1 parity-even scalars and the imaginary parts being pseudoscalars. We can define φ1 √ (S1 + iP1) ≡ 2 1 and φ2 √ (S2+iP2), and can assemble charge and parity eigenstates as S S1+iS2,P P1+iP2. ≡ 2 ≡ ≡ 1 1 † † 1 [81] also make use of the combinations ζ1 √ (φ1 iφ2) = √ S + iP and ζ2 √ (φ1 +iφ2) = ≡ 2 − 2 ≡ 2 √1 (S + iP ). 2 To identify the dual ABJM operators, recall the 70 scalars of the gauged supergravity theory live in a 35v 35c, each of which decomposes into SU(4) U(1)b representations as ⊕ ×
35v 150 102 10−2 , 35c 150 10−2 102 , (4.25) → ⊕ ⊕ → ⊕ ⊕ corresponding to the “Y 2” operators dual to the parity-even scalars,
A † 1 A C † (A B) 2τ † † −2τ 150 : Y Y δ Y Y , 102 : Y Y e , 10−2 : Y Y e , (4.26) B − 4 B C (A B) 98 as well as the “ψ2” operators dual to the pseudoscalars,
†B 1 B †C 2τ †(A †B) −2τ 150 : ψAψ δ ψC ψ , 102 : ψ ψ e , 10−2 : ψ ψ e . (4.27) − 4 A (A B)
Under SU(4) SO(3) SO(3) the 15 does not contain a singlet, while both the 10 and the 10 → × become (3, 3) (1, 1). Thus the complex scalar S and pseudoscalar P living in the SO(3) SO(3)- ⊕ × invariant truncation correspond to the U(1)b-charge 2 operators
A A 2τ 2τ S Y Y e , P ψAψAe . (4.28) O ≡ O ≡
The ansatz (4.1) truncates the scalar sector down to one complex scalar. The truncation can be described in various equivalent forms:
ζ1 = 0 φ1 = iφ2 S = iP S1 = P2,P1 = S2 , (4.29) ↔ ↔ ↔ − and the remaining two degrees of freedom can be identified with λ and α from (4.1) as
iα tanh λ e = ζ2 = √2S = i√2P. (4.30)
We note that while the Lagrangian (4.4) indicates the scalar has charge g in a convention where the gauge field is dimensionless, it is convenient for us to match the natural field theory convention and refer to this as charge 2. Thus our background involves a simultaneous turning on of sources
A A 2τ 2τ and/or expectation values for the ∆ = 1 operator Y Y e and the ∆ = 2 operator ψAψAe , with a fixed relative phase.
All the scalars of the supergravity theory have the asymptotic mass m2L2 = 2, lying in UV − the window where both the leading terms in the near-boundary expansion (4.21) are normalizable deformations of AdS4, and correspondingly the scalars can be quantized in one of two ways. Super- symmetry [63, 64] requires that the pseudoscalars in the 35c have the standard quantization dual to an operator with ∆ = 2, while the scalars in the 35v must have the alternate quantization, and be dual to operators with ∆ = 1. For regular quantization fields, the mode λ1 in (4.21) corresponds to the source, while the subleading term λ2 corresponds to the expectation value; this holds for our
2τ ψAψAe operator. For alternate quantization fields, λ1 is the expectation value, while λ2 is the 99 source, which holds for Y AY Ae2τ . Hence we find each parameter in the solution controls both a source for one operator and an expectation value for the other,2
2 λ1 J 2 = Re Y , (4.31) ∼ Im ψ h i 2 λ2 J 2 = Im ψ , (4.32) ∼ Re Y −h i
2 2 A A 2τ 2τ where Y and ψ are shorthand for Y Y e and ψAψAe respectively. Thus for a solution where
λ1 = 0, the background corresponds to a source for the scalar bilinear, hence the name “Massive
Boson”, as well as an expectation value for the fermion bilinear, while for the “Massive Fermion” solution with λ2 = 0 we have a source for a fermion bilinear and an expectation value for the boson
A A 2τ 2τ bilinear. Since either Re Y Y e or Im ψAψAe is explicitly added to the ABJM field theory
Lagrangian, in either case one breaks the U(1) symmetry explicitly. Hence these domain wall backgrounds are not precisely holographic superconductors, which should involve only spontaneous breaking of the symmetry.
We note in passing that it is possible to cast these backgrounds as true holographic super- conductors, if we pass to an alternate quantization of some of the scalars or pseudoscalars and hence move away from ABJM theory to a non-supersymmetric boundary theory. In a quantiza- tion where all the active scalars are dual to ∆ = 2 operators, λ1 = 0 backgrounds involve only expectation values of the dual operators; conversely λ2 = 0 backgrounds have no sources if all the active scalars are dual to ∆ = 1 operators. In these two scenarios, the solutions are in fact holographic superconductors in the usual sense. The former case can be obtained as the infrared limit of a deformation of ABJM theory by a relevant double trace operator, essentially the square of Y AY Ae2τ . We prefer, however, to work in the quantization dual to the ABJM theory when analyzing fermionic Green’s functions, because there can be no doubt about the operators dual to the supergravity fermions. In this approach, we cannot claim to be analyzing fermionic response in a true holographic superconductor, but in a member of a broader class of symmetry-breaking backgrounds.
2 Additional finite boundary counterterms could in principle shift these relations at the nonlinear level, but the terms required by supersymmetry found in [65] vanish in our background. 100
4.2.4 Conductivities
Before we turn to the task of probing the fermionic properties of these supergravity back- grounds, it is sensible to wonder what lessons we can learn from the linear response of bosonic probes. An obvious candidate is the conductivity of ABJM matter charged under the global U(1)b.
The imaginary part of this conductivity appeared previously in [81], where it was claimed that the
1/ω pole in the imaginary part of the DC conductivity was indicative of superconductivity in the boundary gauge theory. We will briefly revisit this claim before turning to the real part of this
U(1) conductivity.
The linear response of the gauge theory current to an applied electric field is encoded in the retarded Green’s function, which in turn dictates the AC conductivity, σ(ω):
Jx i R σ(ω) = h i = GJxJx (ω) . (4.33) Ex −ω
Roughly speaking, (4.33) shows that the real part of the AC conductivity gives a measure of the density of states for charged matter at zero spatial wavenumber.
From a bulk perspective, computing this conductivity is by now a standard exercise in applied holography. The computation begins by turning on the coupled perturbations A A + δA and → g g + δg where → −iωt −iωt δA = δAx(r)e dx , δg = δgtx(r)e dtdx , (4.34) and continues by solving the equations of motion linearized about these perturbations. The near
R boundary behavior of δAx fully determines the current-current correlator GJxJx . If
(1) (1) (0) δAx R δAx δAx(r ) δAx + + ... then GJxJx = 2 (0) , (4.35) → ∞ ∼ r δAx and using (4.33) results in the conductivity.
In figure 4.4 the real and imaginary parts of the AC conductivity are shown for charge trans- port in both Massive Boson and Massive Fermion backgrounds. From the rightmost plot, it is immediately clear that Im σ 1/ω at low energies. By the Kramers-Kronig relations, this nec- ∼ essarily implies a delta function contribution to the real part of the conductivity. Such a delta 1.5 101 1.4
1.2 1.0
1.0 0.5 Σ
Σ 0.8 0.0 Im
Re 0.6 Ω -0.5 0.4
-1.0 0.2
0.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 ΩΜ ΩΜ
Figure 4.4: The real (left) and imaginary (right) AC conductivity in Massive Boson (darker) and Massive Fermion (lighter) backgrounds. The imaginary part of the conductivity has been multiplied by ω to highlight the 1/ω pole at low energies giving rise to the delta function in Re σ.
function does not imply that the backgrounds we are studying are holographic superconductors, since any translationally invariant background with non-zero charge density will show a similar delta function peak in the conductivity [72]. Indeed, as we saw in the previous subsection, our backgrounds are not true superconductors for ABJM theory, since the U(1) is explicitly broken, al- though the same conductivity calculations apply to the true holographic superconductors associated to non-supersymmetric alternate quantizations.
Re σ(ω) is characterized in both backgrounds by the aforementioned infinite DC contribution separated from the conformal plateau at high energies by a soft gap. The fact that the Massive
Fermion background gives rise to a broader gap can be used to argue that this state has an en- hanced suppression of charge carrying states at intermediate energies relative to the Massive Boson background’s dual. As indicated above, this suppression can be inferred only for the states near the origin in momentum space, and thus is of limited utility for uncovering what is happening to the fermionic degrees of freedom in the dual ABJM state. This is because the natural expectation for a system of fermions at finite density is to organize into a Fermi surface at some finite k = kF .
Thus, to address questions related to the fermionic nature of these states it would be more appro- priate to study current-current correlators at non-zero k along the lines of [85, 86]. An alternative approach, which we adopt here, is to study the fermion response of the ABJM states directly, using appropriate fermion probes. 102
4.3 Fermion Response in the Domain Wall Solutions
We would like to study the linear response of states in the gauge theory to the insertion of various fermionic operators, which is characterized by an assortment of fermionic two-point functions. Holographically, these two-point functions are computed from the linearized fluctuations of supergravity fermions about the classical (bosonic) backgrounds of interest. Here we will describe the calculation of these equations, the mixing between modes, and will discuss a simplified fermion mixing matrix.
4.3.1 Coupled Dirac equations and holographic operator map
We will focus on spin-1/2 fields that cannot mix with the gravitino sector. Under the SO(8) →
SU(4) U(1)b SO(3) SO(3) SO(2) decomposition, we have for the gravitini in the 8s, × → × ×
8s 60 12 1−2 (3, 1)0 (1, 3)0 (1, 1)2 (1, 1)−2 , (4.36) → ⊕ ⊕ → ⊕ ⊕ ⊕ and thus we can avoid mixing in the SO(3) SO(3)-invariant backgrounds as long as we study × fermions in representations other than (3, 1), (1, 3) or (1, 1). The spin-1/2 fields live in the 56s, which decomposes as
56s 152 15−2 100 100 60 → ⊕ ⊕ ⊕ ⊕
(3, 3)2 (3, 1)2 (1, 3)2 (3, 3)−2 (3, 1)−2 (1, 3)−2 (4.37) → ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
2(3, 3)0 2(1, 1)0 (3, 1)0 (1, 3)0 , ⊕ ⊕ ⊕ and thus we see there are fermions in the (3, 3) of SO(3) SO(3) that cannot mix with the × gravitini. They may, however, mix with each other, and in fact do, as we shall see. The different
SO(2) charges of the fermions in the (3, 3) representations are no obstacle to mixing because the
SO(2) symmetry is broken by non-zero λ in our backgrounds. These (3, 3) fermions are not in the fermionic sector of the SO(3) SO(3) truncation discussed previously, and thus to obtain their × dynamics we must return to the full = 8 supergravity theory. N 103
Considering the Dirac equations for the χijk in the (3, 3), we take the first SO(3) to act on greek indices α = 3, 4, 5, the second SO(3) to act on roman indices a = 6, 7, 8, and the SO(2) which corresponds to the active gauge field to act on hatted indicesa ˆ = 1, 2. The fermions that transform as four distinct copies of (3, 3) in (4.37) are then readily seen to be those of the form χαβc, χαbc, and χaβcˆ, where we recall the SO(3) antisymmetric product 3 A 3 = 3. Thus, an example of a ⊗ set of fermions with the same SO(3) SO(3) quantum numbers is χ467, χ538, χ418, χ428 ; these × { } fields may all mix with each other, but not with any others. We will study these four fermions; any other analogous quartet has results related by group theory. We assemble the fermions into charge eigenstates,
χ2 = χ428 + iχ418, χ¯2 = χ428 iχ418, χ0 = χ467 + iχ538, χ¯0 = χ467 iχ538 . (4.38) − −
These fermion fields (4.38) are dual to spinor ABJM operators of the form “Y ψ”, with ∆ = 3/2 and the 56s arising in the product 8v 8c = 56s 8s. Under SU(4) U(1)b the 56s decomposes × ⊕ × as
56s 152 15−2 100 100 60 , (4.39) → ⊕ ⊕ ⊕ ⊕ corresponding to the ABJM operators
A 1 A C 2τ † †B 1 B † †C −2τ 152 : Y ψB δ Y ψC e , 15−2 : Y ψ δ Y ψ e . (4.40) − 4 B A − 4 A C
(A †B) † [A †B] † 100 : Y ψ , 100 : Y(AψB) , 60 : comb. of Y ψ and Y[AψB] , (4.41)
where the other linear combination of the antisymmetric part is part of the 8s. Under SU(4) →
SO(3) SO(3), the 15, 10 and 10 all contain a (3, 3). The fields χ2 andχ ¯2 are charged under ×
U(1)b, and hence will lie in the 152 and 15−2, while χ0 andχ ¯0 sit in the 100 and 100. Tracing 104 through the indices one finds
1 2 3 4 2τ χ2 Y ψ2 Y ψ1 + Y ψ4 Y ψ3 e , (4.42) ↔ − − † †2 † †1 † †4 † †3 −2τ χ¯2 Y ψ Y ψ + Y ψ Y ψ e , (4.43) ↔ 1 − 2 3 − 4 1 †4 4 †1 2 †3 3 †2 χ0 Y ψ + Y ψ Y ψ Y ψ , (4.44) ↔ − − † † † † χ¯0 Y ψ4 + Y ψ1 Y ψ3 Y ψ2 . (4.45) ↔ 1 4 − 2 − 3
The Dirac equation for these fermions can be obtained from (2.40), plugging in the values for the supergravity quantities described in section 2.1 appropriate to the backgrounds discussed in section 4.2.2. The result takes the form
µ iΓ µ 1 + S ~χ = 0 , (4.46) ∇ where 1 is the identity, ~χ is a 4-component vector containing the spinors, and S A + P + M ≡ with A, P, and M describing gauge, Pauli, and mass type couplings, respectively. We find that these matrices fail to commute, and the four spinors mix nontrivially. In this chapter, we study a simplified version of the fermion interactions, which we call the non-chiral mixing matrix, as a step towards the full interactions. These interactions capture many aspects of the full top-down mixing, but differ most significantly in containing no chiral Γ5 terms. The full, chiral interactions will be considered in chapter5.
The non-chiral matrix S is 1 / (3 + cosh 2λ) 0 sinh λ 0 − 4 A − 1 / 0 4 (3 + cosh 2λ) 0 sinh λ S = A − , (4.47) sinh λ 0 √i / 1 / sinh2 λ − 2 2 F 2 A 0 sinh λ 1 / sinh2 λ √i / − 2 A − 2 2 F µ µν where we have written / Γ Aµ, / Γ Fµν. One can see that χ2 andχ ¯2 are charged, and A ≡ F ≡ as with the charged scalar, we identify this as charge 2 in the natural normalization of the field ± theory. Meanwhile χ0 andχ ¯0 are neutral, but have Pauli couplings to the field strength. This basis is diagonal at the ultraviolet fixed point λ = 0, where all the fermions are massless. As the 105 scalar turns on away from the boundary, it rescales the gauge couplings of χ2 andχ ¯2 and, most importantly, introduces couplings between different fermions: there is an interaction between the neutral and the charged fermion of the schematic form φχ2χ0 with φ the charged scalar, and a
2 coupling between the neutral fermion and its conjugate of the form A φ χ0χ¯0. | | It is interesting to compare our Dirac system to other fermionic equations used in holographic superconductors. In [15] an ordinary Dirac equation with tunable charge and mass was studied, and the superconducting character was inherited from interactions with the background. In [16], new terms were added to the Dirac equation to emulate the effects of the Cooper pair condensate by coupling the spinor to its conjugate, with “Majorana” terms
∗ T ∆ φ χ C (η + η5Γ5) χ + h.c., (4.48) L ∼ leading to a Dirac equation of the form
µ µ ∗ (iΓ µ m + qΓ Aµ) χ + (η + η5Γ5)φBχ = 0 , (4.49) ∇ − where η and η5 are coupling constants, Γ5 is the chirality matrix, and B is related to the charge
T 0 conjugation matrix via C B Γ . The scalar must have qφ = 2qχ, and its condensation breaks ≡ the U(1). This leads to terms like χχ + χ∗χ∗ in the effective Lagrangian, analogous to the cc + c†c† terms in a BCS superconductor Lagrangian.
Our system can be viewed as an elaboration of (4.49). Instead of coupling a single charged
field to its conjugate, our system has a “Cooper pair” coupling between the charged scalar, the charged field χ2 and the neutral field χ0, breaking gauge invariance when the scalar condenses, as well as a “Majorana” coupling between the neutral field χ0 and its own conjugateχ ¯0, mediated by the gauge field and the scalar squared. Looking at a coupling like (4.48), it is somewhat natural to think of the scalar field as being dual to the Cooper pair fermion bilinear. Our system is a little more complicated: the two fermionic operators are of the form Y ψ, so the Cooper pair is some part of Y ψY ψ, while the operator dual to the scalar that condenses is either of the form Y 2 or ψ2.
The non-chiral mixing matrix 4.47 lacks the Γ5 terms found in [16], but these are present in the full chiral mixing matrix. 106
In the next subsection we discuss solving the Dirac equations (4.46, 4.47) and relating the results to Green’s functions for these operators, from which we can calculate the normal mode spectrum and the spectral functions.
4.3.2 Solving the Dirac equations and spinor Green’s functions
The analysis of Dirac equations in nonzero density backgrounds is by now standard in the literature, for more details see for example [10, 13, 55, 17]. We rescale the spinors by a factor3
(r4Ge−χ)−1/4 to cancel the spin connection term in the Dirac equations, and Fourier transform as ei(kx−ωt), with frequency ω and spatial momentum k chosen to lie in the x-direction. Next, we follow chapter3 and make the convenient choice of Clifford basis (3.28), where the relevant Γ-matrices are block diagonal, and use the projectors (3.30), which lets us write the four components of the spinor χ as
χα± ΠαP±χ , (4.50) ≡ with α = 1, 2. With this choice of Clifford basis, it is fairly easy to see that the Dirac equations do not mix spinor components with different α, meaning we can split them up into two decoupled sets of equations, and one can show the solutions of the two sets are related simply by k k. We → − also note that our Dirac equations have a discrete conjugation symmetry, being unchanged under the simultaneous substitutions
ω ω , k k , χ+ χ¯+ , χ− χ¯− , (4.51) → − → − ↔ ↔ − where χ± represents the -components of both χ2 and χ0. Thus we can restrict to k > 0 and α = 1 ± (dropping the α-label), using (4.51) to reconstruct k < 0 and obtaining α = 2 simply by changing the sign of k.
Even having restricted to half the spinor components, our system still involves eight coupled
first-order equations. Beginning in the deep IR, as per the usual holographic dictionary we want to impose appropriate boundary conditions to compute retarded Green’s functions. In the infrared
3 The metric function χ appearing in this factor should not be confused with the spinor. 107 limit, the coupling matrix S is off-diagonal in the charge basis χ2, χ¯2, χ0, χ¯0 , but becomes diagonal { } in the “mass basis”
χW = χ538 χ416, χX = χ467 χ426, χY = χ538 + χ416, χZ = χ467 + χ426 , (4.52) − − which diagonalizes the mass matrix M. As r 0 each of the mass basis spinors a = W, X, Y, Z → obeys a second order uncoupled equation of motion of the form
2 00 2 0 p LIR mIR(1 mIR) 0 = χ + χ + + − χa , (4.53) a r a r4 r2 where mIR is the dimensionless fermion mass at the IR fixed point: r 6 mIR mLIR = , (4.54) ≡ ± 7 and p is the 4-momentum combining ω and k in a way respecting the IR Lorentz invariance:
2 2 2 ω p k 2 . (4.55) ≡ − vIR
The character of the solutions depends strongly on whether p is timelike or spacelike. When p is spacelike, (4.53) admits solutions that are either regular or divergent in the IR. The regular solutions are of the form 1 p LIR χa(r) = Na K± 1 −m , (4.56) √r 2 IR r with Na a normalization constant. On the other hand, for timelike p, the solutions are oscillatory in the IR, being either infalling or outgoing.
In the far UV, the charge basis spinors decouple and solve massless second order equations of the form 2 χ00 + χ0 = 0 , (4.57) σ r σ where the index σ now stands for the -components of each charge basis spinor, and we are ± suppressing an index labeling the distinct elements of the charge basis. The leading constant
4 solutions for the χ− modes are associated to the expectation values of the dual operators, hOi 4 The rescaling described above in the beginning of this section removed a leading factor of r−3/2. 108 and those for the χ+ modes are associated to the sources J:
−1 −1 χ+(r) J(ω, k) + (r ) , χ−(r) (ω, k) + (r ) . (4.58) ∼ O ∼ hO i O
The choice of which of χ± is associated with the source and which with the expectation value is determined for ABJM theory by supersymmetry [63, 64, 17].
Were the fermions decoupled, we could solve the Dirac equation for just one of them with the others vanishing; imposing suitable boundary conditions in the IR would compute the relationship between that dual operator’s source and its expectation value. In our system this is not the case; a general solution to the system of equations leads to all four sources J and all four expectation values turning on. Considering the response of the four expectation values to varying the four hOi sources, we we obtain a matrix of Green’s functions, which schematically takes the form
j ij δ GR = hO i . (4.59) δJ i Jk=0
To properly define this matrix Green’s function, we follow a recipe very similar to the one advocated in [80, 87], searching for solutions to the equations of motion with suitable IR boundary conditions in which only one ABJM operator is sourced at a time. Given such a solution, standard application of the AdS/CFT dictionary for spinors allows us to read off the linear response of the operators to this source, and the associated entries in the matrix Green’s function.
In practice we proceed as follows. The IR normalization constants Na (4.56) can be chosen independently for each of the bulk spinors. This is guaranteed by the linearity of the equations of motion combined with the fact that the bulk fermions completely decouple in the IR. Imposing the proper boundary conditions in the IR, one can vary the Na and see how the sources J in the UV change. In this way we can construct a linear map T between the IR data N~ and the UV sources
J~ (J A,J B,J C ,J D): ≡ TN~ = J.~ (4.60)
The inverse of this map allows us to construct the IR data needed to produce a bulk solution with any desired values for the dual sources. Once such a solution is known, sources and expectation 109 values can be read off using (4.58) and plugged into (4.59) to obtain the Green’s function matrix.
From a practical standpoint, constructing the 4 4 matrix T is a straightforward but computa- × tionally tedious affair. One can completely determine the 16 complex entries by integrating the equations of motion four times, with four distinct (but arbitrary) N~ . After each integration the values of J~ are computed from the UV asymptotics of the solution, eventually yielding 16 equations for the unknown entries of T. This process must be repeated for each value of (ω,~k) of interest.
The appropriate boundary condition in the IR depends on whether the IR 4-momentum p
(4.55) is timelike or spacelike. If timelike, the choice of infalling boundary conditions leads to calculating retarded Green’s functions. This boundary condition is complex, leading to a non-
Hermitian matrix of Green’s functions. Solutions that vanish at the boundary are quasinormal modes and are associated with poles in the Green’s functions at complex ω and corresponding excitations with finite lifetime. This occurs “inside” the lightcone in the ω-k plane.
If p is spacelike, the “infalling” boundary condition used to compute retarded correlators can be analytically continued to the regular solution (4.56). This is a real boundary condition on the mode in the IR, which leads to a Hermitian matrix of Green’s functions. Solutions that vanish in the UV as well are normal modes, and are associated with poles in the Green’s functions at real ω, and excitations that are dispersionless.
Thus, the light-cone structure given by (4.55) divides the ω-k plane into a region inside the light-cone where modes decay, and a “stable wedge” outside the light-cone with dispersionless excitations. We will see this in the next subsection, where we will focus on the normal modes and associated dispersionless excitations. This can be contrasted with other types of geometries:
Reissner-Nordstr¨omand its cousins with regular horizons have the light cone fill up the entire k-ω plane, and hence have only unstable modes except potentially at ω = 0 itself, while the IR singular geometries of [55, 17] have a dispersionless region for ω ∆ for a constant ∆, independent of k. | | ≤ These different possibilities were illustrated in figure 1.1 of chapter1. 110
4.3.3 Fermion Normal Modes
The matrix T(ω,~k) contains all of the information necessary for identifying the locations of any fermion normal modes which may appear in the bulk. Any such normal mode can be defined as a solution to the equations of motion which decays in the far IR and whose “source” falloff in the UV vanishes—in other words, a regular solution to the bulk spinor equations at some (ωN ,~kN ) such that J~ = 0. Clearly any non-trivial solution with this property implies a zero eigenvalue of T, and thus one discovers that
det T(ωN ,~kN ) = 0 . (4.61)
This expression provides a powerful method for locating the fermion normal modes, and can be used to determine their location to very high accuracy.
The result of applying the diagnostic tool (4.61) to the fermionic perturbations of the Massive
Boson domain wall solution with the non-chiral mixing matrix 4.47 is shown in figure 4.5, and for the Massive Fermion solution in figure 4.6. We plot the results for k > 0, but due to (4.51) the spectrum is invariant under (k, ω) ( k, ω). As anticipated, the normal modes appear in bands → − − that are confined to the exterior of the IR lightcone, inside the “stable wedge”. Our numerical search reveals two bands for each domain wall solution within the kinematic regions shown5 . In both cases, one of the bands passes through ω = 0, indicating that the non-chirally coupled fermions include gapless fermionic degrees of freedom. Because these gapless fermionic modes appear at
finite momentum, their presence indicates that the dual fermions organize into a Fermi surface.
These ungapped bands in both cases begin at the upper boundary of the light cone, and asymptote along the lower edge of the light cone as far as our numerics can follow. Both cases also possess a gapped band, which appears to both begin and end along the lower light cone edge. In the Massive
Fermion case, the gapped and ungapped bands come close to each other along this edge, but the gapped band appears to terminate before they coincide.
5 While we do not completely exclude bands of normal modes at higher momentum than what is shown in the figures, a rough numerical search along the edges of the IR light cone revealed no further interesting features for k vUV /µ < 10. 111
Thus we see suggestions of both gapless and gapped excitations of ABJM collective fermionic degrees of freedom, which since they correspond to poles at real ω are perfectly non-dissipative, at least at large N. We have set the scales of both figures so that details can be seen, but it should be remembered that the wedge outside the lightcone where such stable fermionic excitations can exist is much smaller for the Massive Boson case than the Massive Fermion case (compare the light cones in figure 4.3). Inside the IR lightcone no additional normal modes exist, but only quasinormal modes at complex frequencies corresponding to excitations that decay; we will get a sense of such modes in the next subsection.
As discussed in section 4.2.2, there is a correlation between the strength of the symmetry breaking source and the size of the IR light cone. When the U(1) symmetry breaking is turned off entirely and only the chemical potential remains, the solution is AdSRN and the “IR light cone” effectively fills the ω-k plane, leaving no space for stable modes; indeed, as first demonstrated in
[10] and shown for top-down ABJM fermionic fluctuations in [47], this geometry supports fermionic zero-energy modes at finite momentum, but no stable (infinitely long-lived) excitations at finite frequency. As the symmetry breaking is turned on weakly in the Massive Boson case, the light cone closes slightly and a kinematic wedge appears where stable modes exist; the Massive Fermion case has symmetry breaking of the same order as the chemical potential and a much larger stable wedge.
It is tempting to conclude that in turning on the symmetry breaking source, some sector of the gauge theory mediating decays of the fermionic excitations has become gapped. That the gap is defined by the boundaries of the IR lightcone and not by the size of the symmetry breaking deformation alone can be understood as a consequence of the emergent IR conformal symmetry.
Far in the IR, the only relevant dimensionless scale is defined by the fluctuation, like ΛIR = ω/k vIR.
Dialing up the strength of the symmetry breaking source closes the IR lightcone further. While the symmetry breaking source is not the same operator in the Massive Boson and Massive Fermion geometries, being a scalar bilinear with monopole operators in one case and a fermionic bilinear with monopole operators in the other, we may speculate that in these geometries it is the size of 112
Figure 4.5: The band structure of fermion normal modes in the Massive Boson background. The shaded blue triangle is the stable wedge where it is possible for normal modes to appear, and the solid blue curves are the locations of fermion normal modes of the bulk theory, as determined by solving (4.61). The intersection of the dashed line with one band indicates the presence of a gapless mode. This band appears to terminate where it reaches the top boundary of the shaded region, but follows it closely along the bottom edge as far as our numerics allow us to compute.
Figure 4.6: The band structure of fermion normal modes in the Massive Fermion background, determined by solving (4.61). Again, there is a gapless mode at finite momentum. At higher momentum the gapped band approaches the ungapped band, but appears to meet the IR lightcone before the two bands coincide. As in the Massive Boson background, the ungapped band traces the bottom edge of the stable region as far as our numerics can reliably follow it. 113 the symmetry breaking rather than the details of its nature that most strongly influences the IR dynamics. This can be inferred from the fact that both sorts of deformations drive the UV theory to the same IR fixed point.
Geometries with “good” IR singularities were studied in [55], and in those cases infinitely long lived fermionic excitations also appeared, again sometimes connected to a zero-energy mode at
finite momentum. In that case the stable region was not a wedge, but a band defined by ω < ∆. | | Beyond the value ∆ (which is proportional to the chemical potential) the normal modes were found to move off the real ω axis and the fluctuations consequently acquired a finite width. It is likely something similar happens in the present case as well.
The normal mode analysis does not, in and of itself, provide any information about which
ABJM fermions are participating in these excitations. By virtue of our top-down holographic approach to this system, we can address this question, and at the same time better understand the fate of the normal modes beyond the boundary of the stable region.
4.3.4 Spectral Functions
The calculation of the normal modes (4.61) treats all four fermions symmetrically. However, the fermions do not all participate in each mode equally. A normal mode may be thought of as a solution for which all the sources vanish; however, the four expectation values may behave differently, as some may vanish in the normal mode and some may not. Equivalently, one may imagine approaching a normal mode in the ω-k plane while keeping a source fixed, and some expectation values will then diverge. Expectation values that are nonzero in a normal mode will thus be associated to poles in the matrix of Green’s functions. It is interesting to determine which fermionic operators participate in which collective normal modes.
A natural way to explore this is to study the spectral function matrix, proportional to the anti-Hermitian part of the retarded matrix Green’s function i(G(ω, k) G†(ω, k)). This spectral − function quantifies the fermionic degrees of freedom at a given frequency and momentum which overlap with the fermionic operators of the ABJM theory. Unlike the normal mode bands found in 114 the last subsection, the spectral function will be nonzero outside the stable wedge, and will provide a sense of the existence of unstable modes in this region. However, inside the stable wedge the spectral function itself is hard to examine. This is because since all excitations there are perfectly stable, the spectral function is zero except for delta function singularities; our numerical solutions cannot pick up these peaks, meaning the plots of these regions would be quite boring. To remedy this problem, we recall the Kramers-Kronig relations require the real parts of the Green’s function matrix to possess 1/ω–type poles when the imaginary parts have delta functions. Hence we choose to study the quantity G†G(ω, k), which will bring together both the real and imaginary parts of the
Green’s functions. We can then plot Tr G†G, which will be a basis independent quantity capturing both excitations in the stable wedge outside the IR lightcone and finite-width excitations inside the
IR lightcone.
Finally, we can define matrices that project onto the subspaces of definite charge:
P+ = diag 1, 0, 0, 0 P− = diag 0, 1, 0, 0 P0 = diag 0, 0, 1, 1 (4.62) { } { } { }
† (c.f. equation (4.38)). Then, Tr P+G G will measure the excitations of χ2 alone, etc.
In figures 4.7 and 4.8 various projections of Tr G†G for the non-chirally mixed fermions are plotted for the Massive Boson and Massive Fermion backgrounds, respectively. Our first observation is that the spectral density inside the stable region is strong along the curves of the normal modes, as we would expect. This density continues outside the stable wedge, pointing to the presence of nearby unstable modes. Most of the bands of density along the normal mode curves are strong; the exception is the region of the gapped band in the Massive Fermion background, which is weak enough to show up in our plot as a series of distinct points. We continue to plot k > 0, but the spectrum is again invariant under (ω, k) ( ω, k); thus the band in the lower-left corner of the → − − Massive Fermion plots is the continuation of the band that exits the plot in the upper left.
By projecting onto the charged and neutral subspaces, we can identify which fermions par- ticipate in which bands of spectral density and associated normal modes. For both backgrounds, it isχ ¯2 andχ ¯0 that dominate both the gapped and gapless bands of normal modes for k > 0. χ2 115
Figure 4.7: Plots of G†G for the Massive Boson background. Within the wedge marked by red edges, all excitations are stable. 116
Figure 4.8: Plots of G†G for the Massive Fermion background. Within the wedge marked by red edges, all excitations are stable. 117
(and χ0) participates only very slightly along these same curves. We stress that this asymmetry is in part arbitrary: the conjugation symmetry (4.51) tells us that χ2 and χ0 will have similar strong excitations for (ω, k) ( ω, k), or in the α = 2 components with ω ω. We note that for → − − → − the Massive Boson background the gapped band continues in the P− projection out of the stable wedge and into the light cone. The symmetry described above also involves charged conjugation in the charged subspaces, and this curve can be seen to finish in a small tail just above the vertex of the light cone in the P+ projection; this feature will be shown to be a remnant of a stronger band when we consider modifying the couplings in the next subsection.
It is interesting to quantify the relative participation of different fermionic modes at a par- ticular point on these bands; we choose to look at the Fermi surface point along the corresponding curve, at zero frequency (relative to the chemical potential) but finite momentum k = kF . One can turn on a unit source at ω = 0 and k = kF for each of the four supergravity fermions and catalog the response of the system to this source in the rows and columns of the retarded Green’s function. Diagonalizing the Green’s function at ω = 0 and k = kF explicitly reveals an eigenmode
with diverging eigenvalue. Denoting this eigenmode ξkF , one can then write
X I ξkF = cI χ (4.63) I where the χI are the bulk supergravity fermions dual to the ABJM operators we study. The amplitudes cI thus quantify the amount in which various ABJM fermions are involved in the fermionic zero-energy mode. This decomposition is shown in figure 4.9 for the Massive Boson and
Massive Fermion backgrounds, quantifying how the normal mode is primarily composed ofχ ¯2 and
χ¯0; this is readily understandable as being a result of the direct mixing of these modes due to the symmetry-breaking “Cooper pair” coupling of the form φχ0χ2. We see also that χ0 and χ2 barely
2 participate at all. This suggests that the Majorana self-coupling term φ χ¯0χ0 does not have much | | effect on the collective mode at the Fermi surface. 118 O Χ2 1.0 1.0 1.0 0.8 0.8 0.8 O Χ2 Χ 0.6 0.6 0.6 O 2 O Χ0 0.4 0.4 O Χ0 0.4 0.2 0.2 0.2 O Χ2 O Χ0 O Χ0 O Χ2 O Χ0 O Χ2 O Χ0 0.0 0.0 0.0
2 Figure 4.9: The squared amplitude cI of each ABJM operator participating in the “Fermi surface” | | zero-energy mode in the AdS-RN background with no scalar (left), and the Massive Boson (middle) and Massive Fermion (right) backgrounds. The normalized Fermi momentum kF vUV/µ in the three cases are 0.53 (AdS-RN), 0.58 (MB), and 0.48 (MF). Note that in all cases the contributions from χ and χ are insignificant. O 2 O 0
4.3.5 Modifying Couplings
It is interesting to ask how these results change as we modify the background or the couplings.
This can give us an idea of which couplings are “responsible” for the effects we see. For example,
figure 4.9 suggests that the charged-neutral coupling is much more important than the neutral- neutral coupling, because theχ ¯2-¯χ0 mixing is strong and the χ0-¯χ0 mixing is weak, and we will see that indeed this is true. It should be kept in mind that all these modified couplings, including the non-chiral mixing matrix 4.47 we have primarily studied, take us outside the top-down approach, since we do not know any explicit embedding in M-theory for these fermion equations. We will comment on three modifications of our system, as follows:
(1) We will keep the Lagrangian the same (i.e. still use = 8 supergravity) but turn off the N scalar field λ while keeping the chemical potential. The result is the AdSRN black brane
corresponding to ABJM theory at zero temperature, deformed only by chemical potentials.
In the terminology of chapter3, this is the four-charge black hole. 6
(2) We will consider massless charged Dirac fermions with no couplings to the scalar fields in
the Massive Boson domain wall background. This is analogous to the approach of [15].
(3) Also in the Massive Boson domain wall background, we will modify the equations of mo-
6 Note that due to a triality rotation carried out in chapter3, the sum of all four gauge fields there corresponds to our single gauge field here. 119
Figure 4.10: Plots of the modulus squared of the Green’s function for massless probe fermions of various charges.
tion 4.46-4.47 for = 8 fermions in only one regard, namely by omitting the off-diagonal N “Cooper pair” and “Majorana” couplings. These couplings are similar to the ones consid-
ered in [16].
Results for the Massive Fermion background are similar. All these interactions can be regarded as steps on the road from the simplest fermion couplings in the simplest finite-density backgrounds to the full chirally mixed interactions.
The existence of the “stable wedge” is a property of domain wall backgrounds with IR AdS regions. If we pass to the AdSRN background, then the stable wedge is closed, and all excitations away from ω = 0 are dissipative. At ω = 0, there is a Fermi surface for the charged fermion [47], and the neutral fermion is at a special transition point between a pole in its Green’s function and a zero as other chemical potentials of the system are varied [17]. We include the observation that
χ¯2 is entirely responsible for this Fermi surface singularity in the AdSRN background in figure 4.9.
Turning the scalar back on leads to the backgrounds studied in this chapter and opens up a stable wedge. We expect there to be stable modes in this wedge for generic charged fermions.
Indeed, in figure 11, we see similar lines of poles for charged, massless fermions in our Massive
Boson background with elementary Dirac equations; only the neutral case does not acquire a band of stable excitations. Thus we conclude the existence of stable fermionic modes is a generic property of the background once the IR AdS region exists and the stable wedge appears. 120
The Dirac equations associated to the non-chiral mixing matrix are substantially more com- plicated than these, involving additionally χ2χ0 and χ0χ¯0 couplings, as well as Pauli terms and a running of the gauge couplings with the scalar. Faulkner et al. [16] also discussed how the turning on of a Yukawa coupling caused bands of excitations that crossed to repel each other, leading to a gap in the dynamics. In that case, there was only a single charged fermion, and the coupling had a Majorana character coupling the fermion to its own conjugate as in (4.49). In general if the particle has a pole at momentum k, the antiparticle will have this pole at k. However, one can − see that the Γ5 factor mixes the α = 1 and α = 2 components, which introduces an additional flip of the sign of k; for this reason the authors of [16] preferred the Γ5 interaction, which couples two modes with poles at the same momentum and leads to eigenvalue repulsion generating a gap. One may ask whether a similar principle of repulsion between bands brought on by a mutual coupling applies in our case.
In figure 4.11, we plot the normal mode structure for our fermions in the Massive Boson background, with the off-diagonal couplings in the mixing matrix (4.47) removed but the diagonal terms preserved, and compare it to the full top-down results previously shown in figure 4.5. In the left plot, describing the decoupled case, there are three bands: the gapless, yellow band stretching from top to bottom is associated toχ ¯2, while the red band crossing this coming from the lower edge of the stable wedge to the left is the neutral fermionχ ¯0. Meanwhile there is a third band in orange, associated to the oppositely charged χ2, crossing theχ ¯2 band below the upper boundary of the wedge and theχ ¯0 band close to the origin in ω-k space. This orange band is also gapless, displaying a zero-energy mode around k vUV/µ = 0.09. We note the resemblance between the χ2 andχ ¯2 bands shown there, and the free fermion q = 2 and q = 2 cases shown in figure 4.10. − By comparing the band structure with and without off-diagonal couplings we can get an idea of how the couplings modify the bands. The lower-right crossing of bands results in both mixing and repulsion, as the crossedχ ¯2 andχ ¯0 bands transform into uncrossed bands involving a mixing of both fermions. This repulsion, however, does not create a gap; unlike the simpler case in [16] there is no reason for the repulsed crossing to exist at ω = 0, since it involves the coupling of two 121
0.02 0.02
0.01 0.01 Ω Ω Μ 0.00 Μ 0.00
-0.01 -0.01
-0.02 -0.02
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k k vUV vUV Μ Μ
Figure 4.11: Normal mode structure in the Massive Boson background with the off-diagonal cou- plings turned off (left), compared to the full top-down result of figure 4.5 (right). With the off- diagonal couplings turned off there are three bands, associated to χ2 (orange),χ ¯2 (yellow), andχ ¯0 (red), intersecting in three places. Turning on the couplings between the different fermions induces repulsion between the bands, as described in the text. 122 distinct fermions instead of a fermion to itself. Meanwhile, the two crossings involving the orange
χ2 band lead to repulsion without mixing; the band of normal modes associated to χ2 is pushed off beyond the stable wedge, ending up as the small tail visible just above the light cone vertex in the
P+ projection of figure 4.7, while theχ ¯2 band only acquires a tiny χ2 component. As the χ2 band is pushed off in this way, its associated zero-energy mode disappears, thus gapping χ2 (for k > 0).
In this case, as in [16], it happens that the coupling has created a gap.
The tiny amount of mixing between the χ2, χ0 sector and theχ ¯2,χ ¯0 sector suggests that the Majorana χ0-¯χ0 coupling is relatively unimportant, and on the whole this proves to be the case; turning it off alone removes the small contribution of the P+ sector to the normal modes, but does not change any of the overall structure. The χ2-χ0 “Cooper pair” coupling is the dominant interaction.
4.4 Discussion
Perhaps the most powerful aspect of our approach is the explicit holographic map between the supergravity modes in our gravitational solutions and various operators in the ABJM theory.
This “top-down” application of gauge/gravity duality opens the door to various interpretations of our results in the context of zero-temperature states of a field theory whose operator content is well understood. The natural next step is to examine the full top-down chiral mixing matrix, including the Γ5 terms. The non-chiral mixing has already gapped out some of the lines of normal modes; a natural question is whether the full chiral mixing will gap out the fermionic fluctuations completely.
This question is answered in the affirmative in the next chapter.
In our setup, we have deformed the ABJM theory in two ways which our analysis makes precise. The first is by the addition of a chemical potential for the U(1)b current, placing charged
ABJM matter at finite density and sourcing a relevant deformation away from the UV fixed point.
A In the standard presentation of ABJM theory, the scalars Y and fermions ψA are neutral under
qτ U(1)b, which is carried only by the monopole operators e . Accordingly, the composite monopole– fermion/scalar operators of (2.45) filling out the 8v,c carry U(1)b charge, and the natural interpre- 123 tation of the states we study is as zero temperature phases of composite matter at finite density.
In fact zero-temperature phases of such composite matter have arisen in other finite-density investigations of 2+1 dimensional field theories, beyond holography in the large-N limit. This is perhaps most famously apparent in the context of the fractional quantum-Hall effect [88], but related phases have also appeared more recently in e.g. [89, 90, 91]. Particle-vortex duality in three dimensions exchanges objects charged under an “ordinary” symmetry with those charged under a current associated to the dualized gauge field of the form (4.24), conserved by virtue of the Bianchi identity. Many theories which permit such a duality are more amenable to calculation in terms of the “magnetic” variables which generate Jb, and thus these variables can often provide a relatively simple description of complicated phases of strongly coupled matter.
Our calculation of the spectral functions for composite fermions can help better understand the nature of these putative phases. One of our main results is the appearance of delta function singularities in the spectral functions within a particular kinematic window controlled by the prop- erties of the IR fixed point. These finite momentum singularities signal the presence of stable excitations which overlap with the fermionic operators written in (4.42-4.45). One plausible expla- nation of these spectral features is that the finite density of composite fermions (those transforming in the 8c) have arranged themselves into a Fermi surface at ω = 0 and k = kF , and the IR excita- tions around this Fermi surface are weakly interacting and thus long-lived. In this picture, the low energy features of these states are qualitatively similar to a Fermi liquid of composite fermions.7
The other deformation we have dealt to the ABJM theory is the addition of a source which explicitly breaks the global U(1)b. In the states that we focus on, this breaking results in a non- vanishing expectation value for composite boson or fermion bilinears. In a sense developed in some detail in section 4.3.3, it is the breaking of the monopole number density that permits stable excitations in the vicinity of the Fermi surface. It is interesting that the fermionic response indicates that the system remains gapless even though the U(1)b has broken.
7 Such a picture differs from the “gaugino” Fermi surfaces discussed in [47], as the Fermi surfaces in this case would be constructed from gauge invariant composite fields. Reconciling this interpretation with the N 3/2 scaling of the correlator and Luttinger’s theorem remains an interesting and unresolved issue. 124
We are now in the position to ask how our results compare to other zero-temperature states of composite matter. One particularly interesting example is = 4 supersymmetric QED, which N is acted on by mirror symmetry and hence like ABJM theory permits a description in terms of magnetic (composite) variables. In [90] it was shown that the IR physics of this theory with a uniform density of magnetic impurities is described by phases in which an “emergent Fermi surface” consisting of composite fermions organizes into a Fermi liquid. Moreover, it was found that this theory permits a phase in which composite bosons acquire an expectation value, yet the
Fermi surface persists. Understanding to what extent our ABJM states match the expectations for these novel phases would be interesting.
In [92], it was argued that the pattern of holographic Fermi surfaces in symmetry-preserving backgrounds of = 4 super-Yang-Mills theory and of ABJM theory can be predicted based on the N form of the dual field theory operators. In particular, the field theory scalars involved in the dual operators may have expectation values, and if they do, then the “boson rule” of [92] predicts the existence of a Fermi surface. A slightly subtle point is that the scalar expectation values do not break symmetries in the large N limit; instead, the eigenvalues of the scalar operators are distributed over the transverse directions in a manner that respects the unbroken R-symmetries. The reasoning behind the boson rule is that a scalar expectation value allows an insertion of the operator dual to a supergravity fermion, generically of the form Y ψ, to deposit all of its momentum into the fermionic component ψ, while the scalar Y is absorbed by the non- symmetry-breaking condensate.
The results of [92] are clearest in cases where at least one of the independent chemical potentials is absent in the black hole background. That is because unequal chemical potentials demand non- zero profiles for supergravity scalars whose field theory duals are expectation values of operators composed entirely from the field theory scalars whose non-symmetry-breaking condensates drive the reasoning behind the boson rule. It is unobvious how to extend the reasoning to the present case, where all four chemical potentials are equal, because then the non-symmetry-breaking supergravity scalars are altogether absent. It would be useful to examine supergravity constructions in which one, two, or three of the chemical potentials are turned off, in order to try to ascertain whether some 125 version of the boson rule can be applied even in the presence of a symmetry-breaking scalar like
λ. In the present case, it is interesting and suggestive to note from figure 4.9 and equations (4.42)-
(4.45) that the field theory operators which contribute to the fermion zero-energy mode at positive
† A k are the ones whose bosonic components involve YA not Y . For the Massive Boson domain wall, it would be in the spirit of the boson rule to speculate that this is because the deforming operator
A † S involves Y but not Y . For the Massive Fermion domain wall, where the deforming operator O A †A P involves only ψA but not ψ , it is not clear how an argument in the style of the boson rule O should go. We hope to report further on field theory interpretation in future work. Chapter 5
Gapped Fermions in Top-down Holographic Superconductors
This chapter is an edited version of [19], written in collaboration with Oliver DeWolfe, Steven
S. Gubser and Christopher Rosen.
5.1 Overview
In this chapter, we continue the line of inquiry centered on top-down fermionic response in strongly coupled phases of matter. The primary objects of interest will again be Green’s functions of fermionic operators in the ABJM field theory. From such correlation functions, one can construct fermionic spectral functions which in turn provide useful data such as the existence, dispersion, and location of fermionic excitations in the phases of interest. As in chapter4, we will be interested in phases which break the global U(1) explicitly or spontaneously. Geometries where the U(1) is broken spontaneously are referred to as holographic superconductors, and were studied first in bottom-up constructions at nonzero temperature [11, 71, 72, 73] and at zero temperature [74, 75].
We will be interested in three particular top-down zero-temperature constructions of this type, all of which appear as flows from the supergravity’s maximally symmetric AdS4 vacuum in the
UV to a distinct AdS4 region in the IR, characterized by a nontrivial extremum of the scalar potential. The flows can be embedded into truncations of the = 8 gauged supergravity that N only keep fields invariant under a subgroup H of the full SO(8) symmetry. The first of these
flows was originally found as the solution to a compactification of 11D supergravity on a generic
Sasaki-Einstein manifold [77, 78, 79], and was later embedded in the H = SU(4)− truncation of 127 four-dimensional gauged supergravity [93]. The other two flows, which were also discussed in some detail in chapter4, were constructed in an H = SO(3) SO(3) truncation [81]. The SU(4)− × case involves only spontaneous breaking of U(1) and hence is a true superconductor, while in the SO(3) SO(3) case the U(1) is explicitly broken. These domain wall geometries represent × holographic candidates for finite density, zero-temperature ground states of the dual ABJM theory with a broken U(1) symmetry.
In the standard BCS theory of superconductivity, the Fermi surface in the normal state of a superconductor is unstable to the formation of Cooper pairs of fermions below the critical temper- ature. When these Cooper pairs condense in the superconducting phase, an effective interaction arises which mixes particle and hole excitations, simultaneously destroying the Fermi surface and gapping the fermionic excitation spectrum. Both the gap and the dispersion relation governing the fermionic excitations of the superconducting phase are visible in Angle Resolved PhotoEmission
Spectroscopy (ARPES) experiments.
It is natural to wonder whether or not the fermionic excitation spectrum is similarly gapped in superconducting phases of holographic matter. Bottom-up “probe” fermions (bulk fermions with constant charge and mass chosen arbitrarily) were studied in the top-down SU(4)− background in [15], where it was noticed that due to the restored Lorentz invariance of the IR fixed point, the dispersion relations of fermionic excitations have a “light-cone” structure with unstable modes inside the light-cone and stable modes outside. Moreover, in contrast to the BCS expectation, the fermionic excitations were not necessarily gapped in the superconducting phase, and it was shown that the greater the fermion’s charge, the more bands of gapless, stable excitations exist.
Faulkner et al. [16] studied bottom-up fermions in bottom-up holographic superconductors, introducing a “Majorana” coupling of a charged fermion ψ to itself along with a scalar φ of twice the charge,
−1 † T e = φ ψ C(η + η5Γ5)ψ + h.c., (5.1) L which has a structure reminiscent of the coupling of a Cooper pair ψψ to the condensate φ in an 128 effective BCS Lagrangian. It was shown in [16] that such an interaction would generically introduce a gap to the band of fermionic excitations as long as the chirality matrix Γ5 was present. The effect of this chiral coupling is to mix “particle” and “hole” states, as we review in section 5.3. In chapter
4 the structure of the top-down fermionic couplings in the SO(3) SO(3) solutions was described, × and a simplified non-chiral fermionic mixing matrix was studied. In this background, similar “Majo- rana” interactions occur, although with a charged fermion coupled to a neutral fermion (suggestive of a charged-neutral “Cooper pair”) and it was shown that the non-chiral mixing introduces gaps into some, but not all, bands that were present for probe fermions.
In this chapter, we present the full top-down fermionic interactions for spin-1/2 fields that do not mix with the gravitino, for both the SU(4)− and SO(3) SO(3) backgrounds. Calculat- × ing fermionic spectral functions, we find that both holographic phases have fully gapped fermionic degrees of freedom, though for different reasons. In the SU(4)− background, the fermion charge is small and probe fermions of this charge already have no bands of stable modes; the full top-down interactions do not change this fact. Moreover, in this background the fermion cannot form the
Majorana couplings analogous to Cooper pairing, as both the SU(4)− group theory and the large charge of the scalar condensate forbid it. In the SO(3) SO(3) background, on the other hand, × we know from chapter4 that the charge is large enough for probe fermions to be ungapped and if a gap appears it will be due to interactions. We indeed find that the complete top-down couplings between charged and neutral fermions, which include a chirality matrix as advocated in [16], fully gap the fermionic modes.
The structure of this chapter is as follows. In section 5.2, we review the SU(4)− solution, determine the top-down Dirac equations, and calculate the holographic Green’s functions and associated spectral functions. Section 5.3 then does the same for the SO(3) SO(3) solutions. We × discuss lessons for strong coupled field theories in section 5.4. 129
5.2 The SU(4)− Flow
In this section we study fermionic response in a zero-temperature geometry solving the equa- tions of an Einstein-Maxwell-scalar theory first obtained in compactifications of 11D SUGRA on
Sasaki-Einstein manifolds [77, 78, 79], and later embedded in the H = SU(4)− truncation of 4D
= 8 gauged SUGRA in [93]; the fermions we study are associated to the latter embedding of the N bulk theory.
5.2.1 The SU(4)− Truncation
The SU(4)−-invariant sector of maximal gauged supergravity in four dimensions [93] is de-
fined as the fields invariant under the SU(4) SO(8) which leaves invariant the four form , ⊂
− − − = + i dz1 dz2 dz3 d¯z4 , (5.2) W23 W2 W3 ≡ ∧ ∧ ∧
4 where the zi are coordinates on C . The sector contains a neutral pseudoscalar, which we can consistently set to zero, and a charged pseudoscalar which is embedded in the coset representative
φ of (2.27) as
i − i − φ = Im ω3 = ω . (5.3) 2 W23 2 W3
iα In the final equality the complex scalar ω3 ω e has been taken to be real. ≡ The 56-bein (2.6) is obtained from the exponential of the generators, as per (2.27). To carry out the matrix exponentiation, it is useful to construct the projector
1 − − Π = where (A B)IJKL AIJMN BMNKL . (5.4) 16W3 ·W3 · ≡
This projector is Hermitian, and squares to itself. Moreover, it satisfies the following useful identi- ties:
φ φ∗ = 4ω2 Π and φ∗ Π = φ∗ . (5.5) · ·
Through explicit computation, one then finds
IJ IJ klIJ i − u = δ + (cosh 2ω 1) ΠijIJ , v = sinh 2ω ( )klIJ . (5.6) ij ij − −4 W3 130
− The single gauge field in the truncation commutes with SU(4) inside SO(8). In terms of the zi,
4 − − one can define a Kahler structure on C with Kahler form J invariant under the SU(4) U(1) × as:
− i J = dz1 d¯z1 + dz2 d¯z2 + dz3 d¯z3 dz4 d¯z4 , (5.7) 2 ∧ ∧ ∧ − ∧ and the U(1) gauge field is then embedded in the AIJ of the SO(8) theory as1
1 A = J −. (5.8) √2A
Inserting these ansatze, and defining ξ (2/√3) tanh 2ω to make contact with the conventions of ≡ [79], we arrive at the Lagrangian governing the SU(4)− invariant sector of the = 8 theory: N 3 ξ 2 24 2e−1 = R 2 |D | ( 1 + ξ2) , (5.9) L − F − 2 (1 3 ξ2)2 − (1 3 ξ2)2 − − 4 − 4 2 where = d . In this section we employ conventions such that g = 2 and GN = 1/(8π). The F A covariant derivative is thus given by Dµξ = ∂µξ 4i µξ, and the scalar has charge 4. − A To see this from the group theory point of view, under the SO(8) SU(4)− U(1) decom- → × position, the gauge fields transform as
28 150 62 6−2 10 , (5.10) → ⊕ ⊕ ⊕ where the 10 is our , and the parity-even and parity-odd pseudoscalars decompose as A
0 35v 150 102 10−2 , 35c 20 0 62 6−2 14 1−4 10 , (5.11) → ⊕ ⊕ → ⊕ ⊕ ⊕ ⊕ ⊕ so our charged scalar ξ (or ω3) is the 14 and its conjugate.
5.2.2 SU(4)−-invariant Domain Wall Solutions
The zero-temperature solution we are interested in corresponds to a flow driven by a relevant deformation from the maximally supersymmetric AdS4 geometry in the UV to the so-called Pope-
Warner AdS4 solution [94] in the IR [77, 78, 79]. This deformation does not involve adding a scalar
1 This A should not be confused with the scalar kinetic tensor defined in (2.14). 131 operator to the dual Lagrangian; the relevant deformation is a spatially uniform chemical potential, and the response of the scalar operator is only to acquire an expectation value, so the geometry is a true holographic superconductor with U(1) broken only spontaneously.
The chemical potential breaks Lorentz invariance as well as conformal invariance, but when it leads to a domain wall solution between two AdS4 vacua, full relativistic conformal invariance is recovered in the infrared as an emergent symmetry. A striking feature is that the speed of light vIR in the infrared is smaller than the speed of light vUV in the ultraviolet—meaning simply that gtt/gxx has different IR and UV limits. Physically, we can think of the ratio vUV /vIR as an index of refraction for the holographic state of matter that we are describing. By rescaling ~x, we can change vUV and vIR by the same factor, but the index of refraction remains invariant. An interesting conjecture [78] states (approximately) that the type of deformation we study, based on a chemical potential and flowing to an infrared conformal fixed point, always exists in holographic theories provided there is an associated renormalization group flow preserving Lorentz invariance throughout with the same UV and IR conformal fixed points.
The SU(4)− holographic superconductor geometry is encapsulated by the ansatz
dr2 ds2 = G(r)e−β(r)dt2 + + r2d~x2, = φ(r) dt, and ξ = ξ(r). (5.12) − G(r) A
2 The maximally supersymmetric AdS4 vacuum has G = 4r and β = φ = ξ = 0, while the PW p AdS4 solution has ξ = 2/3, corresponding to another extremum of the potential 5.9, as well as
G = 16r2/3 and β = φ = 0.
To construct the flow between the AdS4 solutions, it is helpful to characterize the spectrum of irrelevant perturbations of the PW solution, as these can be used to integrate away from the solution towards the maximally symmetric solution in the UV. Linearizing the equations of motion about the PW solution, one finds that there is a scalar mode and a vector mode both with mass m2 = 6 which satisfy the flow criteria. They are holographically dual to scalar and vector operators of the IR conformal field theory with dimension ∆ = (3 + √33)/2 and ∆ = 4 respectively. The 132 linearized analysis fixes the irrelevant perturbations to be of the form
r √ 16 2 2 2 1 (−3+ 33) G = r + . . . , β = 4 + . . . , φ = r + . . . , ξ = + r 2 + ... (5.13) 3 3 J
Scaling symmetries of the equations of motion have been used to fix the amplitudes of the β and
φ perturbations arbitrarily, leaving only a single parameter to be tuned such that the desired J behavior is obtained in the UV.
In the UV, the scalar ξ provides a ∆ = 2 perturbation of the maximally symmetric AdS4.
Since we are interested in the case when the UV fixed point is not deformed by a source for the dual
2 scalar operator, the UV behavior of the scalar is required to be of the form ξ(r ) ξ2/r + ... → ∞ ∼ representing a spontaneously acquired vacuum expectation value for the dual scalar operator.
The desired solution is readily constructed from a numerical shooting technique, tracing the
RG flow upstream to the UV. It appears in figure 5.1.
5.2.3 The Fermionic Sector
The supersymmetries of the gauged supergravity transform in the 8s, which decomposes as
8s 4−1 4¯1 , (5.14) → ⊕ under the SU(4)− U(1). Accordingly no supersymmetries survive the truncation. In fact, since × the spin-1/2 fermions decompose as
56s 20−1 201 43 4¯−3 4−1 4¯1 , (5.15) → ⊕ ⊕ ⊕ ⊕ ⊕ there are no singlets in the fermionic sector of the truncation at all—the SU(4)− invariant theory is entirely bosonic. From our perspective this is not a problem, as we are happy to study any = 8 N gauged supergravity spin-1/2 fields, regardless of whether they are in the SU(4)− truncation. We only wish to avoid gravitino mixing, which we can do as long as we avoid the 4 representations, and study instead the 20−1 and its conjugate. 133
6 5
5 4 4 G 3 2 3 β r 2 2
1 1
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 r r
0.25 1.0
0.20 0.8
0.15 0.6 ϕ ξ 0.10 0.4
0.05 0.2
0.00 0.0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 r r
Figure 5.1: The AdS4 to PW flow. The flow is characterized by an index of refraction n 3.78 2 ≈ and a scalar vev proportional to ξ2/φ 0.33. UV ≈ 134
Churning through the various supergravity tensors in 2.40, one arrives at a Lagrangian for the spin-1/2 modes of the form
−1 e χχ¯ =χ ¯ / + B/ + P/ + M χ , (5.16) L ∇ where the χ are considered as 56 component vectors whose entries are the non-vanishing χijk, is ∇ the derivative covariant on the background geometry, and B, P, M are 56 by 56 dimensional matrices describing the gauge, Pauli, and mass couplings, respectively. We are using a schematic “slashed” notation to indicate the appropriate Lorentz invariant contraction with the Γµ.
We can now isolate the Dirac equations for the fields in the 20. Due to SU(4)-invariance, each member of the 20 cannot mix with anything but itself. This forbids mixing with its own conjugate (which is in the inequivalent 20 representation), ruling out “Majorana” couplings of the type shown in 5.1. It is helpful to note that B/ and P/ commute in this case, and thus the kinetic, gauge, and Pauli terms can be simultaneously diagonalized. In this basis, the decomposition is manifest.
In terms of the χijk, a representative of the 20−1 can be chosen to be the combination
ψ = χ368 + χ467 + i(χ358 + χ457) , (5.17) and this or any other fermion in the 20 can then be seen to satisfy the Dirac equation
4 + 3ξ2 i 3ξ2 i / / + / ψ = 0. (5.18) ∇ − 4 3ξ2 A 4F − 4 3ξ2 − − At the UV fixed point, the scalar ξ vanishes and ψ is massless, and thus from the perspective of the ABJM theory, ψ is dual to operators carrying charge qψ = 1 and having conformal dimension | |
∆ = 3/2. Comparing to the complex scalar ξ of (5.9) with charge qξ = 4, one finds that the | | scalar carries four times the U(1) charge of the decoupled fermions. This is another reason why
“Majorana” couplings of the type 5.1 are forbidden in this case. p Along the flow ξ runs from 0 to the IR value ξ = 2/3. Thus in the IR theory governed by the PW solution, the supergravity mode ψ behaves as though it carries mass mIR = 1. 135
5.2.4 Fermion Response
We now wish to solve the equation (5.18) in the background of Figure 5.1. We use the basis
(3.28) for the generators Γa, and to label the four complex components of our spinors we use the projectors (3.30), again letting us write the four components of a bulk spinor ψ as
ψα± Π˜ αP±ψ . (5.19) ≡
From a 2+1 dimensional point of view, ψ+ and ψ− each transform as Dirac spinors, and α labels the two complex components of these spinors. As discussed in chapter3, supersymmetry fixes ψ+ to be the spinor that asymptotes to a source for the dual fermionic operator.
It is computationally convenient to “square” (5.18) to arrive at second order linear differential equations governing the components of ψ+. We also redefine the spinor as
4 −β − 1 −i(ωt−kx) ψ(t, r, x) (G r e ) 4 ψ(r) e , (5.20) → where we have exploited the background isometries to set the momentum in the x-direction, and the metric factor has been chosen to cancel the spin-connection part of the covariant derivative.
In practice, the basis we adopt allows one to focus on either the α = 1 or α = 2 components independently. Rotational invariance of the background then ensures that ψ1(k) = ψ2( k). − Asymptotically, in the UV the source components behave like
ψ+(r) J(ω, k) + (1/r) , (5.21) ∼ O where J(ω, k) is interpreted holographically as a source for the dual fermionic operator. In the IR, these components obey an equation of the form
2 2 00 2 0 m˜ (1 +m ˜ ) LIRp ψ + ψ + ψ+ = 0 , (5.22) + r + − r2 r4 wherem ˜ = mIRLIR = LIR is the dimensionless mass of the fermion in the IR and LIR of the AdS radius in the PW solution. We have also introduced
2 2 ω 2 p 2 + k , (5.23) ≡ −vIR 136 the Lorentz invariant momentum squared of the mode in the PW background, where vIR is the speed of light in the PW solution.
The features of the solution to (5.22) depend strongly on the sign of p2. The case of spacelike momentum p2 > 0 is particularly interesting. This is because for a system consisting of a finite density of fermions, one might expect to find significant spectral weight at zero energy (as measured from the chemical potential) but non-vanishing momentum. In that situation, (5.22) is solved by a component of the form
1 p LIR ψ+ = K− 1 −m˜ , (5.24) √r 2 r with K the modified Bessel function of the second kind. The IR Green’s function R(ω, k)αβ can be G a useful diagnostic to quantify the fermion response. To construct it, note that the Dirac equation
(5.18) implies that
2 r 1 m˜ 0 ψ− = vIR ψ+ ψ+ LIR k vIR + ω r − p vIR 1 p LIR = K− 1 +m ˜ , (5.25) − √r k vIR + ω 2 r where ψ− is the component of the bulk spinor whose normalizable fall-off encodes the field theory response. Applying the holographic prescription for the dual retarded correlator thus gives
1 2m ˜ 1 Γ( 2 m˜ ) (p LIR) R(ω, k) α,β=1 = m˜ 1 − p vIR (5.26) G −4 Γ( 2 +m ˜ ) k vIR + ω where α, β are spinor indices. To construct a rotationally invariant correlator one can trace over the spinor indices to obtain
R tr R αβ = R(ω, k) 11 + R(ω, k) 11 G ≡ G G G − 1 2m ˜ 1 Γ( 2 m˜ ) (p LIR) = 2m ˜ −1 1 − ω. (5.27) −2 Γ( 2 +m ˜ ) p vIR
In performing the trace, we have exploited the fact that in this system R(ω, k)22 = R(ω, k)11 G G − as a consequence of the dual state’s isotropy.
The domain wall background of figure 5.1 departs fairly quickly from the PW solution which characterizes the IR, and thus one expects that IR Green’s functions such as (5.27) characterize the field theory dynamics only for those bulk fermion solutions which are localized very near r = 0. 137
The solution (5.24) is regular as r 0, and purely real. Its form suggests the interesting → possibility of constructing fermion normal modes in the domain wall solution which behave like
(5.24) in the IR and asymptote to (5.21) in the UV with J = 0 for some choice of (ω, k). Indeed, such fermion normal modes were observed in various bottom-up holographic models, such as [15, 16].
We now attempt to construct these as linearized perturbations of the SU(4)− invariant flow.
Solving the bulk Dirac equation (using numerical shooting from the IR to the UV) and scanning over spacelike momenta reveals a null result: we find no fermion normal modes for the fermions in the 20 or 20. In particular, there is no mode at ω = 0, and thus the fermionic spectral function is gapped in this state of the ABJM theory.
To quantify and better visualize the fermion response one can look to the spectral function of the dual field theory operators, which we define to be
i † A(ω, k) = tr GR G . (5.28) 2 − R
Here GR is the 2 2 matrix of retarded Green’s functions for the two-component fermionic operators. × To extend the domain of the spectral function to timelike momenta, one must modify the IR boundary condition (5.24) to provide the proper notion of “ingoing” necessary to reproduce the causal structure of the retarded correlator. The correct prescription is given in [61], and turns out to be 2 (1) √−p LIR √1 r H− 1 −m˜ r ω > vIR k ψ+ = 2 | | (5.29) (2) −p2 L √1 √ IR r H− 1 −m˜ r ω < vIR k 2 | | with H the Hankel function of the first or second kind as indicated.
The spectral function is shown in figure 5.2. Here and elsewhere in this chapter we plot dimensionful quantities in units of the chemical potential, which can always be set to 1 without loss of generality because of conformal invariance. Notably, from the leftmost plot one observes that for sufficiently large values of ω/µ the spectral weight is confined to the edges of a roughly conical structure in momentum space with slope one in the units of the figure. This is in fact the conformal behavior anticipated from the presence of the maximally symmetric AdS4 in the UV. 138
This can readily be seen from the analytic continuation of (5.27) to timelike momenta, replacing the labels “IR” with “UV”, and evaluatingm ˜ = 0. To wit, for ω > vUV k one obtains | | 2 ω A(ω, k) = p , (5.30) vUV p2 − where the Lorentz contraction implied by p2 is now understood to be with respect to the maximally symmetric AdS4 metric. The right plot in figure 5.2 shows the spectral function zoomed-in around the origin for ω < 0. The spectral function in this region is somewhat diffuse, hence we have added two black dashed lines which trace out the peaks in the spectral weight of the two spinor components of the fermionic operator. These lines clarify that the bands of the two spinor components cross at k = 0 and reach their turning points at some non-zero k; this is reminiscent of a holographic
Rashba effect, as was discussed previously in [95].
To further quantify the properties of any putative fermionic excitations, it is also helpful to consider the spectral weight along several representative momentum slices. Strictly at ω = k = 0, the bulk mode decays in the IR as a power law, and one can explicitly show that the spectral weight vanishes at this point. Extending this computation to finite ω along k = 0 results in the slice shown in the left plot of figure 5.3. Most notably, the spectral weight exhibits a “soft gap”, vanishing like a power law as ω 0. By studying the properties of the IR Green’s functions along this slice, it → is straightforward to demonstrate that
ω A(ω, k = 0) ω2∆IR−3 for 1 , (5.31) ∼ µ
1 where ∆IR = 2 (3 + 2m ˜ ) is the conformal dimension of the fermionic operator in the IR theory. The slices along non-zero momenta are rather more interesting. From the right plot of figure
5.3, one can clearly distinguish the appearance of the hard gap in the spectral weight corresponding to the boundaries of the IR lightcone. As the momentum is increased from zero, the broad peak controlled by the UV fixed point develops a shoulder near the gap, which eventually sharpens into a well defined secondary peak. This secondary peak is present for momenta k k? which is ≈ the momentum at which the maximum of the arcing spectral weight achieves its closest approach 139
Figure 5.2: Spectral function for fermionic operators in the 20. The red lines mark the IR lightcone, while the blue lines show the lightcone of the UV theory. The right figure shows a close-up around the origin for ω < 0. Superimposed on the right figure are black dashed lines, showing the lines of maxima of the spectral weight; black dots, marking the point of closest approach to the ω = 0 axis (k?); and white dots, showing the location of the Fermi surface singularities in the normal phase (kF ). These special points will be discussed in more detail in section 5.4. 140
● ●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●●● 7 ●●●●●●●●● ●●●●●●●● k k ●●●●●●●● ●●●●●●● ●●●●●●● = = ●●●●● v 0.83 v 0.41 ●●●●●●● 8 UV UV ●●●●●●● 0.1 ●●●●●● μ μ ●●●●●●● ●●●●●● ●●●●● ●●●● ●●●●●● ●●●●●● ●●●●● ●●●● 6 ●●●● ●●● k k ●●● ●● v =0.73 v =0.31 ●● UV UV ●● μ μ ● 0.01 ● ● ● ● 5 6 k = k = μ vUV 0.62 μ vUV 0.21 0.001 0.0001 0.001 0.01 4 k = μ vUV 0.52 4
3 A ( ω ,k ) A ( ω ,k = 0 )
2 2
1
0 0 -2 -1 0 1 2 -0.5 0.0 0.5 ω ω μ μ
Figure 5.3: Spectral function for fermionic operators in the 20 as a function of frequency at various momenta. At left, k = 0 and the dashed purple line shows the maximally symmetric AdS4 result as given by (5.30). The inset details the falloff at low frequencies, which asymptotes to a power law with exponent 2∆IR 3 = √3/2 as shown by the pink line. At non-zero momenta (right), the − spectral function develops a hard gap. For momenta in the vicinity of k k? there is a narrow ≈ quasiparticle-like peak just below the gap, as well as a more diffuse hump at larger ω/µ as dictated | | by the UV conformal theory. 141 to ω = 0. Accordingly it is natural to associate this secondary peak with a gapped fermionic excitation in the dual ABJM phase of matter. We will have more to say about this excitation and its holographic interpretation in section 5.4. For now, we note that these spectral functions share similarities with the “peak-dip-hump” structure observed in various ARPES measurements of the high Tc superconductors. This experimental structure has been argued to be a consequence of many-body interactions in the superconducting phase (eg. [96]). Similar line shapes were observed holographically in [97] and [16]. The link between these results and the experimentally observed peak-dip-hump is tenuous; in our current case the pattern is likely a consequence of the previously mentioned Rashba-like crossing of two bands, in combination with the sharpening of the peaks as they approach the IR lightcone.
5.2.5 Field Theory Operator Matching
To make contact with the dual field theory, it is necessary to first employ the holographic dictionary to translate the bulk fields involved in our solutions into field theory operators. These operators are distinguished by their quantum numbers—conformal dimensions and charges under various symmetry groups.
The dual superconformal field theory is most commonly written in terms of ABJM theory [2], a Chern-Simons-matter theory which makes a global SU(4) U(1)b SO(8) manifest, while the × ⊂ full SO(8) is present but not apparent in the Lagrangian. However, this SU(4) subgroup and the
− commuting U(1)b (associated with monopole charge) are different from the SU(4) U(1) subgroup × relevant to our geometry; the two sets of subgroups are related by a triality transformation.
− The supercharges in the 8s decompose under SU(4) U(1) as 8s 4−1 4¯1 but under × → ⊕ the SU(4) U(1)b of ABJM theory as 8s 60 12 1−2. This latter decomposition aligns with × → ⊕ ⊕ 4 the isometries of the moduli space for a stack of M2-branes probing a C /Zk singularity [2]. The former branching, on the other hand, corresponds to the decomposition of the supersymmetries when the sign of the M2-brane charge is reversed. Reversing the sign of the M2-brane charge is realized in the eleven dimensional SUGRA as a “skew-whiffed” solution in which the four-form flux 142 has opposite sign (or, equivalently, the orientation of the S7 is reversed). Indeed, when the PW solution is oxidized to eleven dimensions, the solution is of this skew-whiffed form [94, 93]. The
flows constructed in section 5.2.2 thus connect the PW solution to a skew-whiffed AdS4 in the UV.
7 For Chern-Simons level k = 1, the skew-whiffed AdS4 S still preserves maximal supersymmetry, × and the holographic dual remains the ABJM theory. This is the case relevant for the holographic interpretation of our supergravity results. The skew-whiffing is then realized from the field theory perspective as a triality rotation on the operator spectrum [98], as might be anticipated from the various decompositions of the global symmetries we have considered.
Because the two SU(4) groups do not commute, representations of SU(4)− do not fill out complete representations of the ABJM SU(4). Instead of presenting dual operators in the full
ABJM language, we will instead use the simplified notation introduced in section 2.2.1. As was also done in section 3.2.2, we take complex compbinations of the scalars Zj X2j−1 + iX2j and ≡ fermions Λj λ2j−1 + iλ2j, j = 1, 2, 3, 4. In this notation, the operator dual to the complex scalar ≡ turned on in the background is the ∆ = 2 fermion bilinear,
ξ Λ1Λ1 . (5.32) ↔
The gauge field 5.8 corresponds to the chemical potentials for the four Cartan generators of SO(8) being identified as
µa = µb = µc = µd . (5.33) −
The fermionic supergravity fields are then dual to scalar/fermion composite operators with dimen- sion ∆ = 3/2 of the form ZΛ. The mode 5.17 is the linear combination
ψ Z¯3Λ2 Z¯4Λ4 . (5.34) ↔ −
5.3 The H = SO(3) SO(3) Flows ×
We next turn our attention to a similar pair of domain wall geometries found within an
SO(3) SO(3) invariant truncation of the gauged SUGRA [81]. As before, these backgrounds × 143 are holographically dual to zero temperature phases of ABJM theory with a broken U(1) global symmetry, though in this case it is explicitly as well as spontaneously broken. Again we will discover a gapped fermion excitation spectrum in these states. This time, however, the gapping mechanism relies on a special type of fermion coupling, similar to the “Majorana coupling” previously studied in the bottom-up construction of [16].
These background were discussed in detail in chapter4. Again, we will refer to them as the
“Massive Boson” and the “Massive Fermion” backgrounds. They are shown in figures 4.1 and 4.2, respectively. Note that we make a different choice of units for the SUGRA gauge coupling g in this section relative to section 5.2, setting g = 1. This is visible in the difference in the asymptotic value of the metric function G/r2 1/L2 as r . → UV → ∞
5.3.1 The Fermionic Sector
We now wish to derive the SUGRA Dirac equations in the SO(3) SO(3) domain wall × backgrounds. We first isolate a sector of spin-1/2 fermions that do not mix with the gravitini.
As discussed in chapter4, under SO(8) SU(4) U(1)b SO(3) SO(3) U(1) the gravitini → × → × × transform as
8s 60 12 1−2 (3, 1)0 (1, 3)0 (1, 1)2 (1, 1)−2 , (5.35) → ⊕ ⊕ → ⊕ ⊕ ⊕ and thus we can avoid mixing in the SO(3) SO(3)-invariant backgrounds as long as we study × fermions in representations other than these. The spin-1/2 fields are contained in the 56s of SO(8), which decomposes as
56s 152 15−2 100 100 60 → ⊕ ⊕ ⊕ ⊕
(3, 3)2 (3, 1)2 (1, 3)2 (3, 3)−2 (3, 1)−2 (1, 3)−2 (5.36) → ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
2(3, 3)0 2(1, 1)0 (3, 1)0 (1, 3)0 . ⊕ ⊕ ⊕
We see there are four fermions in the (3, 3) representation of SO(3) SO(3)—a charged fermion, a × neutral fermion and their conjugates—that cannot mix with the gravitini. Group theory does not 144 prevent them from mixing with each other, and generically they do. The different U(1) charges of the fermions in the (3, 3) representations are no obstacle to this mixing because the U(1) symmetry is broken by a non-trivial profile for the charged λ in our backgrounds. Moreover, the fact that
(3, 3) is a real representation means mixing between the spinors and their conjugates is possible, meaning the Majorana coupling of 5.1 can exist.
To derive the explicit Dirac equations, we evaluate the scalar tensors in the fermionic La- grangian (2.40) in the SO(3) SO(3) truncation. As anticipated, we find mixing between sets of × four fermions, corresponding to the four copies of (3, 3). We focus on only one of these sets, say
χ467, χ538, χ418, χ428 , since the other sets are related through group theory. The fermions can be { } assembled into complex combinations that are charge eigenstates,
χ2 = χ428 + iχ418, χ¯2 = χ428 iχ418, χ0 = χ467 + iχ538, χ¯0 = χ467 iχ538. (5.37) − −
The χ2 and χ0 modes have U(1) charges 2 and 0, respectively, and the barred spinors, being charge conjugates of the un-barred ones, have opposite charge. The Dirac equations for these fermions take the form
µ iΓ µ 1 + S ~χ = 0 , (5.38) ∇ where 1 is a 4 4 identity matrix, ~χ χ2, χ¯2, χ0, χ¯0 , and S is a mixing matrix with contributions × ≡ { } from gauge, Pauli, and mass type couplings, whose explicit form is 1 / (cosh 2λ + 3) 0 Γ5 sinh λ sinh λ − 4 A − 1 / 0 4 (cosh 2λ + 3) sinh λ Γ5 sinh λ A − − . i 1 2 Γ5 sinh λ sinh λ √ / / √2 Γ5 sinh λ − − 2 2 F 2 A − 1 2 i sinh λ Γ5 sinh λ / + √2 Γ5 sinh λ √ / − 2 A − 2 2 F (5.39)
µ µν Here / Γ µ, and / Γ µν. This mixing matrix cannot be reduced into smaller blocks, and A ≡ A F ≡ F so we are obliged to solve a coupled system of linear differential equations.
Before solving these Dirac equations numerically, it is instructive to summarize the types of couplings the mixing matrix gives rise to, and the qualitative effects of these couplings on the 145 fermionic spectrum. We will use the same projectors (3.30) as before to label the four spinor components as χα± with α = 1, 2. Writing out the Dirac equations (5.38) at the level of the spinor components, it is easy to see that they split into two independent sets. One set couples together the α = 1 components of χ2 and χ0 with the α = 2 components ofχ ¯2 andχ ¯0. The other set of equations is identical but with α = 1 α = 2 and k k. ↔ → − This coupling is a generalization of the “Majorana coupling” discussed by [16] to involve more than one spinor field. As we described in the introduction, [16] noted that such a coupling effectively forbids the existence of a holographic Fermi surface. This can be understood as a consequence of level repulsion. Imagine that in the absence of such a Majorana coupling, the α = 1 component of χ2 has a band of normal modes that crosses ω = 0 and at a non-zero k = kF ; this crossing is interpreted as a Fermi surface (left part of figure 5.4). Becauseχ ¯2 is the charge conjugate of χ2, it will have a similar normal mode band but with (ω, k) ( ω, k), thus it crosses ω = 0 at → − − k = kF . Moreover, the spectrum of the α = 1 components is related to that of α = 2 components − by k k, as a consequence of the background rotational symmetry. Taken together, this means → − that the α = 2 component ofχ ¯2 has a normal mode band that is related to that of the α = 1 component of χ2 by a reflection across ω = 0 (center of figure 5.4). In particular, these two bands will cross at (ω, k) = (0, kF ). Finally then, turning on the Majorana coupling between them will cause level repulsion at this crossing point, gapping out the Fermi surface (right part of figure 5.4).
This mixing between components can be thought of as analogous to the Bogoliubov transformation mixing particles and holes in BCS theory. This prediction will be confirmed in our numerical results in the next subsection.
In field theory terms, the Majorana coupling in supergravity corresponds to the existence of a three-point function which is schematically of the form λ χ χ among the operator λ dual hO O O i O to the active scalar λ and the fermion. This three-point function is visible in the vacuum state of the dual field theory, and its strength controls how strong the gapping of the Fermi surface will be.
It would be interesting to try to develop a more model-independent, field theoretic account of how similar three-point functions control the size of a superconducting gap. 146
ω k
Figure 5.4: An illustration of the level repulsion induced by the chiral Majorana coupling in the (ω, k)-plane. Left: Without a Majorana coupling, (one of the two spinor components α of) a fermion operator will generically display lines of normal modes (purple) crossing the dashed ω = 0 line, leading to a Fermi surface singularity. Center: Looking at the conjugate fermion, and switching to the other spinor component, gives an identical normal mode line flipped across ω = 0. Right: Turning on the chiral Majorana coupling mixes these two energy bands, causing them to repel. 147
Additionally, a comparison to the results of the previous chapter4 will be helpful. There, fermion response for the same quartet of SUGRA fermions in the same SO(3) SO(3) domain × walls were considered, but the chiral parts of the Majorana couplings in the fermion Lagrangian
(2.40) were neglected. As explained above, the Γ5 matrices in the Majorana couplings are directly responsible for the coupling between spinor components with different α; without them the Ma- jorana terms couple only α = 1 to α = 1 components. Hence, by neglecting these couplings the gapping mechanism described above is no longer present; level repulsion still occurs among the mixed fermions, but it is no longer guaranteed to be localized at ω = 0 where it can create a gap. Thus, when turning on the “non-chiral” Majorana couplings in chapter4, some Fermi surface singularities were lifted, while others remained.
5.3.2 Fermion Response
We now proceed to solve our set of coupled Dirac equations. Many steps are identical to those described in section 5.2 and will not be repeated. The one new ingredient in this system is the mixing between different fermions through the matrix S. As a consequence of this mixing, when sourcing any of the coupled fermions, there will generically be a response in all four of them.
This gives rise to a matrix of Green’s functions, schematically
j ij δ R = hO i , (5.40) G δJ i Jk=0 where i, j 1, 2, 3, 4 label the four coupled fermions, J and denote sources for and responses ∈ { } hOi of the dual operators, respectively, and J k = 0 implies that all sources except J i are zero. The computation of this matrix, including the implementation of correct boundary conditions, requires some care; this is described in detail in [87, 80] and is implemented in a very similar system in chapter4. We will thus proceed directly to a discussion of our results.
First of all, as was the case in section 5.2, the IR geometry controls important aspects of these
Green’s functions. For timelike IR momenta, as defined in (5.23), infalling boundary conditions are imposed. These boundary conditions are complex, which can lead to quasinormal mode solutions. 148
In the dual gauge theory these correspond to excitations with finite lifetimes. In contrast, for spacelike IR momenta one instead imposes regular boundary conditions in the IR. This is a purely real boundary condition, and may give rise to normal mode solutions in the bulk. Such solutions could correspond holographically to stable fermionic excitations in the boundary field theory.
If the fermion spectral weight is non-vanishing at zero frequency, it means that there are gapless fermionic modes in the dual phase. In chapter4, an interesting prediction of the “nearly top-down” model was that spectral weight appeared as a band of delta functions passing through a Fermi surface singularity at ω = 0 and k = kF . However, as anticipated previously by the chiral Majorana coupling-induced level repulsion argument, we find that the true top-down system admits very few normal modes at all. In the Massive Fermion background we find none, while in the Massive Boson background there is a line of normal modes very close to the lightcone, as seen in figure 5.5. These lines sit very close to the lightcone edge, and go on to quite large k and ω.
To get a more detailed picture of the spectrum, we plot the spectral functions as defined in
(5.28) for the fermions of the (3, 3) in the Massive Boson and Massive Fermion backgrounds in
figures 5.6 and 5.7, respectively. As in section 5.2, we observe arcing spectral weight inside the IR lightcone presumably due to the presence of bulk fermion quasinormal modes. Particularly in figure
5.7 we again observe a crossing of the arcs coming from different spinor components. The mixing of charged and neutral fermions is seen in the transfer of spectral weight between arcs as one follows them while varying k (this is most clearly seen in the massive fermion background). Note that the massive boson normal modes are some (k- and ω-dependent) linear combination of the charged and neutral fermions, hence they are drawn in both plots. Importantly, in both holographic phases the spectral weight is only non-zero away from ω = 0, and in nearly every case there is a fairly pronounced gap in the spectral function. Hence we find no sign of a Fermi surface in the fermion correlation functions in these holographic states. The figures show the spectral functions of χ2 and χ0; the spectral functions of their charge conjugate modes are identical but with ω ω, as → − discussed above. 149
0.027 0.15
0.025
0.023 0.10
ω 0.6 0.65 0.7 μ
0.05
0.00 0 1 2 3 4 k v μ UV
Figure 5.5: The band structure of fermion normal modes in the Massive Boson (type 1) background. The normal mode is shown in purple. The inset zooms in on the beginning of this band, emphasizing that it very nearly coincides with the edge of the IR lightcone.
Figure 5.6: The spectrum in the Massive Boson background. The red and blue lines mark the IR and UV lightcones, respectively, and the white lines show the location of the line of normal modes, corresponding to a line of delta function peaks in the spectral weight. The white dots at ω = 0 show the Fermi momentum in the normal phase. 150
Figure 5.7: The spectrum in the Massive Fermion background. The red and blue lines mark the IR and UV lightcones, respectively; for spacelike IR momenta the spectral weight is zero everywhere. The white dots at ω = 0 show the Fermi momentum in the normal phase. 151
5.3.3 Field Theory Operator Matching
Once more exploiting our top-down framework, we can write down exactly which operators in ABJM theory are dual to the quartet of fermions under study. Unlike the SU(4)− case, here the symmetry structure aligns nicely with the ABJM decomposition of SO(8) described above (5.35), with SO(3) SO(3) embedded in SO(6) SU(4) in the natural way. As a result, we can use × ' ABJM operator language directly. This was worked out in chapter4, and for details we refer the reader there. Here, we simply quote the results:
1 2 3 4 2τ χ2 Y ψ2 Y ψ1 + Y ψ4 Y ψ3 e , (5.41) ↔ − − † †2 † †1 † †4 † †3 −2τ χ¯2 Y ψ Y ψ + Y ψ Y ψ e , (5.42) ↔ 1 − 2 3 − 4 1 †4 4 †1 2 †3 3 †2 χ0 Y ψ + Y ψ Y ψ Y ψ , (5.43) ↔ − − † † † † χ¯0 Y ψ4 + Y ψ1 Y ψ3 Y ψ2 . (5.44) ↔ 1 4 − 2 − 3
In this mapping, the Y ’s are ABJM scalars, ψ’s are fermions, and e2τ is a monopole operator which carries all of the charge under the U(1)b.
This identification of symmetries facilitates the field theory description, allowing one to in- terpret the dual state of matter as a phase in which a chemical potential for monopole operators has been turned on. The four Cartan chemical potentials are identified as
µa = µb = µc = µd . (5.45)
This corresponds to the gauge field A12 alone being turned on because 1, 2 are 8s indices, and a triality rotation to the 8v basis reveals all four Cartan charges are turned on equally. The non- trivial bulk scalar signals an explicit breaking of the number density for this composite matter, by an operator of the form
A A 2τ A A 2τ ∆=1 Y Y e or ∆=2 ψ ψ e , (5.46) O ∼ O ∼ for the massive boson (fermion) case, respectively. Viewed in this language, our results for the massive fermion phase demonstrate a novel phase of strongly coupled matter in which there exist perfectly stable composite fermion excitations above a hard gap. 152
5.4 Lessons for Strongly Coupled Systems
One of the most striking lessons from our calculation of fermion spectral functions is that in the broken symmetry phases of ABJM matter that we study, our fermion spectral densities are always gapped. This observation merits further discussion, as it appears to manifest for different reasons in the two cases, and it is not entirely clear how generic this result might be. In an attempt to better understand the absence of Fermi surface singularities in these spectral functions, it proves useful to compare our results against several related calculations which we now describe.
5.4.1 Top-down vs. Bottom-up Fermion Response
In previous investigations of fermion spectral functions in domain wall flows [15], the authors employed non-top-down fermions in an attempt to gain intuition for how the fermionic degrees of freedom behave in the dual phases of matter. A surprising result was the presence of families of bulk fermion normal modes which collectively described ungapped bands of perfectly stable fermionic excitations in the dual field theory.
To realize these bands, it is necessary to deform our top-down system by ignoring the con- straints that D = 4 maximal gauged SUGRA places on the bulk fermion couplings. In the SU(4)−
flow, for example, one can make contact with [15] by setting the scalar to zero in the top-down
Dirac equation (5.18) (so that the fermion couplings do not run), dropping the Pauli coupling, and artificially dialing the bulk fermion’s charge. As explained in [15], for suitably large values of this
“probe” fermion’s U(1) charge, ungapped bands of normal modes appear and a holographic Fermi surface is present. To verify that the formation of a gap is highly dependent on the various couplings in the Dirac equation, we computed the spectral function for a number of deformations of (5.18).
We find that setting the scalar to zero in the Dirac equation, but otherwise leaving the magnitude of the couplings untouched, leaves the results largely unchanged; in particular, the gap remains.
However, if we additionally tune the couplings by (1) factors, for example by doubling the charge O or changing the sign of the Pauli coupling, the gap will in general close. This is consistent with the 153 results of [15], which show that the larger the fermion charge, the more bands of gapless modes are present.
In this context then, it would seem that the fermion spectral functions in the SU(4)− domain wall of section 5.2.3 end up gapped for a fairly straightforward reason: SUGRA demands that in this state, the fermions in the 20 carry a U(1) charge that is too small to support a Fermi surface.
This stands in contrast to the gapping mechanism that appears to be at work in the SO(3) × SO(3) flow. The results of chapter4 demonstrate that in this phase, the U(1) charge carried by the bulk fermion is sufficient to form a holographic Fermi surface, provided that one removes the chiral
Majorana couplings by hand. (Purely bottom-up fermions with the same mass and charge, also studied there, have yet more gapless bands.) In other words, the mechanism of [16], in which the chiral Majorana couplings play the key role, makes the difference in this case between an ungapped
Fermi surface and gapped behavior.
Thus, we find that the SUGRA couplings conspire to gap out the spectral weights in all the cases we study. However, while the resulting spectral weights all have similar features, with gapped, arcing bands, the various bulk Dirac equations have qualitative differences. The Majorana coupling 5.1 acts much like a bulk version of the BCS mechanism, and can therefore be expected to lead to the observed gaps in spectral weights. Yet the fermion spectral functions in the SU(4)− background emphasize that a gap may appear without this coupling. The precise interpretation of these different gapping mechanisms in terms of the physics in the boundary field theory deserves further investigation. Furthermore, it would clearly be interesting to study fermionic spectral weights in other top-down realizations of zero-temperature symmetry-broken states, in order to
find out how general the formation of a gap really is.
5.4.2 Extremal AdSRN and Effects of Broken Symmetry
A complimentary line of insight is directed along comparisons between the spectral functions in our domain wall flows and those in states of unbroken U(1) symmetry. Such states are readily accessible to our decoupled fermions. They are solutions to the bosonic sectors described by (5.9) 154 and (4.4), but with the scalar set to zero. These backgrounds are the familiar AdS4 Reissner-
Nordstr¨om(AdSRN) solution, and its extremal limit is holographically dual to a distinct zero temperature finite density phase.
Although the form of the AdSRN solution is basically the same in both the SU(4)− and
SO(3) SO(3) truncations, their holographic interpretation is slightly different because the U(1) × gauge fields under which the black holes are charged and the associated chemical potentials are embedded differently into SO(8), as is spelled out in (5.33) and (5.45).
Nonetheless, both the fermions in the 20 as well as those of the (3, 3) behave similarly in their respective AdSRN backgrounds. Importantly, both systems display Fermi surface singularities in their dual fermion spectral functions. For the fermions in the (3, 3), the charged modes decouple from their neutral counterparts when the scalars vanish, and unsurprisingly it is the spectral func- tion for the charged operators that exhibits a Fermi surface. It is perhaps helpful to emphasize that these results (unlike the previous subsection) are truly top-down. Both the AdSRN backgrounds and the spin-1/2 Dirac equations can be embedded in the maximal gauged SUGRA theory.
The results of the present chapter show that breaking the U(1) either spontaneously or explicitly destroys this Fermi surface and gaps the corresponding spectral functions. Notably, the new state with broken symmetry appears to “remember” the location of the Fermi surface that was present in the unbroken phase. This is demonstrated by the arcing spectral weights in
figure 5.2 (right plot) and in figures 5.6 and 5.7, which bend towards ω/µ = 0, and achieve their
? closest approach at some finite momentum k vUV/µ. Computation of the fermion response in the unbroken phase reveals a Fermi surface singularity at kF vUV/µ 0.25 for fermions in the 20 of ≈ − 2 SU(4) SO(8), and at kF vUV/µ 0.53 for the fermions in the (3, 3) of SO(3) SO(3) SO(8). ⊂ ≈ × ⊂ ? From the figures, one finds that indeed k /kF 1. ∼ It is interesting to compare this to the gapping that occurs in the fermionic excitation spec- trum of the standard BCS theory. In the normal phase of a superconductor, particles and holes
2 Note that due to the different units employed in sections 5.2 and 5.3, care should be taken in comparing the Fermi momenta between the two phases. 155
8
7
6
5 E(k) 4
3
2 (k⇤ k , ) ⇡ F HSC
1
0 2 0 2 4 6 8 k
Figure 5.8: Illustration of gapped fermionic excitations in BCS theory and holography. In the left panel, the BCS dispersion relation in the superconducting (normal) phase is plotted in blue (dashed black). The parameters are arbitrarily chosen such that vF = kF = 1 and ∆ = 2. | | In the holographic fermion spectral function (cartoon, right), the boundaries of the IR lightcone determines the stability of the fermionic excitations, but the gapping is qualitatively similar. 156 have an approximately linear dispersion about the Fermi surface at k = kF . Thus, in a rotationally invariant system, (k) vF (k kF ) with vF the Fermi velocity. As the superconductor is cooled ≈ − into the superconducting phase, Cooper pairs condense and the mean field BCS Hamiltonian can be rediagonalized via a Bogoliubov transformation that mixes particles and holes. These new Bogoli- ubov modes describe the fermionic excitations in the superconducting phase, and have a dispersion relation of the form q p 2 2 2 2 2 E(k) = (k) + ∆ v (k kF ) + ∆ , (5.47) | | ≈ F − | | which is plotted in the left panel of figure 5.8.
In the right panel of the same figure, a sketch comparing some related features in figures
5.2, 5.6, and 5.7 is shown. The cartoon emphasizes the arcs in the spectral weight for fermionic
? 3 excitations, whose minima at k kF define a gap that is present in the holographic results. ≈ Also depicted is the qualitative effect of the IR critical point, which opens a window of stability for any excitations that may be present in the kinematic region defined by the exterior of the IR lightcone. In the illustration there are no such stable excitations, but such excitations do appear in the spectrum of fluctuations in the Massive Boson background (figure 5.6).
It is worth noting that the peaks of the various spectral weight arcs we observe are in general not sharpest at k = k?, where the gapped excitation achieves its lowest energy; this can be seen particularly well in the right plot of figure 5.3. Instead, the peak representing the gapped excitation typically sharpens further as it nears the IR lightcone. This behavior is natural from the perspective of the dual field theory, where the presence of the IR lightcone can be interpreted as the existence of a kinematic regime in which interactions mediating decays of the fermionic excitations are forbidden.
The holographic spectral densities suggest a suitable (but somewhat rough) estimate for the size of the gap in the holographic broken symmetry phases, ∆HSC. In the examples shown in this chapter, the value of the excitation energy at k? is always close to the boundary provided by the
3 While the spectral functions we compute have a “soft gap” at k = 0 in the sense that the spectral weight vanishes as a power law in ω (see e.g. figure 5.3), the majority of the spectral weight is concentrated into these (gapped) arcs. 157
IR lightcone. Thus we can write
? ∆ HSC E(k ) E(kF ) vIRkF (5.48) | | ≡ ≈ ∼ where vIR is the effective speed of light in the IR theory, and kF is the value of the Fermi momentum in the symmetry unbroken phase dual to the extremal AdSRN solution. This type of estimate, while fairly accurate in our top-down realizations, will generally not be obeyed in an arbitrary bottom-up construction where one is free to tune the different couplings. Again it would be interesting to study other similar top-down embeddings in order to investigate whether this is a standard feature of such states. At the very least, however, (5.48) is a “phenomenological” rule for the finite density states we have studied here, supported by the results in figures 5.2, 5.6, and 5.7.
5.4.3 Stability in Supergravity and Zero Temperature Response
The utility of the fermionic spectral functions is contingent on their ability to quantify and elucidate properties of interesting strongly correlated phases. While we have applied this tool to better understand how some of these phases are constructed from ABJM matter, it is also important to address the possibilities that these zero temperature states have to actually be realized in the phase diagram for ABJM matter at finite density.
Fundamentally, this is a question of stability. A useful example is provided by the SU(4)−- invariant flow of section5 .2 and the AdSRN solution that also solves the equations of motion derived from (5.9). Very generally, both solutions holographically describe zero-temperature phases of strongly interacting ABJM matter at finite density. In both cases, the ABJM theory remains undeformed by the application of any additional sources beyond the chemical potential. Thus, it is natural to wonder which (if either) of these solutions provides the thermodynamically preferred phase for such ABJM matter at low temperatures.
Neither the SU(4)−-invariant flow nor the extremal AdSRN solution preserve any of the supersymmetries of the vacuum AdS4. Accordingly there is no guarantee that either solution is stable at zero temperature, and it is necessary to consider the whole spectrum of SUGRA 158
fluctuations to hunt for instabilities. Unstable modes may, or may not, belong to the consistent truncation that results in the maximal gauged SUGRA of section 2.1, and thus the identification of all possible instabilities is a rather involved task.
It is by now well known that extremal AdSRN solutions exhibit a multitude of instabilities in gauged SUGRA theories. These instabilities are often diagnosed by studying the mass spectrum of supergravity fluctuations around the AdS2 factor of the near horizon geometry of the extremal solution. If the fluctuation’s effective mass lies below the Breitenlohner-Freedman bound [63, 64] of this IR AdS2 region, an instability to the formation of a new branch of solutions with a non-trivial profile for the unstable mode is anticipated.
In the context of the present chapter, this is exemplified in the “superfluid” instability of the extremal AdSRN solution to the formation of scalar ξ hair. The SU(4)−-invariant flow studied in section 5.2 is the zero temperature endpoint of a branch of solutions which extends to finite temperatures via a series of hairy black holes which terminate at some temperature Tc. By com- paring the thermodynamic free energy of the hairy black holes to that of the AdSRN solutions, it is straightforward to demonstrate that the solutions with ξ hair are thermodynamically preferred, and that as the finite density system cools there is a second order phase transition at Tc from the symmetry unbroken “normal” phase to a broken symmetry superfluid phase with a non-vanishing condensate of the operator holographically dual to ξ.
Interestingly, in [93] the authors demonstrate that this superfluid instability is not the end of the story at low temperatures. They show that the PW solution which characterizes the IR of the SU(4)−-invariant flow is itself unstable to fluctuations of scalar modes within the gauged
SUGRA, and identify the origin of these unstable modes from the eleven dimensional perspective.
Consequently, the SU(4)−-invariant flow cannot describe a true ground state for strongly interacting
ABJM matter.
Further instabilities in the finite-temperature generalizations of the SU(4)− flow and its Ad-
SRN companion were identified, and the backreacted geometries corresponding to those instabilities were constructed, by [99] in a larger SU(3)-invariant truncation containing additional scalars that 159 includes the SU(4)−-invariant case as a subtruncation. These other branches of solutions are in thermodynamic competition with the branch we consider, although it is generally not known what their zero-temperature limit is.
Stability of the SO(3) SO(3)-invariant flow has been investigated in [82]. The authors ×
find in this case that despite lacking any supersymmetry, the IR AdS4 solution is stable to scalar perturbations in the gauged SUGRA. While this stability does not automatically extend to the full flow, nor does it guarantee an absence of unstable modes in the eleven dimensional theory, it is nonetheless an interesting observation that distinguishes this flow in the context of holographic phases of matter. Chapter 6
“1kF ” Singularities and Finite Density ABJM Theory at Strong Coupling
This chapter is an edited version of [100], written in collaboration with Christopher Rosen.
6.1 Overview
Within the Landau Fermi liquid paradigm, the existence of a Fermi surface in a finite den- sity phase of interacting matter is reflected in the analytic properties of various susceptibilities.
More specifically, in conventional Fermi liquids a Fermi surface at momentum kF appears as a non-analyticity in (for example) the static charge susceptibility χ(k) at k = 2kF . That such a rela- tionship between fermionic and bosonic response should exist is not surprising. In a Fermi liquid, charge transport is the responsibility of the excitations very near the Fermi surface, with momen- tum k = kF + q where q/kF 1. It is thus natural that the static susceptibility, which quantifies the response of the liquid to a charged impurity, encodes the length scale 1/kF characteristic of these low lying excitations.
These so-called “2kF ” singularities in susceptibilities arise on very general grounds, essentially as a consequence of kinematic constraints relating parallel portions of the Fermi surface. They are also fundamentally quantum mechanical in origin, owing to the fact that fermionic excitations near the Fermi surface have Compton wavelength λ 1/kF . The effects of this finite size on ∼ charge screening can already be anticipated from elementary quantum scattering considerations.
A familiar textbook example (see e.g. [101, 102] for this and related review below) points out that a free quantum mechanical particle (“electron”) with momentum kF incident on a potential 161
(“charged impurity”) will give rise to a deviation in the charge density like
2 2 2 ikF x −i(kF x−δ) ikF x 2 δ Ψ(x) = e + e e = 2 cos(2kF x δ) + ( ) (6.1) | | |R| − |R| − O |R| to the left of the impurity at the origin. These oscillations in the induced charge density, due the quantum mechanical “size” of the relevant excitations, are an example of Friedel oscillations.
Evidently, they are characterized by a wave vector of magnitude 2kF .
The quantum nature of these oscillations is also borne out by more explicit calculation. In the ubiquitous Random Phase Approximation (RPA), the dominant contribution to the susceptibility is captured by the particle–hole polarization diagram—a one loop effect. In 2+1 dimensions, computation of this diagram leads to the well known Lindhard function and static susceptibility of the form s k 2k 2 χ(k) 1 Θ 2 1 F . (6.2) ∝ − kF − − k
Written in this way, the 2kF singularity is easily seen to coincide with the branch points of the square root. In the linear response regime, this non-analyticity in χ will leave its mark on the induced charge density n, giving rise to both oscillations and a characteristic power law decay:
cos (2k R + δ) n(R) F . (6.3) ≈ R2
Here R gives the distance from the impurity (see also the discussion around (6.30) below) and δ is again a phase. This expression makes only very mild assumptions about the form of the impurity potential, and thus retains its validity in a great variety of physical systems.
In particular, its realm of applicability is broadly expected to extend beyond Fermi liquids to include any system with a “well-defined” Fermi surface. An operational definition of the latter could be a sharp surface in momentum space such that gapless fermionic degrees of freedom congregate at this k = kF at zero temperature. In this case wave vectors connecting parallel patches of the Fermi surface ought to dominate the correlation functions, giving rise to 2kF singularities in the susceptibility. Not surprisingly, comparitively little is known about the fate of the Friedel oscillations and other 2kF singularities at strong coupling, or in systems that fall outside the 162 paradigm of Landau Fermi liquidity. Attempts to quantify the effects of interactions on both the
Lindhard function as well as the signature of Friedel oscillations have appeared in a variety of works, relying on a variety of theoretical frameworks (see [103, 104, 105, 106, 107, 108, 109, 110] for a far from exhaustive sample). Although the specifics vary, the take home message seems to be that while interactions may “soften” the 2kF singularities, they typically remain in the response.
Various attempts to clarify the situation have also been made within the framework of gauge/gravity duality [111, 112, 113, 114, 85, 86, 115, 116]. This approach is particularly attrac- tive, in that it allows simple computational access to finite density phases of strongly interacting matter. In practice, no assumption of adiabaticity need apply with respect to the interactions, and thus the reach of this method extends beyond many conventional techniques. What’s more, if the calculations are performed within a supergravity (SUGRA) theory that descends from a low energy limit of string or M theory, there exists the possibility of performing a careful match between the gravitational physics and properties of a known gauge theory in appropriate limits.
In an early attempt to identify an analogue of Friedel oscillations in holographic matter, the authors of [111] forgo the “top-down” string/M-theory pedigree and study the polarization function in a semi-classical bulk model in which fermionic modes fill out a bulk Fermi sea. Although they succeed in constructing a model with a boundary Lindhard function that displays non-analytic behavior at q = 2kF , in this chapter we would like to adopt a slightly different perspective of holographic matter at finite density.
In explicitly known examples of AdS/CFT duality, the holographic (boundary) gauge theories generically describe interacting bosons and fermions charged under various global symmetries. This is typically a consequence of large amounts of supersymmetry, in which case the global symmetries may be R-symmetries, for example. By turning on a chemical potential for matter charged under
(typically the Cartan of) this symmetry, one effects a finite density deformation of the original theory. Generic states in the finite density theory are thus characterized by a large number of strongly interacting bosonic and fermionic modes, at least in the classical SUGRA limit of the duality. 163
By adopting this perspective, our tastes seem to be more closely aligned with those of [115], who search for charge density oscillations in the quintessential finite density phase of holographic matter—the AdS–Reissner-Nordstr¨om(AdS–RN) solution. Under standard application of the holographic dictionary, the bulk U(1) gauge field maps to a conserved global current in the boundary theory. What exactly that global current is depends on the details of the holographic realization. If the solution comes from a sector of the maximal gauged SUGRA in four dimensions, for example, the conserved current might be constructed from ABJM [2] matter charged under a particular U(1) subgroup of the SO(8)R symmetry. In what follows we will be primarily interested in interpreting our results holographically within this maximal gauged SUGRA/ABJM correspondence.
Considered from this vantage point, the finite density ABJM phase dual to the AdS–RN solution is a strongly interacting plasma of bosonic and fermionic degress of freedom charged under the corresponding U(1). Very generally, it is natural to wonder if the finite density of fermions present in this phase have arranged themselves into a Fermi circle in the 2+1 dimensional field theory. If so, then one would expect to find some indication of the length scale 1/kF in various correlation functions, such as the static susceptibility as highlighted above.
In [115] the authors compute this static susceptibility holographically, and find that there is in fact non-analytic behavior tied to one such length scale. Consequently, they observe oscillations in the induced charge density with wavelength controlled by this scale, providing an example of a strongly interacting analogue of the familiar Friedel oscillations. Very interestingly, the authors were able to understand this length scale analytically. They observe that the oscillations are controlled
− by branch points in a certain scaling exponent νk that governs the low energy fluctuations of the charge density.
Explicitly, this exponent calculated in the AdS–RN background turns out to be given by v u s 1u k 2 k 2 ν− = t5 + 8 4 1 + 4 (6.4) k 2 µ − µ and the branch points that appear to be responsible for the induced oscillations are those at k?/µ =
1/2√2 + i/2 where µ is the chemical potential. For future reference, we note that Re k? 0.35µ ≈ 164 sets the scale on which the charge density oscillations take place.
Thus, the situation seems to be that the static susceptibility computed in this phase behaves as though there are charged degrees of freedom with a characteristic momentum Re k?/µ. The ∼ obvious question, then, is whether or not there is good reason to associate this characteristic
? momentum with the existence of a Fermi surface at 2kF = Re k in the finite density phase.
A sensible approach towards an answer to this question is to look for hints of a Fermi surface in other correlation functions evaluated in the same holographic phase. A particularly good candidate might be retarded two point functions of various fermionic operators in the ABJM theory. At zero temperature, such two point functions can be used to diagnose both the existence of a Fermi surface, where
−1 GR (ω = 0, k = kF ) = 0 (6.5) as well as to understand the nature of fermionic excitations near the Fermi surface (via the spectral function, proportional to Im GR).
By fully decoupling 32 of the 56 spin 1/2 modes of the maximal gauged SUGRA from all other fermions, such fermionic two-point functions were computed in chapter3 in the ABJM phase dual to AdS–RN holographically. Two important lessons from that chapter are:
(1) Two-point functions of fermionic ABJM operators can exhibit Fermi surface like singular-
ities at ω = 0 and k = kF in the finite density phase dual to AdS–RN.
(2) For the modes considered, these Fermi surface singularities arise at kF 0.37µ. ≈
? The numerical similarity between Re k computed from the charge susceptibility and kF com- puted from fermion response—two logically independent calculations—is surprising. This similarity has hitherto gone unnoticed in the published literature, and forms the primary motivation for the present chapter. It is perhaps equally surprising that the holographic results appear to suggest the presence of “1kF ” singularities in the charge response, as opposed to the omnipresent 2kF singular- ities that typically accompany a sharp Fermi surface as outlined above. In what follows, the goal will be to further explore the connection between non-analyticities in the static susceptibility and 165 poles in fermionic two-point functions in finite density phases of ABJM matter. Ultimately, we will find some indication that such a connection may persist even in more complicated phases of holographic matter. This evidence will in turn depend strongly on our adherence to a “top-down” framework for our holographic calculations, in which all bulk couplings for bosonic and fermionic
fluctuations are fully determined by the maximal gauged SUGRA.
From the perspective of the fermion response, the holographic ABJM phase dual to AdS–RN is quite unlike a conventional Fermi liquid. Although there is a sharp Fermi surface in the sense of
(6.5), it was found in [47] that the low lying fermionic degrees of freedom can not be interpreted as well defined Landau quasi-particles. Such a peculiarity could conceivably muddle any connection between “1kF ” singularities in the fermion and charge density correlation functions. In an attempt to circumvent this potential obstruction, in this chapter we will focus on a holographic phase of matter that more closely resembles a conventional Fermi liquid from the perspective of top-down fermion response.
Towards this end, we will set the stage for our calculation in section 6.2 where we introduce the particular solution of = 8, D = 4 gauged SUGRA we wish to study — this is the 3QBH N of chapter3. Interpreted holographically, this solution provides a gravitational description of a curious phase of ABJM matter. In particular, the phase is characterized by a specific heat that vanishes linearly at low temperatures, as well as by fermionic spectral functions which suggest the presence of stable charged excitations which reside near a Fermi surface (reviewed in section 6.3.1).
Taken together, these features suggest that in such a phase the physics could in some sense be more
“Fermi liquid-like” and the connection between charge and fermion response might be even more pronounced. It will turn out that this does not seem to be the case.
One reason will be intimately related to the appearance of new channels of charged response in this background relative to the AdS–RN background. The IR portion of this solution and its
fluctuations has previously starred in related holographic investigations of current-current corre- lators as an “η–geometry”, with η = 1 [85, 86]. There the authors also arrive at the conclusion that such geometries are dual to phases that are in some sense more “fermionic”, albeit for rather 166 different reasons. The fluctuation analysis in this background is discussed in section 6.3.2 (and also in more detail in [86]) with some finer points relegated to appendixB.
In contrast to the analysis performed in [86], the computation of the static susceptibility is sensitive to the entirety of the bulk geometry. There is thus little hope of solving exactly the coupled fluctuation equations, and to make progress we resort to numerical integration of these equations throughout the complex momentum plane. The results of this integration are shown in a series of plots in section 6.4, which provide the primary output of this chapter.
We then turn to a discussion of our results in section 6.5, where we highlight the underlying structure responsible for the analytic properties of the static susceptibility, and attempt to rec- oncile those properties with the fermion response. Here we comment on the somewhat surprising similarities between Fermi momenta and branch points of the static susceptibility, and conclude with some avenues for further study.
6.2 The Three-Charge Black Hole
As suggested above, our interest lies primarily in the properties of finite density states of strongly interacting ABJM matter at zero temperature. The holographic dictionary thus directs us towards solutions of D = 4 = 8 gauged supergravity that asymptote to the maximally symmetric N
AdS4 in the UV, but are driven elsewhere in the IR by a non-vanishing profile for an Abelian gauge
field.
Many such solutions are known to exist. To construct them, it is often convenient to consider a truncated subset of the gauged SUGRA which contains fewer fields. In this chapter, we will focus our attention on the properties of a particular solution which resides in the truncation of the SUGRA to singlets under the Cartan U(1)4 SO(8) of the gauge symmetry. The relevant ⊂ truncation (as well as its electrically charged black brane solutions) was worked out in [57], and there the truncation’s consistency was demonstrated by explicitly providing the lift to the eleven dimensional theory.
The truncation to the Cartan leaves an = 2 SUGRA coupled to three vector multiplets. N 167
The so-called three-charge black hole (3QBH) solution is a background in which three of the Abelian gauge fields are set equal to one another, while the fourth vanishes. This restriction also identifies the three dilatons with each other. Additionally, for the purely electric solution we consider, the axions all vanish.
This sector of the SUGRA is thus described by a simple Lagrangian of the form1
φ −1 1 2 3 √ 2 φ 2e = R (∂φ) e 3 F + 6 cosh (6.6) L − 2 − 4 √3 where φ is the remaining dilaton and F = dA is the field strength for the active U(1) gauge fields.
2 We work in units such that the maximally symmetric AdS4 has radius L = 2κ = 1. The electric
3QBH solution is conveniently written in terms of the ansatz
−2χ(r) 2 µ ν 2χ(r) 2 2 e 2 d¯s =g ¯µνdx dx = e f(r)dt + d~x + dr , A¯ = Φ(r)dt, − f(r)
and φ¯ = φ¯(r). (6.7)
Bars have been added to some quantities in anticipation of our impending fluctuation analysis, in which the bar will be used to denote background quantities. The solution is then given by
3 Q r + Q3 χ(r) = log r + log 1 + f(r) = 1 H 4 r − r + Q p rH + Q √3 Q Φ(r) = Q(rH + Q) 1 φ¯(r) = log 1 + (6.8) − r + Q 2 r where rH and Q are integration constants that parametrize the location of the horizon and (roughly) the brane’s charge.
6.2.1 Thermodynamics and Field Theory Interpretation
Under the standard holographic interpretation of this solution, it is easy to see that the dual state of the ABJM theory is characterized by a temperature and entropy
3 p 3/2 T = rH (Q + rH ) and s = 4π√rH (rH + Q) (6.9) 4π √ 1 Note that we have rescaled the scalar field by a factor of 3 compared with section 3.4.2. 168 respectively. The solution contains bulk gauge and matter fields, and their role from the ABJM theory perspective is most conveniently understood from the near boundary behavior of these fields in Fefferman-Graham coordinates. In these coordinates the near-boundary metric locally takes the form 2 2 2 i j du ds = u gijdx dx + , (6.10) u2 while the gauge field and scalar fall off like
ρ Φ(u ) µ + ... (6.11) → ∞ ≈ − u √3Q √3Q2 φ¯(u ) + + .... (6.12) → ∞ ≈ 2u 8u2
The coefficients governing the fall-offs of the gauge field have been labeled suggestively. They are interpreted holographically as a deformation of the ABJM theory by a non-vanishing chemical potential (µ), which in turn places the dual theory in a finite density (ρ) phase. Explicitly,
p p µ = Q(rH + Q) and ρ = (Q + rH ) Q(rH + Q). (6.13)
The thermodynamic quantities are parametrized by two dimensionful parameters, rH and Q.
With an eye towards the grand canonical ensemble, it is helpful to trade these parameters for T and µ, which allows us to express the charge and entropy densities as
µp 16 p ρ = 16π2T 2 + 9µ2 and s = π2T 16π2T 2 + 9µ2. (6.14) 3 9
The correct interpretation of the boundary behavior of the bulk scalar is a bit more subtle.
If we denote the leading 1/u fall-off by α and the subleading 1/u2 fall-off by β, a naive application of the holographic dictionary would relate α to a deformation of the ABJM theory by a scalar operator, and β to that operator’s response. These are Dirichlet (α = 0) boundary conditions.
It is long known, however, that in order to respect the supersymmetry of the gauged SUGRA these scalars must obey an “alternate” (β = 0) quantization in which the roles of the leading and subleading fall-offs are reversed [64, 63]. In either case, that this state of the ABJM theory should be characterized by a deformation of the theory by a source for the scalar operator is surprising. 169
From an eleven dimensional perspective, this solution belongs to a class of spinning M2- branes very analogous to the continuous distributions of D3-branes discussed in [49]. As in that case, this class contains a privileged BPS configuration of M2-branes that is natural to associate holographically with a state on the Coulomb branch of the ABJM theory. From a four dimen- sional perspective, this configuration (called the 2QBH in chapter3) involves a bulk scalar that also appears in the 3QBH solution. That the Coulomb branch solution spontaneously breaks the global symmetry of the boundary theory is reflected holographically in the four dimensional scalar containing only a normalizable fall off near the boundary. Since the 3QBH can be reached from this Coulomb branch solution merely by tuning the rotation parameters of the M2 branes, it seems likely that despite appearances, the bulk scalar profile of the 3–charge solution also encodes only a spontaneous symmetry breaking.
A possible resolution to this puzzle appears in [65, 117]. There the authors demonstrate that by incorporating a manifestly SUSY covariant holographic renormalization scheme, a finite counterterm cubic in the scalars effectively shifts the identification of the source. Explicitly, in the conventions of the present chapter, they supplement the standard holographic counterterm action with a term of the form Z 1 3 3 Sfinite = d x √ γ φ . (6.15) 3√3 − Analyzing the variations of the corresponding on-shell action leads to a quantization condition modified by the finite counterterm, which can be written
β α2 = 0. (6.16) √3 − 6
For further details, see appendixB.
In [65], the authors find that such an improved quantization condition enables a succesful match between calculations of the ABJM free energy on S3 performed both holographically and from the field theory using localization techniques. For our purposes, we note that when (6.16) is applied to the asymptotic behavior of the bulk scalar (6.12), the dual source for the scalar operator vanishes identically. This is in line with our M2-brane intuition developed above. Consequently, 170 we will adopt these boundary conditions for the bulk scalar, and interpret the dual ABJM phase as a finite density state of strongly coupled matter in which a scalar operator has spontaneously acquired an expectation value.
We will primarily be interested in this phase at very low temperatures. In this case, the entropy density at fixed chemical potential (as well as the specific heat Cµ) vanishes like
16 s π2µ T + (T 3). (6.17) ≈ 3 O
The fact that the entropy vanishes at zero temperature is reflected in the absence of an extremal horizon in this limit. Instead, at T = 0 the geometry becomes singular in the IR. This singularity is of the “good” kind in the classification of [118], and it is expected that the singularity is resolved during the oxidation to eleven dimensions. A partial lift of this solution is discussed inA, where
2 it is shown that the singular IR geometry can indeed be resolved to a (non-singular) AdS3 R × region in one higher dimension.
That the singularity is “good” further implies that the zero temperature solution is contin- uously connected to a branch of charged black brane solutions. Thus, we anticipate that we can uncover features of the zero temperature background by taking a small black brane limit along this branch. In what follows, we will employ such a strategy for both computational convenience as well as to help clarify our results.
Another useful susceptibility controlled by the background solution is the uniform static charge susceptibility, χ0, defined at fixed temperature as
∂ρ 2 8π2T 2 + 9µ2 χ0 = . (6.18) p 2 2 2 ≡ ∂µ T 3 16π T + 9µ
At low temperatures, this susceptibility behaves like
64 4 4 6 χˆ0 2 + π Tˆ + (Tˆ ), (6.19) ≈ 81 O where hatted quantities are dimensionless ratios of the un-hatted quantity and the chemical poten- tial. The static charge susceptibility will play a central role in our investigation of the dual finite density phase of ABJM matter, and we will discuss this quantity in considerable detail below. 171
6.3 Fluctuations and Linear Response
The thermodynamic analysis of the preceding section reveals several interesting broad stroke features of the 3QBH’s holographic dual. In particular, the vanishing entropy density at zero temperature encourages questions about the nature of low energy excitations in the dual phase of matter. From the perspective of the field theory, a natural way to investigate the properties of these putative excitations is by way of the correlation functions for field theory operators. For example, in linear response theory considerable mileage can be gained from retarded Green’s functions, which can be used to construct various spectral functions, conductivities, and susceptibilities.
Holography provides an extremely efficient means of obtaining such retarded Green’s func- tions. In what is by now a textbook application of gauge/gravity duality, the field theory calculation is reformulated in the gravitational language as a boundary value problem for linearized bulk fluc- tuations (while not a textbook, a useful exposition appears in [87]). In a top down holographic approach such as the one we follow here, one can identify both fermionic and bosonic fluctuations whose dynamics is entirely determined by the underlying gauged SUGRA. These fluctuations are then related holographically to two point functions for various fermionic and bosonic operators of the ABJM theory, respectively.
Our immediate goal is to understand to what extent some interesting singularities in fermion spectral functions computed holographically in the 3QBH can be related to features in the retarded density-density correlation function. The fermion spectral functions previously appeared in chapter
3. We will quickly summarize the relevant results, before turning our attention to the density- density correlator and the computation of the non-uniform static susceptibility, which are new.
6.3.1 Fermion Spectral Functions
The 4D = 8 gauged SUGRA [25] contains 56 spin-1/2 fermions, and 8 gravitini. In N any given bosonic background, fluctuations of these fermionic modes will generically couple to one another. This mixing greatly complicates the analysis of linearized fermionic fluctuations in 172 supergravity theories.
Fortunately, for particular backgrounds built from bosonic fields that are invariant under some subgroup H SO(8), the situation is often much improved. As we have seen above, the ⊂ 3QBH is a solution within a truncation of the SUGRA to fields neutral under the H = U(1)4
Cartan of SO(8). Thus, by studying the weight vectors of the SUGRA fermions, one can easily distinguish subsets of the spin-1/2 fermions which can not mix either amongst themselves, or with the gravitini. Further details on this disentangling are given in chapter3.
Solving for these decoupled bulk fermions allows one to holographically compute Green’s functions of fermionic operators “ ψ” in the ABJM theory. Because the dual phase of matter is O at finite density, one can wonder whether the dual fermions which are charged under the active chemical potential(s) form Fermi surfaces. The existence of such a Fermi surface can be diagnosed from the analyticity properties of the fermion two point functions. On very general grounds [10,
41, 60], at low energies (measured from the chemical potential) the holographic fermion Green’s function in the vicinity of a bulk zero mode can be written
OψOψ Z GR (ω, k) = , (6.20) ω vF (k kF ) Σ(ω) − − − which mimics the familiar Landau Fermi liquid parametrization. Unlike the Landau Fermi liquid, the quasi-particle weight Z need not remain finite at the Fermi surface, nor does Σ necessarily contain subleading contributions. In fact since Σ is in general complex, it always provides the leading contribution to the width of any fermionic excitations. In any case a Fermi surface at momentum kF appears as a simple pole in the retarded fermionic Green’s function at ω = 0, signalling the presence of gapless fermionic modes at k = kF .
By constructing these Green’s functions holographically, we indeed found Fermi surface sin- gularities dual to bulk finite momentum fermion zero modes in chapter3. Moreover, by studying the properties of the Green’s function (and its related spectral function) at finite frequency, it was noted that in the 3QBH background these holographic Fermi surfaces are accompanied by perfectly stable fermionic excitations at low energies. 173
To characterize these excitations, one can compute the spectral function from the imaginary part of the correlator: i A(k, ω) = Tr (Gψψ Gψψ †). (6.21) 2 R − R The trace is taken over the spinor indices of the retarded Green’s function, which renders the spectral function rotationally invariant. In figure 6.1 an example of this spectral function computed holographically from the 3QBH is shown. This particular spectral function corresponds to the
ABJM fermions which display the largest fermi surface of those studied in this state in chapter3.
The spectral function is characterized by a region of infinitely long-lived fermionic excitations with a dispersion relation marked by the solid purple line. The Fermi surface is marked by large black dots, and is at kˆF kF /µ 0.81. ≡ ≈ For fluctuations with ω/µ > √3/4 (shaded bands in the figure) the normal mode pole in | | the correlator moves off the real axis and the fluctuations acquire a finite width. The fact that this kinematic regime of stability is present in these spectral functions led the authors of [55] to associate the stable region with a gap in which modes mediating interactions with the fermionic excitations are absent, as discussed in chapter3.
The holographic picture that emerges is thus that of a finite density state in which the strongly coupled fermionic degrees of freedom dual to “Class 2” SUGRA modes appear to have organized into a Fermi surface. The low energy fermionic excitations around this Fermi surface behave similarly to those in a Fermi gas: perfectly stable modes with linear dispersion near the
Fermi surface.
6.3.2 The Holographic Static Susceptibility Setup
As discussed in the previous subsection, the holographic fermion spectral function singulari- ties identified in the SUGRA fluctuation analysis can be interpreted as signaling the existence of a
Fermi surface built from ABJM fermions carrying charge under the global U(1) number densities.
In an attempt to better understand these singularities, it is natural to wonder whether or not they leave their mark in other field theory observables. 174
Figure 6.1: The spectral function A(ω, k) computed from “Class 2” fermions in the 3QBH back- ground. The solid lines indicated delta function peaks in the spectral weight. This weight broadens in the shaded bands, and has been omitted from the plot for clarity. 175
ρρ An obvious contender for one such observable is the density-density correlator GR (ω, k). In the static limit (ω 0), this correlation function quantifies the static susceptibility of the state. → In other words, as a function of momentum the static susceptibility measures the response of the charge density to a small amplitude, stationary charged perturbation.
To compute the static susceptibility holographically, we study fluctuations of the time com- ponent of the bulk gauge field subject to the appropriate boundary conditions. In the classical background of the 3QBH, linearized fermionic and bosonic fluctuations clearly decouple. How- ever, the symmetries of the background do not permit one to fully decouple the set of bosonic
fluctuations, and these modes will generically mix amongst each other.
As the bulk scalar is uncharged under the gauged U(1), on general grounds one anticipates a total of 2+2+1 = 5 bosonic gauge invariant modes. The obvious SO(2) isometry of the background will be broken by any finite momentum fluctuation. If we take this momentum to be along the x-direction, then the gauge invariant fluctuations can be organized into representations of the
2 unbroken Z2 SO(2) which labels the parity of the mode under y y. Fluctuations of the ⊂ → − temporal component of the gauge field are clearly even under this Z2, and will mix with the two other parity even modes. We collectively denote these modes +. Z
To construct explicitly the gauge invariant +, it is convenient to first write the bulk fields Z as background plus fluctuation
g =g ¯ + h A = A¯ + a and φ = φ¯ + ϕ (6.22)
and to adopt an appropriate plane wave ansatz for both the fluctuations ψI = h, a, ϕ and the { } gauge transformation parameters I = ξ, λ . Here ξ represents a diffeomorphism and λ denotes a { } U(1) gauge transformation. Schematically,
i(kx−ωt) i(kx−ωt) ψI (t, r, x, y) ψI (r)e and I I (r)e . (6.23) ∼ ∼
By studying the action of the Lie derivative along ξ and the U(1) gauge transformations
2 Note that although the gauged SUGRA action contains a topological term proportional to F ∧ F , this term couples to the axions. These axions vanishes in the 3QBH solution, and thus this term does not affect the linearized fluctuation analysis. 176 parametrized by λ on these plane wave fluctuations, it is easy to construct modes that are invariant under both. In the static limit, one choice is given by + = s, 1, 2 where Z {Z Z Z } φ¯0 s = ϕ 0 hyy (6.24) Z − g¯yy 0 g¯tt 1 = htt 0 hyy (6.25) Z − g¯yy Φ0 2 = at 0 hyy (6.26) Z − g¯yy and a 0 has been used to denote a radial derivative.
In our analysis, the gauge invariant + are primarily important because they demonstrate Z how to properly impose consistent boundary conditions on the bulk fluctuations. In practice, we
2χ work directly in terms of the ψI , changing variables slightly so that h = e h˜. A technical point, examined in more detail in appendixB, is that the radial coordinate in which the background solution (6.7) is presented does not coincide with the Fefferman-Graham coordinate u near the boundary. This slightly complicates the identification of “normalizable” and “non-normalizable” modes of the various fluctuations.
The upshot of this boundary analysis is that for the bulk fluctuations with near boundary behavior
f1 f2 ϕ = + + ... r r2
h˜tt =ht0 + ...
h˜yy =hy0 + ...
a1 at =a0 + + ... (6.27) r a solution which corresponds holographically to turning on a source for only the charge density must obey the UV (r ) boundary conditions → ∞ Q f2 + f1 = (ht0 + hy0) = 0 and a0 = 1. (6.28) 4
In order to better understand the analytic structure of the Green’s function we compute holographically, it is advantageous to construct this correlation function at finite temperature, and 177 study its behavior in the limit Tˆ 0. From the gravitational perspective, this implies that we → will be interested primarily in perturbations to background solutions with a horizon. In this case, the correct IR (r rH ) boundary condition is regularity of the fluctuation at the horizon. Again, → further details are provided in appendixB.
Given a solution to the fluctuation equations of motion that is both regular at the horizon and satisfies (6.28) at the boundary, a straightforward application of the holographic dictionary gives the static susceptibility as
∂u at 1 p χ(k) = lim √ g = a1 (Q + rH ) Q(Q + rH ) hy0 . (6.29) − u→∞ − at − 2
We now turn to the computation of this susceptibility, and a subsequent discussion of our results.
6.4 Results
Our main results are summarized in figure 6.2. The rightmost plot shows the static sus- ceptibility evaluated at real momenta. Importantly, the zero momentum limit is shown to be in excellent agreement with (6.19), demonstrating consistency between the fluctuation analysis and the background thermodynamics. For momenta large compared to the chemical potential, the fluc- tuations are primarily supported at large radii (near the boundary) and thus inherit the conformal characteristics of the UV fixed point. This is manifest in the linear rise of the susceptibility at large kˆ.
The leftmost plot of the same figure shows clearly the non-analyticities present in the static susceptibility at low temperature. At finite temperature these non-analyticities appear as a discrete set of poles which coalesce as the temperature is lowered. At zero temperature, they trace branch cuts in the complex momentum plane, which terminate at branch points naturally distinguished by whether or not Re kˆ = 0.
By eye, the branch points with Re kˆ = 0 appear to lie very close to the location kˆF 0.81 6 ≈ of the Fermi surface identified via the fermion spectral functions evaluated in the same state. This is again reminiscent of the results from [115], in which branch points in the susceptibility occur 178
Figure 6.2: The low temperature static susceptibility along the real kˆ k/µ axis (right) and in ≡ the complex kˆ–plane (left). The susceptibilities pictured are computed at a temperature Tˆ = 10−5. In the rightmost plot, a point at kˆ = 0 shows the location of the uniform static susceptibility computed from the background, in excellent agreement with the fluctuation analysis. 179
0.78 2.5
0.76 * * 2.0 Rek 0.74 Imk
0.72 1.5
0.70 1.0 10-7 10-5 0.001 0.100 10-7 10-5 0.001 0.100 T T
Figure 6.3: The location of the branch points with Re kˆ = 0 as a function of temperature. 6
? at Re kˆ kˆF . As in that case, however, the match is not exact. In figure 6.3, the location of ≈ the branch points corresponding to the Re kˆ = 0 class is plotted against temperature. Evidently, 6 ? at low temperatures these branch points approach kˆ = kˆR + i kˆI 0.75 + 1.15 i. As the | | | | ≈ temperature is increased beyond Tˆ 0.1, the real part of the branch point quickly decreases. This ∼ behavior is anticipated from the analogous calculations in AdS–RN, in which it was shown that for temperatures large compared to the chemical potential the singularities in the static susceptibility are aligned along the imaginary kˆ axis.
Perhaps most interestingly, figure 6.2 indicates a second class of branch cut persisting at very low temperatures along the imaginary kˆ axis. This feature was absent in the state dual to
AdS–RN, and has important implications. Most notably, the magnitude of the imaginary part of these branch points is smaller than those at finite Re kˆ. This observation, coupled with the fact that these branch points have Re kˆ? = 0 signals an absence in long distance oscillations in the charge density induced by a charged perturbation.
One can demonstrate this explicitly by studying this induced charge density n in the linear response limit of the gauge theory. In this limit, a charged impurity δQ deforms the charge density like Z d2k n(R) = eikR χ(k) δQ(k). (6.30) (2π)2
As long as the impurity is relatively well behaved, the induced charge density will be dominated 180
0.3 0.4 ) ) 0.2
0.3 ( ℜ ( ℜ n * n 0.2 0.1 Im k ⅇ
0.1 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 ℜ ℜ
Figure 6.4: The linearized charge density response due to the introduction of a charged impurity at low temperature , Tˆ = 10−5 (left). No oscillations are easily discernible at large distances. To observe the effects of the branch points with Re kˆ? = 0, it is helpful to remove the exponential 6 damping from the branch point along the imaginary axis (right).
by contributions to the integral coming from the branch points in the susceptibility. Choosing for
2 the sake of simplicity a gaussian δQ(R) = e−R /2, the low temperature induced charge density is readily obtained by numerically integrating (6.30), and the result is plotted in figure 6.4. Here R is a radial coordinate in the boundary theory, and should not be confused with the bulk radial direction. Unlike the state dual to AdS–RN, at large distances (R 1) the response is dominated by the branch point on the imaginary axis and no analogue of Friedel oscillations are present.
6.5 Discussion
6.5.1 Singularities and Scaling Exponents
Given the slight mismatch between the poles of fermion two point functions and the non- analyticities present in the static susceptibility, it would be advantageous to have an analytical handle on the heritage of one (or both) of these features. A striking observation made by [115] was an apparent connection between the branch points in their numerical result and those of the
− “semi-local quantum critical scaling exponent”, νk . Although the 3Q-black hole lacks an extremal horizon, it nevertheless inherits some of the interesting properties of extremal AdS–RN as a consequence of an IR geometry that is conformal 181
2 to AdS2 R . Importantly, [86] demonstrated that longitudinal fluctuations of the gauge field in × − this IR geometry are characterized by a set of three scaling exponents that generalize the νk . In the conventions we adopt here, these exponents are given by r 4 ν0(k) = 1 + kˆ2 || 3 r 1 q ν±(k) = 11 + 4kˆ2 8 1 + kˆ2. (6.31) || √3 ±
0 − The exponents ν|| and ν|| have particularly interesting branch points at
i 4 ? √3 ? 57 1 (±π±arctan √ ) kˆ = i and kˆ = ( ) 4 e 2 3 (6.32) ± 2 16 respectively. The signs in the branch points of ν− are not correlated (i.e. there are four such ± || branch points). The argument of these scaling exponents is plotted in figure 6.5, where one can note very good agreement between the numerically obtained analytic structure of static susceptibility at low temperatures and the analytically derived scaling exponents in (6.31).
ˆ? − The similarity between Re k from ν|| and the large kF from the fermion spectral functions in the AdS–RN and 3QBH backgrounds is surprising on several levels. Foremost, from the perspective of the linearized fermion zero mode calculations, the existence and location of a holographic Fermi surface depends intimately on the details of the couplings in the bulk Dirac equation. These couplings in turn are generically functions of the background fields. Changing these couplings by (1) deformations can radically alter the analytic structure of the fermion spectral function. O However the holographic computation of the static susceptibility is agnostic to the details of these couplings, as it obviously depends only on a bosonic subsector of the linearized fluctuations. That these two computations should yield results that possess any commonalities is therefore rather unexpected from the supergravity perspective. We return to this point shortly.
Moreover, the ν|| that characterize the static susceptibility are not the same as the analogous exponents that govern the IR properties of the fermion correlation functions. Nor do they share the same analytic structure in the complex momentum plane. In the extremal AdS–RN background, for example, spin-1/2 supergravity spinors that allow for zero modes are controlled in the IR by a 182
0 − Figure 6.5: The analytic structure of ν|| (left) and ν|| (right) in the complex momentum plane. The plots show the argument of the scaling exponent, which clearly reveals the existence of two classes of branch points. Both of these classes are present in the susceptibility. Numerically, they occur at kˆ? 0.87 i and kˆ? 0.75 1.15 i in excellent agreement with the numerically computed ≈ ± ≈ ± ± susceptibility. 183 scaling exponent of the form r 2 1 νF = 2kˆ . (6.33) − 12
This exponent clearly has branch points along the real kˆ axis, and in fact hints at the existence of an instability of the extremal solution to the formation of an “electron star” (see e.g. [38]). More relevant to the current investigation, these branch points occur at kˆ? = 1/(2√6), which does not ± seem to have much to do with the Fermi momentum—especially when compared to the branch
− points of νk evaluated in the same state (see discussion around (6.4)). Perhaps even more dramatically, analogous supergravity spinors in the 3QBH do not readily appear to possess any such scaling exponent in the deep IR. Their behavior is instead dictated by the appearance of a momentum independent “gap” in the static limit. In neither the 3QBH nor the extremal AdS–RN solution are fermion Dirac equations influenced by the ν|| in their IR limit.
Thus we arrive at one of the most important lessons of our calculation: at leading order in the strong coupling, large N limit, certain fermionic and bosonic correlation functions in these finite
? density states of the ABJM theory appear to identify very similar length scales 1/kF 1/Re k ≈ in holographic computations that are outwardly independent of one another. The details of these correlation functions—a fermion spectral function in one case, and a static susceptibility in the other—encourage an identification between this length scale and the existence of a Fermi surface in the ABJM phase, lending some support to the picture advocated in [47]. We will criticize this identification shortly.
The present calculation illuminates several other interesting features of holographic matter.
Among these, it is somewhat curious that the addition of the bulk scalar to the gravitational
0 theory is at once responsible for opening of a new channel of fluctuation (that associated to ν||) which destroys the long distance oscillations in the charge density, while simultaneously allowing for a stable region in the fermion spectral function where infinitely long-lived fermionic excitations propagate. Naively, one might expect that such a stable region would enhance charge oscillations at long distance, but we find this is not the case. 184
It is important to emphasize that the majority of the lessons we have learned were a conse- quence of our adherence to a “top-down” implementation of the holographic dictionary. By working with bulk equations of motion that are fully constrained by the underlying supergravity theory, we can confidently work within a consistent holographic framework. It is only in this context that we can attempt quantitative matches between the features of fermion spectral functions and static susceptibility in states of ABJM theory.
A trivial manifestation of this was the importance of identifying the correct boundary con- ditions for the fluctuations of the bulk scalar. This identification was especially noteworthy in the present setting, as the quantization of the scalar was influenced by the existence of finite boundary counterterms imposed by the supersymmetry of the boundary theory. Altering these boundary conditions artificially would have led to a mismatch both in the uniform (kˆ 0) limit of the static → susceptibility, as well as the location of the branch points identified numerically relative to those of the ν||.
6.5.2 Commentary
If any connection between the branch points of the ν|| and the Fermi surface poles in the fermionic two-point functions is to hold, then there is some tension to be resolved. To begin, the fact that the branch points in the susceptibility seem most closely related to kF and not 2kF is puzzling and yearns for further explanation. On one hand, it is no great surprise that the singularities in both correlation functions are (µ) in all examples. At zero temperature, the chemical potential O provides the only relevant scale in the background. This is then roughly the scale at which bulk
fluctuations distinguish between IR and UV portions of the geometry. For static fluctuations with momenta sufficiently large compared to the chemical potential, the fluctuation probes the
UV AdS4, a conformally invariant critical point. In the vicinity of this point, static correlation functions are analytic at non-zero momentum. Accordingly, if any zero modes are to arise one typically expects them to do so at small to moderate values of momentum in units of the chemical potential. Thus, while it seems unlikely to us, we can not currently rule out the possibility that the 185
“1kF ” singularities occur at approximately the same momentum simply by numerical coincidence.
If we instead entertain the possibility that the singularities in the fermion correlators and the charge susceptibility are related, then why might it be that Re kˆ kF as opposed to Re kˆ = kF ? ≈ One possibility is that this effect can be attributed to operator mixing between the charge density and the energy and pressure densities, along with (in the present case) the spinless operator dual to the bulk scalar. This mixing is captured holographically by couplings between the linearized
fluctuations of the corresponding bulk modes. Such couplings can in principle open new modes as well as deform normal modes that were present in the decoupled system, shifting their eigenvalues.
Some examples of this well known phenomenon with a holographic lean appear in e.g. [87].
On the other hand, as reviewed in the introduction, the “2” in 2kF is essentially a geometric consequence of gapless excitations at a sharp Fermi surface. What might it mean for holographic response that the branch points in the static susceptibility have a real part that is closer to kF than
2kF from the perspective of fermionic correlators in the same state?
One possibility is that the “Fermi surface singularities” in the fermion two-point functions are not in fact indicative of a sharp Fermi surface in the dual phase. This could be the case if, for example, the zero mode they detect is actually due to an ABJM scalar that is bound to an ABJM fermion in the gauge invariant operator dual to the bulk fermion. In the schematic classification of chapter3, for example, the “Class 2” bulk fermions emphasized in this chapter are dual to a gauge theory operator of the form Trλ1Zi for i = 2, 3, 4. In this labeling, λ is a complex combination of gauge theory fermions while the Z are complex scalars.
In charged black brane backgrounds, scalar zero modes are very often marginal modes that sit precisely at the boundary of a spatially modulated instability. Examples of this behavior are plentiful—some noteworthy realizations include [119, 120, 10]. However the bulk fermion zero modes do not typically suffer the same fate. As demonstrated in [10], in contrast to the scalar instabilities, tuning k away from kF does not in general result in the fermionic zero mode migrating into the upper half frequency plane. Thus, if the singularities in the fermionic correlators are actually reflecting the behavior of the Zi, the dual ABJM phase would be characterized by gapless 186 scalar modes gathered at the “1kF ” momentum. This seems like a strange state of affairs in a translationally invariant phase that is stable against these modulated scalar fluctuations (at least in the large N limit).
Another possibility is that while the 1kF singularities in the fermion two-point functions are in fact signaling the existence of a sharp Fermi surface, the large bath of charged scalars radically alters the dominant interaction channels available to charged excitations. If the interactions were such that parallel patches of the Fermi surface were effectively inaccessible to one another, it is plausible that the susceptibilities would be dominated instead by wave vectors at or very near kF . Alternatively, it might be that the important physics of the fermionic correlation function computed holographically is actually that of a “two particle” correlator. Such an interpretation would appeal to the holographic identification of the bulk spin-1/2 mode with a boundary operator of the form ψ TrλZ. In this light, bulk fermion zero modes might themselves simply be O ∼ signals of 2kF singularities in the “ ψ static susceptibility”. It would be very interesting to further O explore the tenability of such phenomena in a toy model of interacting bosons and fermions outside of holography.
In chapter3, in addition to the large Fermi surface that held our attention above, a further two
Fermi momenta were detected in the set of fermion spectral functions from the 3QBH. Evidently,
− no accompanying branch points arise in either ν|| or the static susceptibility. A sketch towards one possible resolution of this apparent discrepancy might begin with the following observation: the additional Fermi momenta identified in chapter3 are much smaller than the large Fermi surface pictured in section 6.3.1. Thus, by a Luttinger-like argument, it is natural
< to anticipate that these smaller Fermi surfaces, kF , control only
< < < 2 gf q kF 1 2 > > > = 3 (0.34) 0.1 (6.34) gf · q · kF · 3 · ≈
> of the fermionic charge density relative to the larger kF . The multiplicative factor of 3 accounts for the relative degeneracy of the Fermi surfaces, while the factor of 1/3 accounts for the differing charges of the ABJM fermions under the active U(1). These factors are chosen to correspond to 187 the larger of the two small Fermi surfaces. It is thus conceivable that the influence of the charged degrees of freedom around these smaller Fermi surfaces on the static susceptibility is completely overshadowed by the effects of the larger Fermi surface and/or other charged modes in the dual state.
Yet another curious observation is that in both the AdS–RN and 3QBH backgrounds the non-analyticities in the static susceptibility appear to be controlled exclusively by the IR region of the geometry. On general grounds, this is not guaranteed to be the case. As emphasized in [86], the static susceptibility is holographically realized as a static fluctuation calculation, and is thus in principle sensitive to the entirety of the bulk geometry. This a priori leaves open the possibility for the existence of (normalizable) zero modes in the longitudinal fluctuations at real values of spatial momentum. Evidently such zero modes do not exist in these backgrounds. The presence of such zero modes would generically imply both long distance oscillations and power law decay in the induced charge density, and thus it would be very interesting to find holographic examples of this behavior.
Additionally, it is interesting to wonder to what extent the somewhat rough connection we observed between fermion response and static susceptibility is present in other holographic phases of matter. One modest point in favor of such a connection appears in the (top-down) computation of fermion spectral functions in certain ground states of ABJM that break the global U(1) associated with conserved charge density [19]. In stark contrast to analogous bottom-up calculations, it was found that such fermion spectral functions showed no sign of a Fermi surface in any of the states studied.
Although the symmetry breaking was spontaneous in some instances and explicit in others, a common feature of the bulk description of the holographic phase was a horizonless geometry that took the form of a domain wall interpolating between the maximally symmetric AdS4 in the UV and a distinct AdS4 in the IR. Importantly, an IR AdS4 does not permit fluctuations controlled by a momentum dependent critical scaling exponent—there is no analogue of ν|| in such geometries— and one might be tempted to guess that the static susceptibility computed holographically in such 188 a background would show no non-analytic behavior at momenta with Re kˆ? = 0. If such a scenario 6 were born out by explicit computation it would provide further support for a link between poles in the fermionic two-point functions and certain branch points in the static susceptibility.
Other avenues to pursue might include generalizing the analysis to move along the one pa- rameter branch of solutions that interpolate between AdS–RN and the 3QBH. These are the 3+1
Q–black brane solutions of [57] in which all four generators of the Cartan of SO(8) are active. The naming reflects the fact that along this branch, generically 3 of the charges are set equal to one another, while the fourth is distinct. In this language, the “+1” charge takes one from the 3+0
Q–black brane studied here to the 4Q (AdS–RN) solution in which all four charges are set equal.
Fermion response from supergravity was studied in these backgrounds in chapter3, where a fairly rich array of behaviors in the dual states of the ABJM theory was described. These include states with multiple Fermi surfaces, or even thick Fermi “shells”. Understanding if or how these features are reflected in charge density correlation functions would be a logical and potentially illuminating continuation of the line of research we’ve advanced here.
Alternatively, from the perspective of the gauged SUGRA, one may wonder if there is a deeper symmetry at play which (approximately) relates branch points in the susceptibility to poles in the fermion Green’s functions. In a supersymmetric background, bulk fermion and boson modes related by supersymmetry will share the same analytic structure in their dual correlation functions.
A nice example of this appears in [50]. Although the 3QBH preserves no supersymmetry, one can nevertheless ask whether it is possible that the supersymmetry is broken sufficiently softly such that an approximate relationship still holds in these backgrounds. Again, this question could be explored in more detail by moving along a branch of solutions in the U(1)4 SO(8) truncation of ⊂ the gauged SUGRA. Starting from the “2+0” solution, which is supersymmetric, and turning on a small amount of “+2” charge may allow one to gather some intuition for the effects of broken supersymmetry on the analytic structure of holographic correlation functions. Chapter 7
Concluding Comments
As we now reach the end of this dissertation, let us take stock of some of the things we have learned. Starting in chapter1, we argued that string theory reveals a quite unexpected property of certain QFTs: they describe, and are described by, a gravitational theory in one more (non- compact) dimension. These holographic dualities can probably be counted among the stranger things to emerge from QFT. Considering the time it took for them to be discovered, it is tempting to wonder what other secrets might still be hiding deep inside various path integrals.
Having introduced these ideas, we reviewed some important properties of the two dual theories used in this dissertation in chapter2, and then proceeded to applications. The questions we asked were inspired mainly by open problems in condensed matter physics, and led us to study states of ABJM theory at strong interaction and non-zero density. Here, our holographic tools offered a controlled perturbative expansion in a limit where conventional quasiparticle descriptions are lacking. This makes them potentially well-suited for answering questions about, for example, the strange metals appearing in the phase diagrams of many high-temperature superconductors.
With this in mind, we used holographic techniques to study fermionic two-point functions, and associated spectral functions, in a variety of holographic states. Unlike much of the literature on applied holography, we mainly stayed within the top-down framework provided to us by string theory. This allowed us to discuss the dual QFT from a more concrete point of view; for example, we were able to specify more or less precisely which operators were dual to the different supergravity
fields. Furthermore, it led us to some generalizations of bottom-up constructions, and also seemed 190 to select for certain particularly interesting behaviors. Examples of such behaviors are the non-
Fermi liquid fermions of chapter3, and the gapped fermions of chapter5.
One recurring theme throughout the dissertation was the sharp effect that the deep interior of the bulk geometry had on fermionic spectral functions. As we have discussed, this interior region
(often, but not always, a black hole horizon) can be thought of as an emergent low-energy theory that acts as a bath into which the fermions can decay. These observations have previously been distilled into the semi-holographic construction of [121]. There it was shown that several key features of holographic two-point functions can be reproduced by starting from a free fermion and coupling it to a large-N strongly interacting sector. It is interesting to ask if such a construction might be applicable to a more general class of strongly interacting systems.
Many open questions about the nature of holographic states of matter remain to be answered.
In chapter6, we investigated one such question from a top-down perspective, namely, whether there exists any traces of Fermi surfaces in the susceptibility of holographic states. While some interesting connections were observed, a clear link could not be established. Perhaps there is none at leading order in N — at strong coupling, one has to be prepared to give up on some intuition originating at weak coupling. If a link exist, however, it seems clear that it can only be found from a top-down point of view. Another question that might benefit from further top-down studies concerns the nature of the fermions responsible for the Fermi surface singularities. Since the fermionic operators under study are by necessity gauge invariant composites of “elementary” fields, it is unclear if the Fermi surfaces should be thought of as made up of the elementary fermions (“gauginos”) or fermion-scalar bound states (“mesinos”) [47, 122]. It has been argued that top-down computations support the gaugino picture [92], but further work is needed to settle the matter.
Overall, holographic dualities have proven able to capture a fascinating range of physical phenomena. There are many questions left to attack using these techniques, and we believe that the answers are likely to increase our understanding of quantum field theory in general, and strongly interacting quantum matter in particular, while also informing experimental work. And on that positive note, we end. Bibliography
[1] Neil Lambert. M-Theory and Maximally Supersymmetric Gauge Theories. Ann. Rev. Nucl. Part. Sci., 62:285–313, 2012.
[2] Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. N=6 supercon- formal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP, 10:091, 2008.
[3] Juan Martin Maldacena. The Large N limit of superconformal field theories and supergravity. Adv.Theor.Math.Phys., 2:231–252, 1998.
[4] S.S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. Gauge theory correlators from noncritical string theory. Phys.Lett., B428:105–114, 1998.
[5] Edward Witten. Anti-de Sitter space and holography. Adv.Theor.Math.Phys., 2:253–291, 1998.
[6] P. Kovtun, Dan T. Son, and Andrei O. Starinets. Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett., 94:111601, 2005.
[7] Michal P. Heller. Holography, Hydrodynamization and Heavy-Ion Collisions. Acta Phys. Polon., B47:2581, 2016.
[8] Joseph Polchinski. Effective field theory and the Fermi surface. In Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, pages 235–276, 1992.
[9] John McGreevy. Viewpoint: In pursuit of a nameless metal. APS Physics, 3:83, 2010.
[10] Thomas Faulkner, Hong Liu, John McGreevy, and David Vegh. Emergent quantum criticality, Fermi surfaces, and AdS2. Phys. Rev., D83:125002, 2011.
[11] Steven S. Gubser. Breaking an Abelian gauge symmetry near a black hole horizon. Phys. Rev., D78:065034, 2008.
[12] Steven Weinberg. Superconductivity for Particular Theorists. Prog. Theor. Phys. Suppl., 86:43, 1986.
[13] Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen. Fermi surfaces in N=4 Super- Yang-Mills theory. Phys.Rev., D86:106002, 2012. 192
[14] Nabil Iqbal, Hong Liu, and Mark Mezei. Semi-local quantum liquids. JHEP, 1204:086, 2012.
[15] Steven S. Gubser, Fabio D. Rocha, and P. Talavera. Normalizable fermion modes in a holo- graphic superconductor. JHEP, 10:087, 2010.
[16] Thomas Faulkner, Gary T. Horowitz, John McGreevy, Matthew M. Roberts, and David Vegh. Photoemission ’experiments’ on holographic superconductors. JHEP, 03:121, 2010.
[17] Oliver DeWolfe, Oscar Henriksson, and Christopher Rosen. Fermi surface behavior in the ABJM M2-brane theory. Phys. Rev., D91(12):126017, 2015.
[18] Oliver DeWolfe, Steven S. Gubser, Oscar Henriksson, and Christopher Rosen. Fermionic Re- sponse in Finite-Density ABJM Theory with Broken Symmetry. Phys. Rev., D93(2):026001, 2016.
[19] Oliver DeWolfe, Steven S. Gubser, Oscar Henriksson, and Christopher Rosen. Gapped Fermions in Top-down Holographic Superconductors. Phys. Rev., D95(8):086005, 2017.
[20] Oliver DeWolfe. Private communication (unpublished).
[21] B. Biran, A. Casher, F. Englert, M. Rooman, and P. Spindel. The Fluctuating Seven Sphere in Eleven-dimensional Supergravity. Phys.Lett., B134:179, 1984.
[22] L. Castellani, R. D’Auria, P. Fre, K. Pilch, and P. van Nieuwenhuizen. The Bosonic Mass Formula for Freund-rubin Solutions of d = 11 Supergravity on General Coset Manifolds. Class.Quant.Grav., 1:339–348, 1984.
[23] A. Casher, F. Englert, H. Nicolai, and M. Rooman. The Mass Spectrum of Supergravity on the Round Seven Sphere. Nucl.Phys., B243:173, 1984.
[24] B. de Wit and H. Nicolai. N=8 Supergravity with Local SO(8) x SU(8) Invariance. Phys.Lett., B108:285, 1982.
[25] B. de Wit and H. Nicolai. N=8 Supergravity. Nucl.Phys., B208:323, 1982.
[26] B. de Wit and H. Nicolai. The Consistency of the S**7 Truncation in D=11 Supergravity. Nucl.Phys., B281:211, 1987.
[27] Jonathan Bagger and Neil Lambert. Modeling Multiple M2’s. Phys.Rev., D75:045020, 2007.
[28] Andreas Gustavsson. Algebraic structures on parallel M2-branes. Nucl. Phys., B811:66–76, 2009.
[29] Jonathan Bagger and Neil Lambert. Gauge Symmetry and Supersymmetry of Multiple M2- Branes. Phys. Rev., D77:065008, 2008.
[30] Jonathan Bagger and Neil Lambert. Comments on multiple M2-branes. JHEP, 0802:105, 2008.
[31] Igor R. Klebanov and Giuseppe Torri. M2-branes and AdS/CFT. Int. J. Mod. Phys., A25:332– 350, 2010.
[32] Jonathan Bagger, Neil Lambert, Sunil Mukhi, and Constantinos Papageorgakis. Multiple Membranes in M-theory. Phys.Rept., 527:1–100, 2013. 193
[33] Marcus Benna, Igor Klebanov, Thomas Klose, and Mikael Smedback. Superconformal Chern- Simons Theories and AdS(4)/CFT(3) Correspondence. JHEP, 09:072, 2008.
[34] Eric D’Hoker and Boris Pioline. Near extremal correlators and generalized consistent trun- cation for AdS(4—7) x S**(7—4). JHEP, 0007:021, 2000.
[35] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein. Phenomenology of the normal state of Cu-O high-temperature superconductors. Phys. Rev. Lett., 63:1996, 1989.
[36] P. W. Anderson. ”Luttinger-liquid” behavior of the normal metallic state of the 2D Hubbard model. Phys. Rev. Lett., 64:1839, 1990.
[37] P. Gegenwart, Q. Si, and F. Steglich. Quantum criticality in heavy-fermion metals. Nature Physics, 4:186, 2008.
[38] Sean A. Hartnoll. Horizons, holography and condensed matter. eprint arXiv:1106.4324 [hep-th], 2011.
[39] Sung-Sik Lee. A Non-Fermi Liquid from a Charged Black Hole: A Critical Fermi Ball. Phys.Rev., D79:086006, 2009.
[40] Hong Liu, John McGreevy, and David Vegh. Non-Fermi liquids from holography. Phys. Rev., D83:065029, 2011.
[41] Mihailo Cubrovic, Jan Zaanen, and Koenraad Schalm. String Theory, Quantum Phase Tran- sitions and the Emergent Fermi-Liquid. Science, 325:439–444, 2009.
[42] Mohammad Edalati, Robert G. Leigh, Ka Wai Lo, and Philip W. Phillips. Dynamical Gap and Cuprate-like Physics from Holography. Phys.Rev., D83:046012, 2011.
[43] Mohammad Edalati, Robert G. Leigh, and Philip W. Phillips. Dynamically Generated Mott Gap from Holography. Phys.Rev.Lett., 106:091602, 2011.
[44] Jerome P. Gauntlett, Julian Sonner, and Daniel Waldram. Universal fermionic spectral func- tions from string theory. Phys. Rev. Lett., 107:241601, 2011.
[45] Raphael Belliard, Steven S. Gubser, and Amos Yarom. Absence of a Fermi surface in classical minimal four-dimensional gauged supergravity. JHEP, 1110:055, 2011.
[46] Jerome P. Gauntlett, Julian Sonner, and Daniel Waldram. Spectral function of the super- symmetry current. JHEP, 11:153, 2011.
[47] Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen. Fermi Surfaces in Maximal Gauged Supergravity. Phys.Rev.Lett., 108:251601, 2012.
[48] A. Brandhuber and K. Sfetsos. Wilson loops from multicenter and rotating branes, mass gaps and phase structure in gauge theories. Adv.Theor.Math.Phys., 3:851–887, 1999.
[49] D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner. Continuous distributions of D3-branes and gauged supergravity. JHEP, 07:038, 2000. 194
[50] Massimo Bianchi, Oliver DeWolfe, Daniel Z. Freedman, and Krzysztof Pilch. Anatomy of two holographic renormalization group flows. JHEP, 0101:021, 2001.
[51] Massimo Bianchi, Daniel Z. Freedman, and Kostas Skenderis. How to go with an RG flow. JHEP, 0108:041, 2001.
[52] Massimo Bianchi, Daniel Z. Freedman, and Kostas Skenderis. Holographic Renormalization. Nucl. Phys., B631:159–194, 2002.
[53] Steven S. Gubser and Fabio D. Rocha. Peculiar properties of a charged dilatonic black hole in AdS5. Phys. Rev., D81:046001, 2010.
[54] Norihiro Iizuka, Nilay Kundu, Prithvi Narayan, and Sandip P. Trivedi. Holographic Fermi and Non-Fermi Liquids with Transitions in Dilaton Gravity. JHEP, 1201:094, 2012.
[55] Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen. Fermionic response in a zero entropy state of = 4 super-Yang-Mills. Phys. Rev., D91(4):046011, 2015. N [56] M. J. Duff and James T. Liu. Anti-de Sitter black holes in gauged N = 8 supergravity. Nucl. Phys., B554:237–253, 1999.
[57] Mirjam Cvetic, M.J. Duff, P. Hoxha, James T. Liu, Hong Lu, et al. Embedding AdS black holes in ten-dimensions and eleven-dimensions. Nucl.Phys., B558:96–126, 1999.
[58] James Alsup, Eleftherios Papantonopoulos, George Siopsis, and Kubra Yeter. Duality be- tween zeroes and poles in holographic systems with massless fermions and a dipole coupling. Phys. Rev., D90(12):126013, 2014.
[59] Garrett Vanacore and Philip W. Phillips. Minding the Gap in Holographic Models of Inter- acting Fermions. eprint arXiv:1405.1041 [cond-mat.str-el], 2014.
[60] Steven S. Gubser and Jie Ren. Analytic fermionic Green’s functions from holography. Phys.Rev., D86:046004, 2012.
[61] Nabil Iqbal and Hong Liu. Real-time response in AdS/CFT with application to spinors. Fortsch.Phys., 57:367–384, 2009.
[62] Igor R. Klebanov and Edward Witten. AdS / CFT correspondence and symmetry breaking. Nucl.Phys., B556:89–114, 1999.
[63] Peter Breitenlohner and Daniel Z. Freedman. Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity. Phys.Lett., B115:197, 1982.
[64] Peter Breitenlohner and Daniel Z. Freedman. Stability in Gauged Extended Supergravity. Annals Phys., 144:249, 1982.
[65] Daniel Z. Freedman and Silviu S. Pufu. The holography of F -maximization. JHEP, 03:135, 2014.
[66] B. Pioline and J. Troost. Schwinger pair production in AdS(2). JHEP, 0503:043, 2005.
[67] Vijay Balasubramanian, Jan de Boer, Vishnu Jejjala, and Joan Simon. Entropy of near- extremal black holes in AdS(5). JHEP, 0805:067, 2008. 195
[68] R. Fareghbal, C.N. Gowdigere, A.E. Mosaffa, and M.M. Sheikh-Jabbari. Nearing Extremal Intersecting Giants and New Decoupled Sectors in N = 4 SYM. JHEP, 0808:070, 2008.
[69] M.M. Sheikh-Jabbari and Hossein Yavartanoo. EVH Black Holes, AdS3 Throats and EVH/CFT Proposal. JHEP, 1110:013, 2011.
[70] Maria Johnstone, M.M. Sheikh-Jabbari, Joan Simon, and H. Yavartanoo. Near-Extremal Vanishing Horizon AdS5 Black Holes and Their CFT Duals. JHEP, 1304:045, 2013.
[71] Sean A. Hartnoll, Christopher P. Herzog, and Gary T. Horowitz. Building a Holographic Superconductor. Phys. Rev. Lett., 101:031601, 2008.
[72] Sean A. Hartnoll, Christopher P. Herzog, and Gary T. Horowitz. Holographic Superconduc- tors. JHEP, 12:015, 2008.
[73] Steven S. Gubser and Abhinav Nellore. Low-temperature behavior of the Abelian Higgs model in anti-de Sitter space. JHEP, 04:008, 2009.
[74] Steven S. Gubser and Abhinav Nellore. Ground states of holographic superconductors. Phys. Rev., D80:105007, 2009.
[75] Gary T. Horowitz and Matthew M. Roberts. Zero Temperature Limit of Holographic Super- conductors. JHEP, 11:015, 2009.
[76] Steven S. Gubser, Christopher P. Herzog, Silviu S. Pufu, and Tiberiu Tesileanu. Supercon- ductors from Superstrings. Phys. Rev. Lett., 103:141601, 2009.
[77] Jerome P. Gauntlett, Julian Sonner, and Toby Wiseman. Holographic superconductivity in M-Theory. Phys. Rev. Lett., 103:151601, 2009.
[78] Steven S. Gubser, Silviu S. Pufu, and Fabio D. Rocha. Quantum critical superconductors in string theory and M-theory. Phys. Lett., B683:201–204, 2010.
[79] Jerome P. Gauntlett, Julian Sonner, and Toby Wiseman. Quantum Criticality and Holo- graphic Superconductors in M-theory. JHEP, 02:060, 2010.
[80] Martin Ammon, Johanna Erdmenger, Matthias Kaminski, and Andy O’Bannon. Fermionic Operator Mixing in Holographic p-wave Superfluids. JHEP, 05:053, 2010.
[81] Nikolay Bobev, Arnab Kundu, Krzysztof Pilch, and Nicholas P. Warner. Minimal Holographic Superconductors from Maximal Supergravity. JHEP, 03:064, 2012.
[82] Thomas Fischbacher, Krzysztof Pilch, and Nicholas P. Warner. New Supersymmetric and Stable, Non-Supersymmetric Phases in Supergravity and Holographic Field Theory. eprint arXiv:1010.49101 [hep-th], 2010.
[83] Hadi Godazgar, Mahdi Godazgar, Olaf Krger, Hermann Nicolai, and Krzysztof Pilch. An SO(3) SO(3) invariant solution of D = 11 supergravity. JHEP, 01:056, 2015. × [84] Nabil Iqbal, Hong Liu, and Mark Mezei. Quantum phase transitions in semilocal quantum liquids. Phys. Rev., D91(2):025024, 2015. 196
[85] Sean A. Hartnoll and Edgar Shaghoulian. Spectral weight in holographic scaling geometries. JHEP, 07:078, 2012.
[86] Richard J. Anantua, Sean A. Hartnoll, Victoria L. Martin, and David M. Ramirez. The Pauli exclusion principle at strong coupling: Holographic matter and momentum space. JHEP, 03:104, 2013.
[87] Matthias Kaminski, Karl Landsteiner, Javier Mas, Jonathan P. Shock, and Javier Tarrio. Holographic Operator Mixing and Quasinormal Modes on the Brane. JHEP, 02:021, 2010.
[88] J. K. Jain. Composite-fermion approach for the fractional quantum Hall effect. Physical Review Letters, 63:199–202, July 1989.
[89] Subir Sachdev. Compressible quantum phases from conformal field theories in 2+1 dimen- sions. Phys. Rev., D86:126003, 2012.
[90] Anson Hook, Shamit Kachru, Gonzalo Torroba, and Huajia Wang. Emergent Fermi surfaces, fractionalization and duality in supersymmetric QED. JHEP, 08:031, 2014.
[91] Shamit Kachru, Michael Mulligan, Gonzalo Torroba, and Huajia Wang. Mirror symmetry and the half-filled Landau level. Phys. Rev., B92:235105, 2015.
[92] Charles Cosnier-Horeau and Steven S. Gubser. Holographic Fermi surfaces at finite temper- ature in top-down constructions. Phys. Rev., D91(6):066002, 2015.
[93] Nikolay Bobev, Nick Halmagyi, Krzysztof Pilch, and Nicholas P. Warner. Supergravity In- stabilities of Non-Supersymmetric Quantum Critical Points. Class. Quant. Grav., 27:235013, 2010.
[94] C. N. Pope and N. P. Warner. An SU(4) Invariant Compactification of d = 11 Supergravity on a Stretched Seven Sphere. Phys. Lett., B150:352–356, 1985.
[95] Christopher P. Herzog and Jie Ren. The Spin of Holographic Electrons at Nonzero Density and Temperature. JHEP, 06:078, 2012.
[96] J. C. Campuzano, M. R. Norman, and M. Randeria. Photoemission in the High Tc Super- conductors. eprint arXiv:cond-mat/0209476, September 2002.
[97] Jiunn-Wei Chen, Ying-Jer Kao, and Wen-Yu Wen. Peak-Dip-Hump from Holographic Super- conductivity. Phys. Rev., D82:026007, 2010.
[98] Davide Forcella and Alberto Zaffaroni. Non-supersymmetric CS-matter theories with known AdS duals. Adv. High Energy Phys., 2011:393645, 2011.
7 [99] Aristomenis Donos and Jerome P. Gauntlett. Superfluid black branes in AdS4 S . JHEP, × 06:053, 2011.
[100] Oscar Henriksson and Christopher Rosen. “1kF ” Singularities and Finite Density ABJM Theory at Strong Coupling. eprint arXiv:1612.06823 [hep-th], 2016.
[101] Alexander Altland and Ben Simons. Condensed Matter Field Theory. Cambridge University Press, 2010. 197
[102] Piers Coleman. Introduction to Many Body Physics. Cambridge University Press, 2015.
[103] E. G. Dalla Torre, D. Benjamin, Y. He, D. Dentelski, and E. Demler. Friedel oscillations as a probe of fermionic quasiparticles. Phys. Rev. B, 93(20):205117, May 2016.
[104] George E. Simion and Gabriele F. Giuliani. Friedel oscillations in a fermi liquid. Phys. Rev. B, 72:045127, Jul 2005.
[105] E. C. Andrade, E. Miranda, and V. Dobrosavljevi´c.Quantum Ripples in Strongly Correlated Metals. Physical Review Letters, 104(23):236401, June 2010.
[106] Chetan Nayak and Frank Wilczek. Renormalization group approach to low temperature properties of a nonFermi liquid metal. Nucl. Phys., B430:534–562, 1994.
[107] David F. Mross, John McGreevy, Hong Liu, and T. Senthil. A controlled expansion for certain non-Fermi liquid metals. Phys. Rev., B82:045121, 2010.
[108] B. L. Altshuler, L. B. Ioffe, and A. J. Millis. Low-energy properties of fermions with singular interactions. Phys. Rev. B, 50:14048–14064, Nov 1994.
[109] T. Senthil. Critical Fermi surfaces and non-Fermi liquid metals. Phys. Rev. B, 78(3):035103, July 2008.
[110] A. Liam Fitzpatrick, Gonzalo Torroba, and Huajia Wang. Aspects of Renormalization in Finite Density Field Theory. Phys. Rev., B91:195135, 2015.
[111] V. G. M. Puletti, S. Nowling, L. Thorlacius, and T. Zingg. Friedel Oscillations in Holographic Metals. JHEP, 01:073, 2012.
[112] Thomas Faulkner and Nabil Iqbal. Friedel oscillations and horizon charge in 1D holographic liquids. JHEP, 07:060, 2013.
[113] Joseph Polchinski and Eva Silverstein. Large-density field theory, viscosity, and ’2kF ’ singu- larities from string duals. Class. Quant. Grav., 29:194008, 2012.
[114] Mikhail Goykhman, Andrei Parnachev, and Jan Zaanen. Fluctuations in finite density holo- graphic quantum liquids. JHEP, 10:045, 2012.
[115] Mike Blake, Aristomenis Donos, and David Tong. Holographic Charge Oscillations. JHEP, 04:019, 2015.
[116] Blaise Goutraux and Victoria L. Martin. Spectral weight and spatially modulated instabilities in holographic superfluids. eprint arXiv:1612.03466 [hep-th], 2016.
[117] Daniel Z. Freedman, Krzysztof Pilch, Silviu S. Pufu, and Nicholas P. Warner. Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved. eprint arXiv:1611.01888 [hep-th], 2016.
[118] Steven S. Gubser. Curvature singularities: The Good, the bad, and the naked. Adv. Theor. Math. Phys., 4:679–745, 2000.
[119] Steven S. Gubser and Indrajit Mitra. Instability of charged black holes in Anti-de Sitter space. eprint arXiv:hep-th/0009126, 2000. 198
[120] Aristomenis Donos and Jerome P. Gauntlett. Holographic striped phases. JHEP, 08:140, 2011.
[121] Thomas Faulkner and Joseph Polchinski. Semi-Holographic Fermi Liquids. JHEP, 1106:012, 2011.
[122] Sean A. Hartnoll, Andrew Lucas, and Subir Sachdev. Holographic quantum matter. eprint arXiv:1612.07324 [hep-th], 2016.
[123] Nikolay Bobev, Henriette Elvang, Daniel Z. Freedman, and Silviu S. Pufu. Holography for N = 2∗ on S4. JHEP, 07:001, 2014.
[124] Andreas Karch, Brandon Robinson, and Christoph F. Uhlemann. Precision Test of Gauge- Gravity Duality with Flavor. Phys. Rev. Lett., 115(26):261601, 2015. Appendix A
Lifting the Three-Charge Black Hole to Five Dimensions
Here we summarize how the near-horizon region of the extremal 3QBH may be lifted to
2 a non-singular five-dimensional AdS3 R geometry, with the inert gauge field identified as the × graviphoton. Additionally, the lift of a fermion action to five dimensions implies a relation between the fermion eigenvalues, which is satisfied by all the fermions we work with. This analysis is similar to what is done in [55], section 6, and we refer the reader there for additional detail.
The near-horizon (r 0) limit of the metric and scalar 3.113, 3.114 in the extremal 3QBH → gives r 3√Qr3/2 Q3/2√r L2 Q ds2 = dt2 + d~x2 + dr2 , eφ = . (A.1) − L2 L2 3√Qr3/2 r
A five-dimensional ansatz of the form
dsˆ2 = eφds2 + e−2φ(dz + )2 . (A.2) A dimensionally reduces to a four-dimensional action
3 2 1 −3φ 2 4 = √ g R (∂φ) e , (A.3) L − − 2 − 4 F which matches the Einstein term, scalar kinetic term, and second gauge kinetic term in 3.79 if we identify the a gauge field with the graviphoton . Then using this lift, the five-dimensional metric A arising from A.1 is 3Qr L2 r Q2 dsˆ2 = dt2 + dr2 + dz2 + d~x2 , (A.4) − L2 3r2 Q L2 200 which with the change of coordinates r ρ2 becomes ≡ 3Qρ2 4L2 ρ2 Q2 dsˆ2 = dt2 + dρ2 + dz2 + d~x2 , (A.5) − L2 3ρ2 Q L2
2 which is of the form AdS3 R , as promised. The leading order term in the four-dimensional × potential in 3.79 arises by dimensional reduction of a cosmological constant term,
p √ϕ 5Λ = gˆΛˆ 4Λ = √ ge 3 Λˆ , (A.6) L − → L − with Λˆ = 3/L2.
Consider now reducing a massless, four-component Dirac fermion λ from five dimensions. Its action is
p M N 5λ = giˆ λγ¯ eˆ ˆ N λ , (A.7) L − M ∇
1 PQ where ˆ N = ∂N ωˆ γPQ. We make an ansatz for this spinor in terms of a four-dimensional ∇ − 4 N Dirac spinor χ
4 λ(xµ, z) = einz/Re−φ/4e−πγ /4χ(xµ) , (A.8) where R is the radius of the compact z coordinate. Here we have rescaled by a power of φ to obtain canonical kinetic terms, and performed a chiral rotation to remove factors of γ4 (which is i times the four-dimensional chirality matrix) from the mass and Pauli terms. We arrive at
−1 µ n 3φ/2 n µ i −3φ/2 µν e =χ ¯ iγ µ e + γ µ e γ µν χ . (A.9) L ∇ − R R A − 8 F
Comparing to the 3QBH fermion Lagrangian (3.87) with the identification a = we find agreement A for the appropriate terms (including the term in the potential of leading order as r 0) given the → identifications
m n q1 n = , = , p1 = 1 , (A.10) 4L ±R 4L R ∓ where the second choice of sign can be obtained by doing a chirality flip 3.40. Thus both the mass parameter m and the charge q1 are given by the momentum in the compact direction, and we have the relation
m = q1p1 , (A.11) − 201 as given in 3.118. Checking the eigenvalue table for the (3+1)QBH in section 3.4.2, we find perfect agreement with this constraint in all cases. Appendix B
Further Notes on the Static Susceptibility Computation
This appendix provides some additional important details regarding the computation of the static susceptibility in chapter6. First, the subject of holographic renormalization through the addition of boundary counterterms is discussed, and it is pointed out that finite counterterms required by supersymmetry are important for the correct holographic interpretation. Then, it is discussed how to impose correct boundary conditions in the IR (at the black brane horizon) and the UV (at the AdS boundary), and how to use numerical methods to integrate the full equations of motion.
B.1 Finite Counterterms and the Alternate Quantization
The importance of a holographic renormalization procedure in the computation of many gauge theory quantitities via AdS/CFT is well known. Such a procedure is necessary to remove various near boundary divergences, and also plays a role in precision matches between gravity and gauge theory (recent examples include [65, 123, 124]). Comparatively, the role of finite boundary counterterms in holography has received much less attention. In part, this is because in many cases
(including “bottom-up” implementations of holography) there is no obvious method by which the coefficients of such terms can be unambiguously determined. At the same time, such terms have little physical significance in a broad class of holographic computations.
The situation changes somewhat in certain well known examples of top-down gauge/gravity duality. In these cases, the large amount of supersymmetry in the boundary gauge theory places 203 strong constraints on the structure of finite counterterms, and this symmetry can sometimes be used to fix these terms completely. This procedure has recently been performed in detail in the
= 8 gauged SUGRA dual to the ABJM theory in the limit we study in the body of this work N [117].
Interestingly, finite boundary counterterms in this gauged SUGRA have profound conse- quences for the holographic dictionary. This is perhaps most evident for the scalars transforming as the 35v of the SO(8) gauge symmetry. Consistency of the SUGRA theory requires that such modes obey an “alternate” quantization, and in this case the addition of a finite counterterm to the on-shell boundary action can effectively redefine the identification of the source for the dual operator from the perspective of the gauge theory.
To see this in more detail1 , it is convenient to start from the renormalized action for a scalar in the 35v which obeys alternate boundary conditions. Expanding (6.6) to quadratic order, the scalar sector is given by Z 4 1 2 2 Sφ = d x√ g (∂φ) + φ (B.1) − −2 plus the usual (infinite) alternate quantization boundary counterterms
Z 2 3 φ 0 Sc.t. = d x√ γ + rφφ (B.2) − 2 where 0 is a derivative with respect to r. The counterterm action is understood to be evaluated at some cutoff near the boundary which will eventually be taken to infinity. To these conventional terms, we now include the finite boundary counterterm
Z 3 3 Sf.c.t. = λ d x√ γφ (B.3) − and seek boundary conditions on the bulk scalar such that the variational problem is well defined.
Near the boundary the scalar falls off like
α β φ(r ) + + ... (B.4) → ∞ ≈ r r2 1 We are indebted to O. DeWolfe and J. Gauntlett for discussions that greatly clarified these details. 204
o.s. and thus a variation of the renormalized on-shell action S˜o.s. S + Sc.t. + Sf.c.t. leads to ≡ φ
2 δS˜o.s. = α δβ + 3λα δα. (B.5) −
Note that the conventional alternate boundary conditions, which identify the source for the scalar operator dual to φ as β such that δβ = 0, no longer extremize the action in the presence of the
finite counterterm. Instead one can rewrite (B.5) as 3 2 δS˜o.s. = α δ β λα (B.6) − − 2 which, upon choosing λ = 1/3√3, yields the quantization condition (6.16).
B.2 Boundary Conditions and Numerical Methods
In order to compute the static susceptibility, we must solve the linearized fluctuation equations for at, and the modes that couple to at, while imposing the correct boundary conditions. The setup is very similar to [115], with the added complication of the scalar field ϕ. We make the gauge choice hµr = ar = 0. Inserting the “background+fluctuation” and plane wave ansatzes of (6.22) and (6.23) into the bulk Einstein, Maxwell, and Klein-Gordon equations, we arrive at a set of ODEs that couple together the modes at, htt, hyy, ϕ , as anticipated from the set of gauge invariant modes { }
(6.26). These equations are first order in htt and second order in the other three modes. Hence there are seven constants of integration that must be fixed by boundary conditions.
We solve the equations exclusively in the non-zero temperature 3-charge black brane geome- try. The IR thus has an event horizon, at which we impose regular boundary conditions:
2 at c1(r rH ) + ((r rH ) ) (B.7) ≈ − O −
hyy c2 + (r rH ) (B.8) ≈ O − 2 htt c3(r rH ) + ((r rH ) ) (B.9) ≈ − O −
ϕ c4 + (r rH ) (B.10) ≈ O −
The equations of motion fix e.g. c3 in terms of the other ci. Hence we are left with three undeter- mined constants, which are to be fixed by UV boundary conditions. 205
When prescribing appropriate boundary conditions in the UV, extra care must be taken since we are working in terms of gauge dependent fluctuations. To compute the static susceptibility, one must ensure that the only non-normalizable fluctuation turned on belongs to at.
The scalar is quantized according to (6.16), and thus normalizable linearized fluctuations of the scalar obey δβ 1 α¯ δα = 0. (B.11) √3 − 3 In Fefferman-Graham coordinates, the fluctuation fall-offs in the UV (r ) are related to those → ∞ of (6.27) by
3Q f1 f2 + f1 ϕ = + 4 + ... (B.12) u u2
h˜tt = ht0 + ... (B.13)
h˜yy = hy0 + ... (B.14)
a1 at = a0 + + ... (B.15) u
Thus, to compute the static susceptibility it would naively seem as though one should seek solutions such that Q f2 + f1 = ht0 = hy0 = 0 and a0 = 1. (B.16) 4 However, since the earlier near-horizon analysis only provided us with three undetermined constants, it is clear that fixing these four boundary conditions would overconstrain the system.
Instead, one is only permitted to enforce boundary conditions that are gauge equivalent to (B.16).
Turning to the near boundary behavior of the gauge invariant modes, 1 1 3Q 1 2 1 s = f1 + √3Q hy0 + f1 + f2 + √3Q hy0 + ... (B.17) Z 4 u 4 8 u2
2 1 = (ht0 + hy0) u + ... (B.18) Z p ! (Q + rH ) Q(Q + rH ) 1 2 = a0 hy0 a1 + ... (B.19) Z − 2 − u one finds that the correct prescription is given by
Q f2 + f1 = ht0 + hy0 = 0 and a0 = 1. (B.20) 4 206
Any solution satisfying (B.20) can be brought to the form (B.16) by an appropriate coordinate transformation.
Having determined the correct boundary conditions for the IR and the UV, we employ nu- merical “shooting” to integrate the fluctuations between the two. Starting from the IR near-horizon
T expansion, we organize the undetermined constants into a vector ~c (c1, c2, c4) , and set them to ≡ some arbitrary values, say ~c = (1, 1, 1)T . Then, performing the numerical integration out to the − UV boundary, we read off the values of the resulting sources—which we also collect into a vector
Q T J~ (f2 + f1, ht0 + hy0, a0) . In general, this procedure will not lead to the desired UV bound- ≡ 4 ary behavior J~ = (0, 0, 1)T . To remedy this, we use the fact that our fluctuation equations are linear. Repeating the numerical integration three times, with three linearly independent ~c, allows us construct the linear map T from the set of IR data to the UV sources, defined by
T~c = J.~ (B.21)
With T in hand, it is easy to find the vector ~c which corresponds to any desired J~, and to then compute the susceptibility using (6.29).
The matrix T is especially useful when searching for poles in the static susceptibility, which on the gravity side correspond to zero modes (ZMs) in the fluctuation spectrum. As explained in more detail in [87], a ZM implies that T has a zero eigenvalue, and hence
det T(~kZM ) = 0 (B.22)
gives the condition for a pole in χ(k) at ~k = ~kZM . Combining this criterion with an efficient numerical root-finding algorithm (such as Mathematica’s FindRoot) provides a powerful method for locating ZMs and tracking their trajectory while changing e.g. the temperature.