PHYSICAL REVIEW D 91, 046011 (2015) Fermionic response in a zero entropy state of N ¼ 4 super-Yang-Mills

Oliver DeWolfe,1 Steven S. Gubser,2 and Christopher Rosen3 1Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309, USA 2Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544, USA 3Crete Center for Theoretical Physics, University of Crete, 71003 Heraklion, Greece (Received 9 July 2014; published 26 February 2015) We analyze fermionic response in the geometry holographically dual to zero-temperature N ¼ 4 super- Yang-Mills theory with two equal nonvanishing chemical potentials, which is characterized by a singular horizon and zero ground state entropy. We show that fermionic fluctuations are completely stable within a gap in energy around a Fermi surface singularity, beyond which non-Fermi liquid behavior returns. This gap disappears abruptly once the final charge is turned on and is associated to a discontinuity in the corresponding chemical potential. We also show that the singular near-horizon geometry lifts to a 3 smooth AdS3 × R and interpret the gap as a region where the quasiparticle momentum is spacelike in six dimensions due to the momentum component in the Kaluza-Klein direction, corresponding to the final charge.

DOI: 10.1103/PhysRevD.91.046011 PACS numbers: 11.25.Tq, 04.50.Gh, 71.10.Hf

I. INTRODUCTION AND SUMMARY this and other studies of nonzero-density systems using the gauge-gravity correspondence, see [1–4] and many other A. Fermionic response in gauge-gravity systems papers [5–52]; recent reviews appear in [53,54]. Depending Strongly coupled systems are challenging to study on parameters, these systems may manifest Fermi liquid, theoretically, yet they describe a number of vital and non-Fermi liquid, or marginal Fermi-liquid-like behaviors. interesting phenomena in diverse areas of physics. To In a Fermi liquid, quasiparticles become arbitrarily well better understand such systems, it is valuable to develop defined as one approaches the Fermi surface at zero alternate theoretical tools in which the physics is temperature. In a non-Fermi liquid, this does not happen; presented in a dual description using weakly coupled instead, the width of excitations remains comparable to or variables. One such framework is the gauge-gravity parametrically greater than their energies. Marginal Fermi correspondence, in which strongly coupled field theories liquids lie on the border between the two classes. In the with a large number of degrees of freedom may be realized “strange metals” arising in high-Tc cuprates [55,56] and in holographically as smooth gravity backgrounds in a heavy fermion systems [57], Fermi surfaces can be iden- higher dimension. tified from photoemission experiments, but the associated Field theory states of nonzero temperature and/or density gapless excitations are not long lived, suggesting an are realized in the correspondence as geometries including association with non-Fermi liquids. a black-brane horizon, with equilibrium properties deter- It is natural to wish to embed this kind of calculation in mined by black hole thermodynamics. We will be focused supergravity and string theory; such a “top-down” con- on zero-temperature, nonzero-density systems, which cor- struction has the advantages of explicit knowledge of the respond to extremal black branes. Generically, the event field theory dual and the operator map, as well as putting to horizon for such an extremal brane is smooth, and, due to rest concerns that there might be something unphysical the relationship between the horizon area and entropy, such about the bottom-up construction. Such a top-down study geometries are dual to zero-temperature field theory states of fermionic response was investigated in [58] for the with a nonzero, macroscopic entropy density. This large four- and five-dimensional geometries dual to the maximally entropy is somewhat problematic for physical applications, supersymmetric conformal theories in three and four dimen- as well as difficult to understand; explaining and potentially sions, respectively, Aharony-Bergman-Jafferis-Maldacena circumventing this feature is an important task. theory and N ¼ 4 super-Yang-Mills (SYM) theory, and To characterize the properties of these systems beyond Fermi surfaces were found. The N ¼ 4 SYM case was basic thermodynamics, one may study linear response. studied in much more detail in [59]. Earlier top-down studies Here we will be concerned with the response of the system of the gravitino sector, which did not find Fermi surface to probe fermionic operators; one may identify Fermi singularities, can be found in [60–62]. surface singularities and study the spectrum of nearby N ¼ 4 SYM is characterized by three independent fluctuations. This has been studied substantially from the chemical potentials. Extremal black holes with all three “bottom-up” perspective, where the gravity background is chemical potentials nonzero have nonsingular horizons and not derived directly from string theory or supergravity; for the associated nonzero entropy at zero temperature; we will

1550-7998=2015=91(4)=046011(26) 046011-1 © 2015 American Physical Society DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) call these “regular” cases. These were exhaustively studied surface, and poles for nonzero ω describe the dispersion for the case of two equal chemical potentials in [59], where relation of nearby fluctuations. For regular cases, the 1 2 ω ≡ ω − Γ the Fermi surface behavior for all spin- = fermionic fluctuation energy is complex, i , including a ω Γ modes not mixing with a gravitino was obtained. The real energy and a width indicating the fluctuations are results were all characterized by Fermi surfaces with non- unstable. For Fermi liquids, the imaginary part of the Fermi-liquid-like excitations, with a single case asymptoti- Green’s function vanishes faster than the real part as one Γ ω → 0 cally approaching a marginal Fermi-liquid-like state; no approaches the Fermi surface, and one has = . excitations with ordinary Fermi liquid properties were in In cases where the real and imaginary parts scale identically evidence. as one approaches the Fermi surface, on the other hand, Γ ω → It is desirable to understand the entropy that shows up in one has = const, indicating a non-Fermi liquid. a generic zero-temperature background of N ¼ 4 SYM. The essential novel feature of the extremal two-charge Moreover, it is interesting to consider limits where this black hole is that there exists an energy scale Δ of order the zero-temperature entropy is absent. There is such a class of chemical potential, which we call the “gap,” such that geometries in the N ¼ 4 SYM family, where one of the fluctuations within Δ of the Fermi surface have no decay chemical potentials vanishes; the simplest example is when width and are thus precisely stable.1 This is not a gap in the other two charges are equal, which we will call the the conventional sense of the absence of any low energy “two-charge black hole” (2QBH). The vanishing entropy excitations, but rather a low energy regime in which for the extremal two-charge black hole is associated to a excitations mediating decays of the fermionic modes near zero-area horizon, which is singular [8]. Probe fermions the Fermi surface appear to be suppressed. On the other were studied [8] and the Dirac equation solved exactly in side of the line ω ¼Δ, fluctuations reacquire a width and this background in [40]. As with the bottom-up models, behave analogously to excitations near the Fermi surface in Fermi liquid, marginal Fermi liquid and non-Fermi-liquid- regular cases. We find that the real and imaginary parts of like cases were all possible in these studies. Other recent the Green’s functions for N ¼ 4 SYM scale identically in studies of the 2QBH include [43,44,46,51,52]. this region, characteristic of non-Fermi liquids, consistent Γ ω However, the full story is more subtle. The 2QBH with the results of [59]; the constant that = approaches, background is a solution of the 5D gauged supergravity however, is zero, matching onto the stable region. This coming from the reduction of type IIB supergravity on situation is summarized in cartoon form in Fig. 1. 5 AdS5 × S and, hence, corresponds to a specific state in A possible qualitative interpretation of this phenomenon N ¼ 4 super-Yang-Mills. However, the fermion actions runs as follows. In addition to whatever fermionic fluctua- studied in [8,40] were not deduced directly from super- tions we create, one postulates that there exists an addi- gravity but instead were postulated to take a simple form tional sector with a large density of states. For a generic with a constant mass, making them bottom-up excitations background in N ¼ 4 SYM with all charges, this sector is in a top-down background. In [59], the proper top-down not gapped and extends down to the Fermi surface, where fluctuation equations of the fermions of N ¼ 8 super- its states are perceived as the nonzero ground state entropy. gravity were worked out, and they displayed a property Fermionic fluctuations couple to this sector, and it mediates not covered by the analysis of [40]: the masses depended on their ability to decay. In the special case of the two-charge the running scalar field, which diverges at the horizon or black hole, however, this sector is gapped and cannot be singularity. Such scalar couplings are common in super- excited by energies less than Δ away from the Fermi gravity, and one might imagine that if a zero-entropy surface. This removes the large entropy from zero temper- extremal geometry always has a singularity, then it will ature and simultaneously removes the mechanism by which be generic for fermionic fluctuation equations to have such the fermionic fluctuations decay, rendering them stable in divergences. Thus, it is possible that the true nature of this energy range. We note that this implies that the fermionic response in such a background can only be fermionic fluctuations have no intrinsic self-interaction; uncovered by including these couplings, and one would one might speculate that this is a large-N effect. like to study them in more detail. This process was begun in [59], and we continue it here. 1The gap we are discussing does not appear to be a manifes- tation of superconductivity, in that the Uð1Þ gauge fields under B. The gap in the two-charge black hole which the black hole is charged remain unbroken by any condensate, as does the final Uð1Þ gauge field under which To analyze the fermionic response in finite density the black hole is not charged. Conductivities in these states with ’ systems, we calculate retarded Green s functions for respect to the two Uð1Þ gauge fieldsp wereffiffiffi computed in [59] and gauge-invariant fermionic operators; this is done on the displayed a soft gap of size Δσ ∼ 2 2Δ. The conductivity with 1 gravity side by solving Dirac fluctuation equations with respect to the Uð Þ under which the black hole is charged does exhibit infinite dc conductivity, and while at least some part of suitable infalling boundary conditions at the horizon. this feature is attributable to unbroken translational invariance in A pole in the Green’s function at the Fermi energy (defined a state with finite charge density, it is nonetheless possible that the as ω ¼ 0) for some momentum k ¼ kF defines a Fermi gap we observe is in some sense related to superconductivity.

046011-2 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015)

Non-Fermi liquid Non-Fermi liquid = Stable modes = 0 = 0 Fermi surface Fermi surface Stable modes

Non-Fermi liquid Non-Fermi liquid

FIG. 1 (color online). Cartoon comparison between a regular extremal black-brane background (left) and the extremal two-charge black hole (right). In the former, fluctuations characteristic of a non-Fermi liquid surround the Fermi surface; in the latter, they are separated from the Fermi surface by a gap of size 2Δ wherein the fluctuations are stable.

As one moves away from the 2QBH by adding even a One can also derive consistency conditions on the tiny amount of the charge Q1 that had been zero (corre- charges and couplings of fermions that can be lifted to sponding to adding a charge density ρ1), the gap disappears six dimensions. The discontinuity in the chemical potential entirely. This behavior is associated to another discontinu- μ1 is associated to a failure to commute of two different ous phenomenon: the μ1 chemical potential jumps through paths to the extremal 2QBH in the black hole parameter zero from a positive to a negative constant as one varies space, and the form of the Dirac equation will depend on from infinitesimal positive ρ1 to infinitesimal negative ρ1 the path, leading to contradictory expressions, unless the (see Fig. 4). Such a discontinuity in the chemical potential fermions satisfy these conditions. All the fermions of can be characteristic of moving through a gap in the density maximal gauged supergravity satisfy them, while a generic of states, as the lack of states precludes the charge density Dirac action will not, once again suggesting that top-down from increasing even as the chemical potential is raised. considerations are an important ingredient, whereas a On the gravity side, the technical explanation for the purely generic bottom-up fermion action may fail to gapped region is that the solution to the fluctuation equation properly capture the physics of these backgrounds. becomes purely real. Associated to this, a true infalling Turning to specific examples, we analyze a number of boundary condition at the horizon is no longer possible; distinct fermion species corresponding to specific operators instead we impose a regularity condition, as is characteristic of N ¼ 4 SYM in the 2QBH and numerically identify the of Euclidean gauge-gravity calculations. We show this is the dispersion relations through the stable region. We find two proper continuation of the infalling condition. classes of poles in the Green’s function. When there is a In the regular cases, the near-horizon and small-ω limits Fermi surface singularity at ω ¼ 0, one may follow this do not commute, necessitating the introduction of inner and singularity through the ω-k plane to the gap on either side. outer regions which are then patched together [4]. The inner In addition, in some cases new pairs of poles nucleate at a 3 region has the geometry of AdS2 × R and determines the nonzero value of ω and move towards the gap. These poles universal parts of the fermionic Green’s function, including seem to be associated with the presence of an oscillatory the decay width of quasiparticles. For the 2QBH, analogous region at ω ¼ Δ—a region where the Green’s function phenomena occur at jωj¼Δ. The near-horizon geometry is has log periodic behavior in ω and gapless excitations singular in five dimensions and does not naively resemble for a range of k, arising from an instability towards pair anything as simple as AdS2. However, we show that this production in the near-horizon region of the geometry 3 geometry lifts to a smooth AdS3 × R in six dimensions, [4,63]. These new poles generally reach the gap inside the demonstrating that the singularity is of a “good” type and oscillatory region, at which point they cease to exist, the geometry is sensible. The Kaluza-Klein charge of though in some cases one may miss it and survive. the reduction is the charge Q1 turned off in the 2QBH The plan of this paper is as follows. We review five- backgrounds, and the six-dimensional lift provides a new dimensional supergravity, the black-brane solutions and perspective on the gap Δ, which can be understood as the their thermodynamics, and the fermionic fluctuation spec- minimum energy to turn a momentum vector with a fixed trum in Sec. II. The analysis of fermion fluctuations in amount of compact momentum (q1 charge) timelike. This regular extremal black holes is recapped in Sec. III, before suggests that a field theory explanation of the gap in turning to the main subject of the paper in Sec. IV, the Green’s functions of appropriately charged fermionic fermionic fluctuations in the 2QBH. We apply these results operators could be developed based on the emergence of a to a number of fermionic operators of N ¼ 4 super-Yang- nonchiral Virasoro algebra in the infrared. Mills in Sec. V. In Sec. VI we present the lift of the 2QBH

046011-3 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) 3 near-horizon geometry to AdS3 × R in six dimensions with the functions     and demonstrate how fermions are constrained by the lift. 2 2 r 1 Q1 1 Q2 Some technical details are left for the Appendixes. AðrÞ¼log þ log 1 þ þ log 1 þ ; L 6 r2 3 r2     2 2 II. GRAVITY DUAL OF N 4 SUPER-YANG- r 1 Q1 2 Q2 ¼ BðrÞ¼− log − log 1 þ − log 1 þ ; MILLS AT FINITE DENSITY L 3 r2 3 r2 2 2 2 2 2 2 A. Supergravity and black-brane backgrounds 1 − r ðrH þ Q1ÞðrH þ Q2Þ hðrÞ¼ 2 2 2 2 2 2 ; N ¼ 4 super-Yang-Mills theory is the most symmetric rHðr þ Q1Þðr þ Q2Þ rffiffiffi   rffiffiffi   avatar of four-dimensional gauge theory and as such is often 2 2 2 2 ϕ − 1 Q1 1 Q2 the gauge theory most amenable to study, leading it to ðrÞ¼ log þ 2 þ log þ 2 ; 3 r 3 r become the simple harmonic oscillator of the twenty-first   2 2 2 2 century study of quantum field theory. We are interested in Φ Q1ðrpHffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ Q2Þ 1 − rH þ Q1 1ðrÞ¼ 2 2 2 2 ; states of finite density, associated to global conserved 2LrH r þ Q1 r þ Q1 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH   charges. The theory possesses an SOð Þ R-symmetry group; 2 2 2 2 since this group is rank 3, there are three independent Φ Q2 rH þ Q1 1 − rH þ Q2 2ðrÞ¼ 2 2 2 : ð3Þ associated conserved quantities, and correspondingly one LrH r þ Q2 may turn on three distinct chemical potentials. Following The solutions are characterized by two charges Q1, Q2 and [59], in what follows we will always simplify matters by a horizon radius r (the latter may be traded for a mass taking two of the three chemical potentials equal; since the H parameter). We refer to these as (2 þ 1)-charge black holes major distinct behaviors are associated with whether a given [ð2 þ 1ÞQBHs]. The associated thermodynamics are chemical potential is zero or not, this simplification seems still to capture the most interesting possibilities. 2 4 2 2 − 2 2 2 2 rH þ Q1rH Q1Q2 Q1ðrH þ Q2Þ N ¼ 4 SYM with SUðNÞ gauge group is holographically T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; μ1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 5 2π 2 2 2 2 2 2 2 dual to type IIB string theory compactified on AdS5 × S L rH rH þ Q1 L rH rH þ Q1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with N units of self-dual five-form flux, with SOð6Þ 2 2 2 μ Q2 rH þ Q1 manifested as the isometry group of the five-sphere. 2 ¼ 2 ; L rH The dynamics of the lowest-mass modes arising from the 1 5 2 2 1=2 2 2 Q1s Kaluza-Klein reduction on S are given by five-dimensional s r Q r Q ; ρ1 ; ¼ 4 3 ð H þ 1Þ ð H þ 2Þ ¼ 2π maximally supersymmetric (N 8) gauged supergravity, GL rH ¼ pffiffiffi where SOð6Þ becomes the gauge group. There exists a 2 ρ Q2s consistent truncation of the maximal gauged supergravity 2 ¼ ; ð4Þ 2πrH to a bosonic sector containing just the three gauge fields corresponding to the Cartan generators of SOð6Þ,alongwith where T and s are the temperature and entropy density, the metric and two neutral scalars, the so-called STU model respectively, μ1 and ρ1 are the chemical potential and [64]. Simplifying matters as described above, we set two of charge density for the single charge, respectively, and μ2 the three gauge fields equal; in this case only one scalar is and ρ2 are likewise for the two charges set equal. sourced, and we are left with the effective gravity Lagrangian Geometries corresponding to zero temperature are extremal, with the constraint on the three parameters 1 8 ϕ 4 −2ϕ −4ϕ −1 2 pffiffi pffiffi pffiffi μν L − ∂ϕ 6 6 − 6 e ¼ R ð Þ þ 2 e þ 2 e e fμνf 2 4 2 2 − 2 2 0 2 1 2 L L rH þ Q1rH Q1Q2 ¼ ½extremal ð þ ÞQBH; ð5Þ 2ϕ pffiffi μν μνρστ − 2e 6FμνF − 2ϵ fμνFρσAτ: ð1Þ which has solutions for jQ2j >rH.ForQ1;Q2 ≠ 0, the extremal black holes are regular, possessing a double pole Here Aμ is the gauge field associated to the two equal at the horizon in the function hðrÞ and no singularity. All charges, and aμ goes with the remaining one. these geometries asymptotically approach (the boundary N 4 R3;1 States of ¼ SYM on with nonzero temperature of) AdS5 at r → ∞: and chemical potentials are dual to geometries solving (1) with the black-brane form r2 L2 e2A → ;e2B → ;h→ 1; ϕ → 0; L2 r2 2BðrÞ 2 2AðrÞ 2 2 e 2 Φ → ds ¼ e ð−hðrÞdt þ dx~ Þþ dr ; i const: ð6Þ hðrÞ μ μ aμdx ¼ Φ1ðrÞdt; Aμdx ¼ Φ2ðrÞdt; ϕ ¼ ϕðrÞ; We will be focused on the special case where we take Q1 → 0 strictly; these are called the 2QBHs. In this case 2 ð Þ Φ1 ¼ 0. The 2QBH solution is

046011-4 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015)   2 r 1 Q2 since Q1 is taken strictly to zero first, while the latter AðrÞ¼log þ log 1 þ ; L 3 r2 instead gives   r 2 Q2 ffiffiffi − − 1 2 Q1Q2 p BðrÞ¼ log log þ 2 ; → 2 ðextremal 2 þ 1Þ: ð11Þ L 3 r r2 rffiffiffi   H 2 2 2 2 2 ðrH þ Q2Þ Q2 hðrÞ¼1 − ; ϕðrÞ¼ log 1 þ ; This distinction affects only Φ1ðrÞ. In the former case, it ðr2 þ Q2Þ2 3 r2  2  becomes zero strictly. In the latter case, however, one finds 2 2 Φ Q2 1 − rH þ Q2 it approaching the nonzero constant (see Fig. 3) 2ðrÞ¼ 2 2 ; ð7Þ 2L r þ Q2 Q2 Φ1 → pffiffiffi : ð12Þ with thermodynamics 2L pffiffiffi Φ 0 2 2 2 While this is gauge equivalent to 1 ¼ , its form none- rH Q2 rHðrH þ Q2Þ T ¼ ; μ2 ¼ ;s¼ ; theless affects the thermodynamics. In general the chemical πL2 L2 4GL3 pffiffiffi potential μ1 is realized holographically as (proportional to) 2 ρ Q2s the difference in potential between the boundary and the 2 ¼ ; ð8Þ μ ∝ Φ ∞ − Φ Φ 2πrH horizon: 1 1ð Þ 1ðrHÞ. The form (3) for 1ðrÞ is constructed to have Φ1ðrHÞ¼0, so in general we simply μ ∝ Φ ∞ and μ1 ¼ ρ1 ¼ 0; we will sometimes refer to μ2 simply as have 1 1ðpffiffiffiÞ.AsQ1 approaches zero, one finds μ. Thus extremality occurs for Φ1ð∞Þ → Q2= 2L with Φ1ðrHÞ¼0; this corresponds to the chemical potential μ1 approaching the value of μ2 from below: rH ¼ 0 ðextremal 2QBHÞ: ð9Þ

μ1 The order of limits in reaching the extremal 2QBHs we are lim → 1: ð13Þ Q1→0;T¼0 μ studying can be somewhat subtle. We have so far described 2 → 0 the 2QBH by first setting Q1 and then imposing At Q1 ¼ 0,however,Φ1 degenerates to the constant (12) → 0 extremality rH . One may instead consider another and no longer satisfies Φ1ðr Þ¼0; see Fig. 3. Since the 2 1 H path: first achieve extremality for ð þ ÞQBHs by impos- difference between the boundary and horizon values is ing the relation (5) between rH, Q1 and Q2 and then tune zero, the chemical potential μ1 vanishes, consistent with the Q1 to zero while maintaining extremality, which also original 2QBH limit. Thus the extremal 2QBH is a state → 0 requires adjusting rH (see Fig. 2). From the field with μ1 ¼ 0, which nonetheless appears as the limit of a set ρ theory point of view, this is tuning the charge density 1 to of extremal geometries approaching μ1 ¼ μ2. We display 0 zero while maintaining T ¼ . this behavior in Fig. 4, where we plot μ1=μ2 as a function of As discussed in [59], these different limits are not the charge density ratio ρ1=ρ2. Note that as one can show precisely equivalent; the former implies

Q1Q2 → 0 2 2 ðextremal Þ; ð10Þ 0.8 rH

0.6

Nonextremal Q1 Nonextremal (2+1)QBH 2QBH 0.4

Q1= Q1(rH, Q2) rH 0.2

0.0 Extremal Extremal 0 1 2 3 4 (2+1)QBH Q1, rH 2QBH

FIG. 3 (color online). Plots of Φ1ðrÞ for various Q1 as Q1 → 0 with T ¼ 0 held fixed. For all nonzero Q1 one has Φ1ðrHÞ¼0; FIG. 2 (color online). Two approaches to the extremal two- note rH moves left as Q1 → 0. For Q1 ¼ 0, Φ1 degenerates to a charge black hole. nonzero constant.

046011-5 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) Μ1 We note in addition that the fact that μ1=μ2 is decreasing Μ2 1.0 as ρ1=ρ2 increases is a signal of a negative susceptibility and, correspondingly, of a thermodynamic instability. Such an instability is associated with a perturbative 0.5 instability in the spectrum of bosonic fluctuations. We do not study such instabilities here, but they were consid- ered originally in [65–67], and subsequent related work Ρ1 has included [68–71]. 30 20 10 10 20 30 Ρ2

B. Fermion spectrum and Dirac equation 0.5 The 5D Dirac equation in the class of backgrounds just reviewed has been studied for general regular backgrounds 1.0 in [4] and was solved in the 2QBH background with bottom-up fermions in [40]. The actual spectrum of μ μ FIG. 4 (color online). The ratio of chemical potentials 1= 2 fermionic fluctuations in N ¼ 4 SYM was obtained in compared with the charge density ratios ρ1=ρ2.Asρ1 approaches [59] by embedding the black-brane solutions (2) into N ¼ 8 zero from above, μ1 approaches μ2 from below, but at ρ1 ¼ 0 gauged supergravity and then studying the fermionic strictly we have μ1 ¼ 0 (heavy dot at origin). fluctuations of that theory. The spin-1=2 fermions of pffiffiffi Q1 2ρ1 maximal gauged supergravity that do not mix with the ¼ ; ð14Þ gravitino all satisfy a linearized equation of the form Q2 ρ2 −2ffiffiϕ one can just as easily think of the x axis as being the black μ μ μ p μν ðiγ ∇μ − mðϕÞþgq1γ aμ þ gq2γ Aμ þ ip1e 6 fμνγ hole charge ratio Q1=Q2 up to a rescaling. ϕ pffiffi μν One can argue that such a jump in a chemical potential þ ip2e 6Fμνγ Þχ ¼ 0; ð15Þ as the charge density is varied is indicative of the presence of a gap in the density of states. As a chemical potential is where the mass term depends on the scalar: moved through the gap between bands in a semiconductor,

ϕ 2ϕ for example, no new states are filled and the density −pffiffi pffiffi remains constant. Only upon reaching the other side of mðϕÞ ≡ gðm1e 6 þ m2e 6Þ; ð16Þ the gap will the density begin to change again as the nonzero density of states causes new states to be occupied. with m1, m2, q1, q2, p1, and p2 rational numbers and In studying fermionic fluctuations around the 2QBH, we g ≡ 2=L. The list of so-called “maximal” mode fermions will find evidence for a gap from another perspective. found in [59] is

Fermions are labeled by the three Cartan charges of Antiparticles with opposite sign charges exist for each SOð6Þ, with χ and χ¯ dual to left- and right-handed row in the table as well. spinors in four dimensions. The second column describes We note that, with the exception of the fermion the composition in terms of the N ¼ 4 gauginos λa, neutral under q2, the fermions come in pairs, the two a ¼ 1…4, and adjoint scalars Zj ≡ X2j−1 þ iX2j, j ¼ 1, having the same charge under one of the gauge fields 2, 3; the trace over the gauge group is implied. and the opposite charges under the other. It is thus

046011-6 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) straightforward to imagine how a Fermi surface with It what follows we will suppress the label α on ψ;itis ρ1 ¼ 0 but ρ2 ≠ 0 as in the 2QBH could exist: one can clear from (22) that the solution for one two-component have equal numbers of excitations with opposite charges spinor will be identical to that for the other with k → −k. under aμ but the same charge under Aμ filling the We will now pass to decoupled second-order equations, ground state. in a fashion outlined in [40].2 Define the combinations A few relations can be seen to hold for each fermion: q1 ≡ ψ ψ is half-integer quantized, the magnitude of p1 is fixed: U − i þ: ð25Þ

1 The first-order equations then take the form jp1j¼4 ; ð18Þ 0 − U− þ iYU− ¼ð X þ iZÞUþ; and the parameters m2, q1 and p1 are related: 0 − − − Uþ iYUþ ¼ð X iZÞU− ð26Þ

m2 −2q1p1: 19 ¼ ð Þ and lead to the second-order equations

In Sec. VI, we will demonstrate that all these relations are 00 0 0 2 2 2 U− þ pU− þðiY − X þ Y − Z þ iYpÞU− ¼ 0; required by the consistency of lifting the fermion to six 00 ¯ 0 − 0 − 2 2 − 2 − ¯ 0 dimensions and show that the q1 charge corresponds to Uþ þ pUþ þð iY X þ Y Z iYpÞUþ ¼ ; Kaluza-Klein momentum in the sixth dimension. ð27Þ One may process the Dirac equation as follows. First we rescale the spinor χ as where

−2A −1=4 −iωtþikx χ ¼ e h e Ψ; ð20Þ p ≡ −∂r logð−X þ iZÞ: ð28Þ where ω is the frequency and k is the spatial momentum, To calculate a retarded Green’s function, one solves this chosen to lie in the x direction, and the e−2Ah−1=4 factor equation with infalling boundary conditions imposed at the exactly cancels the spin connection term coming from ∇μ horizon. The response is then read off from the behavior of in (15). Next, each field Ψ is a four-component spinor; solutions near the boundary r → ∞: by suitably choosing the Clifford basis, one may decom- ψ ∼ Aðω;kÞrmL þ Bðω;kÞr−mL−1; pose this into a pair of two-component spinors ψ α, þ α 1 mL−1 −mL ¼ , 2, which decouple from one another [4,59]. The ψ − ∼ Cðω;kÞr þ Dðω;kÞr ; ð29Þ Dirac equation for each two-component spinor is then of the form with mL ≡ 2ðm1 þ m2Þ.Form>0, A is the source term, and D the response (the case m<0 exchanges their roles, ð∂r þ Xσ3 þ Yiσ2 þ Zσ1Þψ α ¼ 0; ð21Þ corresponding to a dual fermion of opposite chirality). The Green’s function is then defined as the ratio of the response where to the source:

− D meB eB A G ¼ : ð30Þ X ≡ pffiffiffi ;Y≡ − pffiffiffi u; R A h h B−A A Fermi surface is identified as the momentum k where e α F Z ≡ − pffiffiffi ðð−1Þ k − vÞ; ð22Þ there is a pole in the retarded Green’s function at zero ω: h A ω 0;k k ≡ 0: and we have defined (following [4,59]) ð ¼ ¼ FÞ ð31Þ 1 To make progress, one should study the near-horizon u ≡ pffiffiffi ðω þ gq1Φ1 þ gq2Φ2Þ; behavior of the Dirac equation, where the infalling con- h dition is imposed. We first review the well-known “regular” 2ϕ ϕ − −pffiffi pffiffi B 6 6 cases in the next section, before turning to the 2QBH case v ≡ 2e ðp1e ∂rΦ1 þ p2e ∂rΦ2Þ: ð23Þ in the section following. Let us write each two-component spinor as 2   For the numerical solutions of Sec. V we used a different, ψ α− equivalent second-order equation, derived in [59] and discussed ψ α ¼ : ð24Þ ψ in Appendix A. The choice given here is more convenient for the αþ analysis of Secs. III and IV.

046011-7 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) III. GREEN’S FUNCTIONS FOR REGULAR One sees that, near the horizon, the terms involving ω are EXTREMAL BLACK HOLES dominant. This is a sign that interesting physics happens near ω ¼ 0. We may study the small-ω limit, but for any fixed, In this section we review the near-horizon analysis of the tiny ω there will always be values of r sufficiently close to Dirac equation in regular extremal black holes, the calcu- the horizon that the first two terms in parentheses dominate. lation of the IR Green’s function and using it to obtain an Thus the small-frequency and near-horizon limits do not expression for the full Green’s function, and the extraction commute. Yet, we must understand the behavior of a mode of the dispersion relation for fluctuations. This will be near the horizon to impose infalling boundary conditions. useful as a warm-up to the case of the two-charge black For this reason, we must consider inner and outer regions. hole. This analysis was originally performed from a Consider the inner region first. This is the region bottom-up perspective by [4] and was extended to the designed to take into account the values of the radial maximal gauged supergravity Dirac equation (15) in [59]. coordinate that are so close to the horizon that ω cannot be Generic (2 þ 1)-charge extremal black holes are regular, ignored, even though it is small. For the inner region, we having a double pole at the horizon but no singularity there. scale ω and r − rH at the same time: r → rH, ω → 0 with One has, in the near-horizon limit r → rH, ω=ðr − rHÞ held fixed. The inner region equation then takes   τ τ 2 precisely the form of (34) with all neglected terms scaled A B 0L2 0 2 e → k0;e→ ;h→ ðr − rHÞ ; to zero: k0 k0  4 4 4 2 Φ → β ðr − r Þ; ϕ → ϕ0; ð32Þ 1 L ðQ2 − r Þω i i H U00 U0 H þ − þ 16 2 2 − 2 − 4 r rH ð Q2 rHÞðr rHÞ τ β ϕ  where k0, 0, L2, i and 0 are all constants defined in [59], #ω ν2 3 − 0 leading to the near-horizon geometry AdS2 × R [4], þ 3 2 U ¼ : ð37Þ ðr − rHÞ ðr − rHÞ L2dr2 ds2 −τ2 r − r 2dt2 2 k2dx~2: In the inner region we then solve this equation for all r with ¼ 0ð HÞ þ − 2 þ 0 ð33Þ ðr rHÞ fixed ω. In general there is a solution in the inner region that is purely infalling as r → rH; this can be obtained explicitly We see in the metric everything is regular and approaches a in terms of Whittaker functions, but we will not record the constant except for the horizon function h; the gauge fields form here. have a single zero. The second-order Dirac equation (27) The near-boundary limit of AdS2 is where the inner then has the form as r → r : H region glues onto the outer region, corresponding to r − rH    large (with ω kept fixed), which becomes 1 4 4 − 4 ω2 00 0 L ðQ2 rHÞ U þ þ U þ 2 r − r 16ð2Q2 − r2 Þðr − r Þ4 1 ν H  2 H H U00 þ U0 − U ¼ 0: ð38Þ 2 − 2 #ω ν r rH ðr − rHÞ þ − þ U ¼ 0; ð34Þ ðr − r Þ3 ðr − r Þ2 H H This equation has power law solutions 2 3 where we have neglected terms of order ω =ðr − r Þ and 1 H −2ν 2 −1 U ∼ ðr − rHÞ : ð39Þ ω=ðr − rHÞ as well as ðr − rHÞ in the no-derivative term, # is a constant whose form we will not record, and ν2 is given as In general the solution of the whole inner region that is infalling at small r − rH can be expanded at large r − rH in 1 − μ 2 μ2 2 ~2 1 some linear combination of (39): ν2 ðm1ð RÞ þ m2 RÞ k ¼ 4 þ 2 2 1 − μ μ 2 1 μ −1 ν −1−ν ð Þ 2 ð þ Þ ∼ − 2þ G ω − 2 ffiffiffi R R U ðr rHÞ þ ð Þðr rHÞ ; ð40Þ p 3 2 2 ð 2q1μ þ q2ð1 − μ ÞÞ − R R ; ð35Þ G ω 4ð1 − μ2 Þð1 þ μ2 Þ2 where the relative weighting ð Þ between the two R R solutions depends on k and the other parameters as well; 3 thinking of the two terms in (40) as the source and response with μR ≡ μ1=μ2 and where in an AdS2 fluctuation, we may consider GðωÞ to be an ffiffiffi ’ ~ α p AdS2 Green s function. Since the AdS2 region corresponds k ≡ k − −1 2p1μ1 2p2μ2 : ð Þ ð þ Þ ð36Þ to the IR part of the full geometry, this is also called the IR Green’s function. Note that the only dependence on ω in 3Note there is an overall sign typo in the second line of (123) (40) is in GðωÞ. The full form of GðωÞ is recorded in [4]; for in [59]. small ω it takes the form

046011-8 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) γ 2ν GðωÞ¼jcðkÞjei k ð2ωÞ ; ð41Þ behavior is associated to an instability in the IR of the geometry towards pair production due to the strong electric for real quantities jcðkÞj and γk. field [63], and the corresponding behavior in scalars leads Now consider the outer region. Here by definition r − rH to unstable modes. is large enough that the ω terms in (34) can be neglected. When ν is real, on the other hand, we may find a Fermi The near-horizon equation for the outer region is then surface at a particular Fermi momentum k ¼ kF; determin- ing kF requires solving the Dirac equation out from the 1 ν2 horizon to the boundary to find (31), and thus the value U00 þ U0 − U ¼ 0; ð42Þ r − r ðr − r Þ2 of the Fermi momentum requires knowledge of the UV H H 0 0 physics. One then has aþðkFÞ¼ , and near the Fermi which precisely matches (38); the two regions have the surface the Green’s function takes the form same solutions on the overlap region. Thus in the outer ω → 0 η0 h1 region we have in general solutions with near- ω ∼ GRðk; Þ 1 iγ 2ν : ð48Þ boundary behavior k⊥ − ω þ− h2e kF ð2ωÞ kF vF

0 −1 ν η → − 2 ≡ − ðr rHÞ ; ð43Þ Here k⊥ k kF, and we have plugged in the form (41) for GðωÞ. The real constants h1, h2 and vF are more difficult and thus the solution that has infalling boundary conditions to compute, but once kF is known some physics can be ν γ in the outer region takes the form extracted just from knowledge of kF and kF . The dispersion relation for excitations near the Fermi ∼ η0 G ω η0 ω U þ þ ð Þ −: ð44Þ surface is defined by the values of ðk⊥; Þ for which the denominator of (48) vanishes and thus has the form In general as we move away from ω → 0 we will have corrections 1 2ν 2ν ω γ 2ω kF γ 2ω kF k⊥ ¼ þþh2 cos kF ð Þ þ ih2 sin kF ð Þ : vF η η0 ωη1 ¼ þ þ; ð45Þ ð49Þ and thus the small-ω solutions are Since vF is real, the properties of the IR Green’s function GðωÞ determine the leading imaginary part in the U ∼ η þ GðωÞη−: ð46Þ 2νk þ dispersion relation to scale as ω F ; if we have ν 1 2 kF < = , this is the leading real part as well, beating Both components of the two-component spinors take this the ω=v term. For ν > 1=2 we have a Fermi liquid; for ψ F kF generic form. Thus passing from U to and expanding ν 1 2 kF < = , a non-Fermi liquid, with marginal Fermi liquid near the boundary, we obtain a general expression for the sitting in between. ’ ω 0 retarded Green s function near ¼ as For a Fermi liquid, fluctuations near the Fermi surface 0 1 0 1 become asymptotically stable as the fluctuation energy D b þ ωb þþGðωÞðb− þ ωb− þÞ G ¼ ¼ þ þ ; goes to zero. For a non-Fermi liquid, on the other hand, the 0 ω 1 G ω 0 ω 1 Γ ω A aþ þ aþ þþ ð Þða− þ a− þÞ ratio of the width to the energy of a fluctuation near ð47Þ the Fermi surface approaches the constant

Γ γν where the ai and bi are k dependent. Calculating the F ω ¼ tan 2ν : ð50Þ i i F coefficients a and b in general requires numerical solution of the Dirac equation over the entire space, but The residue of the (would-be) quasiparticle pole is then a the IR Green’s function GðωÞ and the associated parameter constant for a Fermi liquid, and for the non-Fermi liquid ν may be determined solely from the IR physics. The quantity ν2 may be positive or negative depending scales as on the parameters, including k, and thus ν may be real or 1 2ν −1 imaginary. When ν is imaginary, the boundary conditions Z ∼ ðk⊥Þ kF : ð51Þ (43) become complex, and it is not possible to find a Fermi surface as one cannot make the denominator of the In [59], the fermionic fluctuations of all spin-1=2 modes not Green’s function (47) vanish at ω ¼ 0 while keeping k real; mixing with the gravitino were studied for all values of μR in general ImGRðω ¼ 0;kÞ ≠ 0. In this regime, called the in the extremal ð2 þ 1ÞQBH backgrounds. A number of oscillatory region, the Green’s function is periodic in log ω, Fermi surface singularities were discovered, all associated and one has gapless excitations for a range of k [4]. This to non-Fermi-liquid-like states.

046011-9 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015)   ’ 1 1 2 1 IV. GREEN S FUNCTIONS FOR THE EXTREMAL ν2 ≡ − ω 2 TWO-CHARGE BLACK HOLE 4 P 2 sgnð m2Þ þ 4 m2 þ m1m2

We now turn to our main interest, the extremal 2QBH. sgnðωm2Þ − pffiffiffi q2m2: ð59Þ The horizon in this case is at r ¼ 0, but this case is different 2 2 because it is singular as r → 0 as well. The functions X, Y and Z that appear in the Dirac equation, expanded This quantity has been given the same name as (35) for near r → 0, are the ð2 þ 1ÞQBH case; as we shall see, they will play an analogous role and even coincide in the appropriate limit. a cω P Equation (57) has obvious similarities to the ð2þ1ÞQBH X ¼ þ b; Y ¼ þ dω þ f; Z ¼ ; ð52Þ r2 r2 r case (34) but differences as well. Most salient is that both 4 share 1=ðr − rHÞ terms that dominate near the horizon but where a, b, c, d, and f are constants depending on the are suppressed by factors of energy. For the ð2 þ 1ÞQBH 4 background and the type of fermion, case, it was near ω ¼ 0 that the 1=ðr − rHÞ terms were   suppressed, leading to the need for inner and outer regions 2 2 1 3 L there. For the extremal 2QBH, this happens instead at a ¼ m2μL ;b¼ 2m1 þ m2 ;c¼ − ; μL2 2 2 jωj¼Δ. This difference lies at the heart of the novel structure for this case. 1 q2 1 d ¼ − ;f¼ − pffiffiffi ; ð53Þ 2μ2L2 2 μL2 A. Solutions away from jωj¼Δ 2 2 and P includes the spatial momentum: As long as ω − Δ is not small, the first term in parentheses in (57) dominates over all other zero-derivative pffiffiffi terms at small r, and we get − −1 α k 2 P ¼ ð Þ μ þ p2: ð54Þ 2 c2ðω2 − Δ2Þ ψ 00 þ ψ 0 þ ψ ¼ 0: ð60Þ We are neglecting terms of Oðr2Þ from X and Y and OðrÞ r r4 from Z. One can show that the quantity p defined in (28) is There is no need in these cases to worry about inner and 2 iP outer regions. The solutions of (60) were considered in p ¼ þ þ OðrÞ; ð55Þ [59]. Let us define the quantity ϵ which measures the r a deviation of jωj away from Δ: which implies that ω2   ϵ2 ≡ − 1: ð61Þ cωP 1 Δ2 0 − O iY þ iYp ¼ 2 þ ; ð56Þ ar r Then for jωj > Δ, we have the complex solutions O −3   where two terms of ðr Þ canceled. We then find that the iΔL2ϵ near-horizon second-order equation (27) takes the same ψ ∼ exp ; ð62Þ 2r form for either U, with differences subleading in r, and ψ thus can also be written as an equation for either ,as corresponding to infalling and outgoing waves at the   2 horizon. In this case the usual infalling boundary conditions ψ 00 þ þ ψ 0 are easy to impose, and since the infalling wave is complex, r the corresponding fluctuation is complex as well.  1 2  c2ðω2 − Δ2Þ − 4ν þ Oðjωj − ΔÞ On the other hand, for jωj < Δ we have instead growing þ þ 4 þ ψ ¼ 0; r4 r2 and dying solutions   ð57Þ ΔL2jϵj ψ ∼ exp : ð63Þ 2r where we have defined the scale Δ, closely related to the chemical potential, As was noted in [59], there is no obvious solution here corresponding to an infalling wave. Instead, it was sug-

a gested there to adopt the standard boundary condition of Δ≡ ¼ 2jm2jμ; ð58Þ c Euclidean AdS/CFT, keeping the regular solution at the horizon, corresponding to keeping the dying exponential and the quantity ν2 is and removing the diverging exponential. Choosing the

046011-10 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015)     nondiverging solution is a real boundary condition, and 1 ΔL2ϵ 1 ΔL2ϵ ψ ∼ pffiffiffi J2ν ; pffiffiffi Y2ν ; ω > Δ; since the differential equation is real, we will in this case r 2rÞ r 2r have purely real solutions to the fluctuation equations. The ð67Þ Fermi surface can be defined as usual (31) as the vanishing of the retarded Green’s function at ω ¼ 0; however, since ω Δ ϵ ω Δ while for < , we have imaginary , and the above this lies in the j j < region, to do this we must accept become modified Bessel functions: this modified prescription for the boundary conditions. It     appears that the Fermi surface is surrounded by an energy 1 ΔL2jϵj 1 ΔL2jϵj gap of size 2Δ, within which the modes are purely real; as ψ ∼ pffiffiffi K2ν ; pffiffiffi I2ν ; ω < Δ: r 2r r 2r we shall describe, the corresponding fluctuations have no decay width and so are stable. In the next subsection we ð68Þ look at the region jωj ≈ Δ, where this unusual behavior ω Δ crosses back over to more typical behavior. For > , one may impose infalling boundary conditions at the horizon by choosing the combination of (67) to be the Hankel function of the first kind: B. Solutions near jωj¼Δ   rffiffiffiffiffi   1 Δ 2ϵ 2 Δ 2ϵ Looking at the near-horizon equation (57), it is evident ð1Þ L −iπ−2πiν i L “ ” ψ ¼ pffiffiffi H2ν → e 4 2 exp ; ð69Þ that the kind of interesting behavior that showed up in the r 2r ϵπ 2r regular case (34) at ω ¼ 0 happens here instead at jωj → Δ; this is the case for which the first term in parentheses is which indeed takes in the infalling form at r → 0, matching suppressed except for very close to the horizon. The analog (62). We can take this solution and examine the inner of studying small ω for the regular cases is to study small ϵ, region near-boundary (r → ∞) behavior and extract the that is, to study energies near Δ. Recall that ω ¼ 0 already “IR Green’s function” G.4 Consider for simplicity the case ε μ corresponds to fluctuations at the Fermi surface F ¼ q2 ; of nonintegral 2ν; this will be generic except for special so we are now discussing fluctuations around a scale values of the momentum. As x → 0 we have Δ ¼ 2jm2jμ above or below the Fermi energy.   −2ν Let us analyze (57) for energies near Δ. As with regular 1 Γ 2ν ð Þ − i ð Þ x cases near ω ¼ 0, we have the issue that for fixed small ϵ H2ν ðxÞ¼ π 2 þ there will always be sufficiently small r where the 1=r2   iΓð−2νÞ x 2ν term dominates, and again we can solve this with inner and − −2πiν π e 2 þ; ð70Þ outer regions. For the outer region, r is large enough that the relevant term can be neglected, and we set ω ¼ Δ—that where each leading term is corrected by a power series in is, ϵ ¼ 0. The second-order equation in the outer region even powers of x; since 2ν is nonintegral here these do not then has the near-horizon limit overlap. Thus we see that 1 2    2 4 − 4ν 2 −2ν ψ 00 ψ 0 ψ 0 i ΔL ϵ 2ν−1 þ þ 2 ¼ : ð64Þ ψ − Γ 2ν 2 r r ¼ π ð Þ 4 r þ    2 2ν The solutions to (64) are power laws: −2π ν ΔL ϵ −2ν−1 Γ −2ν i 2 þ ð Þe 4 r : ð71Þ −1 2ν ψ ∼ r 2 : ð65Þ These power laws indeed match the two solutions for the → 0 near-boundary limit of the inner region from (65). The ratio Meanwhile the inner region is defined by taking r , −2ν−1 2ν−1 ϵ → 0, with ϵ=r fixed. We obtain the inner region equation between the r 2 and r 2 terms is     2 4ν 2 2 1 2 Γ −2ν ΔL ϵ 2 a ϵ − 4ν G ϵ −2πiν ð Þ ψ 00 ψ 0 4 ψ 0 þð Þ¼e þ þ 4 þ 2 ¼ : ð66Þ Γð2νÞ 4 r r r   Γð−2νÞ L4ðω2 − Δ2Þ 2ν ¼ e−2πiν ; ð72Þ The near-boundary limit of the inner region involves Γð2νÞ 16 r → ∞ with ϵ fixed and results in the same equation (64) as the near-horizon limit of the outer region; as expected 4 ’ there is a match in the transition region. In the regular case, the term IR Green s function was well motivated by the fact that the near-horizon region was AdS2 and We can solve the inner region equation in general, −1=2 hence had its own holographic interpretation. We will continue obtaining r times Bessel functions. For jωj > Δ we to use the terminology; later we will discuss the relation of this have near-horizon region and AdS3.

046011-11 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) ω Δ G ϵ G ϵ where we use the label þ to indicate > . For the case showing the expressions for þð Þ and −ð Þ are the same 2ν−1 of ν real, we take ν > 0 and then the r 2 term always formula, with ϵ continued from real to imaginary values; −2ν−1 dominates over r 2, so we may think of them as the we drop the distinction from here on. This provides further source and the response, respectively; then GðϵÞ may confidence in our choice of boundary condition for the usefully be thought of as the IR Green’s function. jωj < Δ region. For jωj − Δ small, we have As in the regular cases, the outer region solution will   match to the inner solution on their overlap, and thus the Γð−2νÞ L4Δðjωj − Δ 2ν continuation of our boundary condition to the outer region G ðϵÞ ≈ e−2πiν : ð73Þ þ Γð2νÞ 8Þ gives solutions ψ η G ϵ η Comparing this to the form (41) for the regular cases, we ¼ þ þ ð Þ −; ð80Þ see both scale like the deviation from the “special energy” −1 2ν 2ν ν η ∼ 2 to the power. Furthermore, in this case for real the where r and, as in the regular case, the solutions phase of G is simply can be extended away from ϵ ¼ 0 in a power series in ϵ2;for small deviations ϵ2 is proportional to the linear deviation: γ ≡ arg G ¼ −2πν: ð74Þ 2ðjωj − ΔÞ ϵ2 ≈ : ð81Þ Consider now the case of ω < Δ,sopϵ isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imaginary; here we Δ will give results in terms of jϵj¼ 1 − ω2=Δ2. Now the solutions (68) are purely real, and consequently they do not Meanwhile the outer region near-boundary behavior still has show infalling or outgoing behavior at the horizon; instead, the form (29), and since both components have the same one solution is regular there and one is divergent. Not near-horizon behavior, we get for the full Green’s function having the infalling prescription available, we will instead near jωj¼Δ follow the prescription used for Euclidean AdS/CFT and ð0Þ ϵ2 ð2Þ G ϵ ð0Þ ϵ2 ð2Þ choose the regular solution, which is D bþ þ bþ þþ ð Þðb− þ b− þÞ GR ¼ ¼ : ð0Þ 2 ð2Þ ð0Þ 2 ð2Þ   A a þ ϵ a þþGðϵÞða− þ ϵ a− þÞ 1 ΔL2jϵj þ þ ψ ¼ pffiffiffi K2ν : ð75Þ r 2r ð82Þ

The expansion near the boundary is Thus the IR Green’sfunctionGðϵÞ plays the same role near ω Δ G ω ω 0   j j¼ as its cousin ð Þ does near ¼ in the 2 −2ν Γð2νÞ ΔL jϵj 2ν−1 regular case. ψ 2 ¼ 2 4 r þ As with the regular case, there are different situations   depending on the sign of ν2 (59). Note that the momentum 2 2ν 2 Γð−2νÞ ΔL jϵj −2ν−1 ν þ r 2 þ; ð76Þ dependence of enters in the positive-definite term 2 4 ðP 1=2Þ2; this is played against a constant term depending on the fermion in question, which may be negative. If ν2 < 0, and so now the IR Green’s function is the power law solutions (65) are not real, and we again find     an oscillatory region where there are no Fermi surface Γ −2ν Δ 2 ϵ 4ν Γ −2ν 4 Δ2 − ω2 2ν 2 ð Þ L j j ð Þ L ð singularities. For ν > 0, Fermi surfaces may be found. G−ðϵÞ¼ ¼ ; Γð2νÞ 4 Γð2νÞ 16Þ Note that the form (82) is not useful near ω ¼ 0, so it does ð77Þ not tell us anything about fluctuations near the Fermi surface. Instead, it tells us about dynamics near the gap Δ. The ≡ which for Δ − jωj small is denominator may vanish for a particular value of k kΔ, corresponding to a pole in the Green’s function at ω Δ.   ¼ Γ −2ν 4Δ Δ − ω 2ν (One may also define a k−Δ where the denominator vanishes G ϵ ≈ ð Þ L ð j jÞ −ð Þ : ð78Þ for ω ¼ −Δ, and kΔ and k−Δ need not coincide.) The Γð2νÞ 8 spectrum of nearby fluctuations may then be worked out. ω Δ We are now in a position to further justify the regular For j j > , the case of complex fluctuations, the prescription: it is the continuation of the infalling pres- behavior of the Greens function near k ¼ kΔ is cription as jωj moves past Δ, corresponding to imaginary ϵ. h1 This can be seen using the Bessel function identity: G ∼ ; R jωj−Δ −2πiν 2ν ðk − kΔÞ − þ− h2e kΔ ðjωj − ΔÞ kΔ π vF 2νþ1 ð1Þ K2νðxÞ¼2 i H2ν ðixÞ; ð79Þ ð83Þ

046011-12 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) pffiffiffi and we find a story very similar to the regular case (48). The 2q1Q2 ν ω ¼ ω2 1 þ energy dependence scales with the quantity in an identical þ L2 fashion, justifying our choice for the analogous notation. ¼ ω2 1 þ q1μ2: ð87Þ The phase is determined by ν as well, and once again we þ ν 1 2 will have a Fermi-liquid-like state if kΔ > = , non-Fermi ω ν 1 2 Here is the energy as defined by the Dirac equation that liquid behavior for kΔ < = , and marginal Fermi-liquid- approaches the 2QBH directly, while ω2 1 is the energy in like features for ν ¼ 1=2. For the non-Fermi case, the þ kΔ the Dirac equation approaching via extremal ð2 þ 1ÞQBHs. ratio of the width to the energy is Since μ1 → μ2 in this limit, we may interpret the shift as

Γ −2πνΔ 0 ω ¼ ω2 1 þ q1μ1; ð88Þ ω ¼ tan 2ν ¼ : ð84Þ þ Δ which is precisely the shift in the energy that would Thus unlike the generic regular case, we find that even for accompany a particle of charge q1 with the first chemical ν 1 2 cases with < = , the dispersion of the modes vanishes potential shifted from 0 to μ1 ¼ μ2. as the special energy is approached. We can still consider One may also interpret this shift in terms of the gap Δ. these most similar to non-Fermi liquids, as the real and Given the relations (18) and (19) (which we will show are ’ imaginary parts of the Green s function denominator scale necessary for the lift to six dimensions in Sec. VI), we have with the same power law, and other quantities such as the residue consequently keep the non-Fermi liquid form jq1j¼2jm2j: ð89Þ

1 2ν −1 Z ∼ ðk − kΔÞ Δ : ð85Þ Consequently, the shift between the two scales of energy is precisely the magnitude of the gap: One might say that Γ=ω approaches a constant as with other ω ω Δ non-Fermi liquids, but the constant in this case is zero. ¼ 2þ1 þ sgnðq1Þ : ð90Þ ω Δ ’ For < , we have instead the form for the Green s 2 1 function In [59], many fermions were analyzed in ð þ ÞQBHs for various values of 0 ≤ μR ≡ μ1=μ2 ≤ 1, at what we are now ω 0 h1 calling 2þ1 ¼ . What we have now learned is that the G ∼ ; 86 μ → 1 R jωj−Δ 2ν ð Þ R limit of those results should match on to the results − − Δ − ω kΔ ðk kΔÞþ v þ h2ð j jÞ ω 0 F found here for the 2QBH. Moreover, 2þ1 ¼ implies ω ¼Δ; thus the “Fermi surface” results found there which is just the continuation of the first form to negative match up with the physics not in the middle of the stable ω − Δ j j . The primary difference is that everything is real. region but at one of its edges. Thus the edge of the gapped ω As a result, the energy of the fluctuation has no region coincides with the limit of a series of Fermi surfaces. imaginary part; the corresponding modes are stable every- The gapped region is unusual in the way it appears ω Δ where for j j < . This connects smoothly to the result suddenly for the 2QBH, with energy width 2Δ, while if one (84) for jωj > Δ. turns on any nonzero amount of Q1 to give a ð2 þ 1ÞQBH, this region vanishes entirely. The results of this section C. Connection with extremal (2 þ 1)-charge black holes relate this “sudden” appearance of the band to the order of In this subsection, we show how the boundary of the limits issue in the parameter space. By choosing how one stable region jωj ≈ Δ can be probed by studying the limit of approaches the 2QBH, one can choose to “zoom in” on the Fermi surfaces at ω ≈ 0 in extremal (2 þ 1)-charge black energy region at the center of the stable region (if Q1 ¼ 0 holes. When we study results for specific fermions of is taken first), or instead one may zoom in on the edge of maximal gauged supergravity in 2QBH backgrounds in the this band by approaching via the extremal ð2 þ 1ÞQBHs. next section, this will connect those results to the results A limit of Q1, Q2 and rH that lies in between can of [59]. presumably be taken to approach some other energy As described in Sec. II, approaching the extremal 2QBH location in the middle of the band. via extremal ð2 þ 1ÞQBHs shifts Φ1 by a constant (12). The sudden disappearance of the gap under the addition This constant shift, while not a true shift in the chemical of Q1 charge is notable in that for any nonzero Q1, the potential, does correspond to a simple shift in the zero vanishing of the gap is accompanied by a finite zero-point point of the energy for q1-charged particles. Consider the entropy density in the dual field theory state. This non- Dirac equation, approaching the 2QBH via extremal vanishing entropy at zero temperature is a well known ð2 þ 1ÞQBHs. Φ1 enters in both u and v (23), but because feature of states dual to geometries with AdS2 regions in the the shift (12) is constant, only u is affected. Indeed, one infrared and is challenging to interpret. From the point of may absorb the resulting shift into a redefinition of the zero view of N ¼ 4 SYM, one possible explanation is that the point of the energy ω by gq1Φ1: shift in energy in (90) is suggesting that the gauge theory

046011-13 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) ffiffiffi p 3 2 2 state dual to the extremal 2QBH can be interpreted as a ð 2q1μ þ q2ð1 − μ ÞÞ ðqeÞ2 ¼ R R : ð97Þ zero-temperature, zero-entropy state characterized by an eff 4ð1 − μ2 Þð1 þ μ2 Þ2 intermediate energy scaling regime which can be matched R R to the infrared physics of certain states dual to ð2 þ 1ÞQBHs. It is easy to see that this goes to infinity as μR → 1 (as long This interpretation is similar in spirit to the semilocal as q1 ≠ 0, which holds for all our modes; in fact it is quantum liquid picture advocated for extremal black holes necessarily half-integer quantized by the lift to 6D). For real with regular horizons in [27]. νk we then have Given that the Dirac equations coincide, various related quantities should coincide also. We can see this for ν2. γk → −2πνk; ð98Þ The definition of this quantity in the ð2 þ 1ÞQBHs is (35). μ → 1 We are interested in the R limit; expanding we find which indeed matches the phase (74). It is interesting that the limit is only sensible for fermions ~ pffiffiffi k → k − −1 α 2 2 − −1 α 2 that can be lifted to six dimensions. The apparent diver- μ μ ð Þ ð p1 þ p2Þ¼ ð Þ ðP þ p1Þ; 2 2 gence in (92) at μR → 1 is related to a term in the Dirac 2 1 ð91Þ equation: if one takes the limit via extremal ð þ ÞQBHs ω 0 with 2þ1 ¼ , one expects to get the 2QBH equation (57) −4 and thus with jωj¼Δ, which should have no r term and thus should coincide with the outer region equation (64). 4 2 − 2 However, one finds an extra term ν2 → m2 q1 16ð1 − μRÞ     2 q2 − 4m2 1 − 4ν2 1 7 2 5 2 ψ 00 ψ 0 1 2 4 ψ 0 2 q1 m2 q1q2 þ þ 4 þ 2 ¼ ; ð99Þ þ ðP þ 2p1Þ þ 4m1m2 þ − − pffiffiffi r 2r r 4 8 2 2

þ Oð1 − μRÞ: ð92Þ which vanishes only for fermions obeying (19). Thus we learn that a generic fermion action, not derived from We see there is a leading term that apparently diverges in maximal gauged supergravity, is not consistent throughout the limit. However, this term is absent for all our fermions, the parameter space; approaching the same extremal given the relation (89). Using this relation to eliminate q1 in 2QBHs in two different ways it acquires two inequivalent favor of sgnðq1Þ and m2, we find equations unless (89) is obeyed. ffiffiffi 2 1 2 2 p ν → ððP þ 2p1Þ þ 4m1m2 þ m − sgnðq1m2Þ 2q2m2Þ: 4 2 V. FERMIONS IN N ¼ 4 SUPER-YANG-MILLS ð93Þ We turn now to studying a number of fermionic The relation (19) also implies a relationship between signs: fluctuations of maximal gauged supergravity, dual to operators in N ¼ 4 super-Yang-Mills, in the 2QBH back- sgnðm2Þ¼−sgnðq1Þsgnðp1Þ: ð94Þ grounds. The Dirac equations for all spin-1=2 fields were worked out in [59]. Thanks to (90), we may take sgnðωÞ¼sgnðq1Þ in this Here we work out the dispersion relation between k limit. Putting these together with (18), we have and ω in the entire stable region for each fermion. The   Fermi surface is at ω ¼ 0; we find such a surface and the 1 1 2 2 2 corresponding k and then vary ω away from zero and find ν → P − sgnðωm2Þ þ 4m1m2 þ m F 4 2 2 ’  the k giving a pole in the Green s function in each case. pffiffiffi Close to the Fermi surface pole the dispersion is linear − sgnðωm2Þ 2q2m2 ; ð95Þ and primarily deviates from this behavior in the vicinity of jωj¼Δ.Atjωj¼Δ, the results match on to the μR → 1 2 1 ω 0 which exactly matches the expression (59) for the 2QBH limit of the study of ð þ ÞQBHs at 2þ1 ¼ in [59],as version of ν2. Thus the dispersion relation for fluctuations described in the last section. has the same scaling in both cases, as it must. In addition to the poles connected to the Fermi surface Another quantity providing information about the fluc- singularity, we also find other pairs of poles nucleating at nonzero ω. These additional fluctuations always seem tuation spectrum is γk ≡ argðGÞ. For the ð2 þ 1ÞQBHs, this was determined by [4] to be to appear near an oscillatory region living at the gap, and generically the poles fall into the region and cease to exist. −2πiνk −2πðqeÞ γk ≡ arg ðΓð−2νkÞðe − e eff ÞÞ; ð96Þ In a few cases, however, they miss the oscillatory region and survive as excitations at the gap, which also match with 2 2 1 with ðqeÞeff given by the last term in (35) without the sign: Fermi surfaces seen in the ð þ ÞQBHs in [59].

046011-14 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) A. Fermion A Consider first the pair of distinct fermions [59]

In the 2QBH background, where q1 and p1 are irrelevant, k−Δ − k F ≈−0.92513; ð104Þ these fermions have the same Dirac equation with μ ðm1;m2Þ¼ð1=2; −1=4Þ and ðq2;p2Þ¼ð2; 0Þ. In this case Δ is just half the chemical potential: with

μ ν 0 33211 Δ ≡ 2 μ k−Δ ¼ . ; ð105Þ jm2j ¼ 2 : ð101Þ associated to non-Fermi-liquid-like behavior. For ω ¼ Δ It was found in [59] that there is a Fermi surface (imposing − the regular boundary condition) at kΔ kF ≈ 0 743052 μ . ; ð106Þ kF ≈ 0 83934 μ . : ð102Þ implying

kΔ 1 58239 Continuing to use the regular boundary condition, we can μ ¼ . ; ð107Þ numerically solve for poles in the Green’s function from ω ¼ 0 up to ω ¼Δ; in so doing k varies continuously at which point from kF up to kΔ. The result of this procedure is depicted as the blue dots in Fig. 5. The dispersion is approximately ν ¼ 0.08211: ð108Þ linear at small values of ω=μ, with the form kΔ For fluctuations above ω ¼ Δ, we again have properties ω ≈ ≈ 0 724 vFk⊥;vF . ; ð103Þ characteristic of a non-Fermi liquid. Outside the stable region, ω will develop an imaginary part as well and move which is aboutffiffiffi 5=4 times larger than the speed of sound off into the complex plane. p ω Δ cs ¼ 1= 3 in this (conformal) background. At ω ¼ −Δ, The behavior at ¼ should match on to results from we have the extremal ð2 þ 1ÞQBHs found in [59]. Here the two fermions in (100) behave differently according to (90) since 2.0 they have opposite signs of q1: the first line in the table, called case 1 in [59], has ω ¼ ω2 1 þ Δ and thus the   þ  1.5        2 1 ω Δ   extremal ð þ ÞQBH limit zooms in on the ¼ side of                  the gapped region, while the second line in the table, case 3 1.0             ω −Δ    in [59], zooms in on the ¼ side.        0.5      We reproduce the plots from [59] showing locations of k           0 ≤ μ ≤ 1  Fermi surfaces at various values of for cases 3   R  0.0   and 1 in Fig. 6. The values of k at the μ → 1 limit must   F R     0.5  match to values of k−Δ and kΔ in Fig. 5, respectively, and indeed the Fermi surfaces indicated by blue dots do in each 1.0 case. Values of νk match as well.

1.5 There is a new feature here, however: case 1 has two 0.4 0.2 0.0 0.2 0.4 distinct Fermi surfaces, one on each side of an oscillatory region. (Since the oscillatory region is defined by ν2 < 0, and ν necessarily matches between the two limits, the FIG. 5 (color online). Poles in the retarded Green’s function oscillatory region is also present in the 2QBH case and is from ω ¼ 0 to ω ¼Δ (red dashed lines) for fermion A. indicated by a green line in Fig. 5.) Thus there ought to be

046011-15 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015)

FIG. 6 (color online). Fermi surfaces in (2 þ 1)-charge black holes as a function of μR from [59]; on the left is case 3, on the right case 1. The μR → 1 limits (purple dashed lines) match onto the left and right edges of the gapped region in Fig. 5. another pole in the 2QBH Green’s function at ω ¼ Δ, one purple, reaches the ω ¼ Δ axis inside the oscillatory not connected to the pole at ω ¼ 0. region and does not survive. Thus the overall picture is Indeed we find this is the case. Near ω ¼ 0, the only pole consistent between the ð2 þ 1ÞQBHs from [59] and the in the Green’s function is the one connected to the Fermi current work. surface pole at ω ¼ 0. But at a certain value 0 < ω < Δ,a As one moves even closer to ω ¼ Δ, several further pairs new pair of poles nucleates. As ω is increased further, they of poles nucleate, all of which disappear into the oscillatory spread apart with different values of k. One pole (indicated region; these continue to appear to the limits of our in red in Fig. 5) reaches the ω ¼ Δ line just below the numerical precision. These pairs lie very close to ω ¼ Δ oscillatory region and matches the red pole from Fig. 6. and are not indicated in the figure. Thus there appears to be This pole has an accumulation of additional states near the oscillatory region. kΔ ≈−0 58239 μ . ð109Þ B. Fermion B ν ω Δ and the same value of kΔ as the other pole at ¼ Consider now a second pair of fermions whose charges (108). Meanwhile, the partner pole, indicated in Fig. 5 in coincide when q1, p1 are neglected:

These behave the same for the 2QBH and in [59] a Fermi 1.5 surface was found at

kF ≈ 0 05202 μ . : ð111Þ k Again Δ ¼ μ=2, we can follow the pole away from ω ¼ 0 all the way to ω ¼Δ; this is plotted in Fig. 7. As with fermion A, the dispersion at small ω=μ is linear, and the 0.5 corresponding Fermi velocity v ≈ 0.678 is similar. At F 1.0 ω ¼ Δ, the pole hits the axis at

1.5 kΔ ≈ 0 79289 μ . ; ð112Þ 0.4 0.2 resulting in FIG. 7 (color online). Poles in the retarded Green’s function ν 0 22855; kΔ ¼ . ð113Þ from ω ¼ 0 to ω ¼Δ (red dashed lines) for fermion B.

046011-16 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015)

FIG. 8 (color online). Fermi surfaces in (2 þ 1)-charge black holes as a function of μR from [59] for cases 4 and 5. The μR → 1 limit (purple dashed lines) matches onto the right edges of the gapped region in Figs. 7 and 9, respectively. yet another non-Fermi-liquid-like region. Approaching Fig. 8. We again see that the value of kF=μ at μR → 1 ω ¼ −Δ, however, the line of poles reaches the axis inside matches the pole at ω ¼ Δ in Fig. 7. an oscillatory region, and there is no solution at ω ¼ −Δ There are oscillatory regions on both sides of the gap precisely. ω ¼Δ. Again we find a proliferation of additional poles Again we expect a match to results from [59] for near the ω ¼ Δ oscillatory region; the first such pair is ð2 þ 1ÞQBHs. The first fermion listed in the table (110) plotted in Fig. 7 and lies almost on top of the region. No has q1 < 0 and hence zooms in on ω ¼ −Δ, where we such poles are found for the ω ¼ −Δ oscillatory region. found no pole. Indeed, no Fermi surfaces were found for this fermion in [59]. For the second fermion in the table, C. Neutral fermion q1 > 0 and it zooms in on ω ¼ Δ. This fermion was called case 4 in [59], and its Fermi surfaces are reproduced in One may also consider the fermion

called case 5 in [59]. This case is neutral as far as the 2QBH lower pole falls into the oscillatory region while the upper is concerned, with Δ ¼ 3μ=2, and no Fermi surface was pole escapes, matching the pole in Fig. 8. Because of the found at ω ¼ 0. However, the results of [59] shown in vanishing of all gauge couplings for this case, the Dirac Fig. 8 indicate there should be a singularity at ω ¼ Δ to equation is symmetric under ðk; ωÞ → ð−k; −ωÞ, and we match the Fermi surface at μR → 1. This can occur if a pair find identical behavior at ω ¼ −Δ. of singularities nucleate at some nonzero ω, one makes it to The sole remaining fermion for which a Fermi surface the edge of the gap while the other falls into the oscillatory was found in [59], there called case 2, was characterized by region. This is indeed what we find in Fig. 9, where the the charges

While this fermion has Fermi surface singularities for many nature of the inner region a little better. Here we show that μ values of R, the line of Fermi surfaces falls into the in the near-horizon limit, the geometry can be lifted to μ 1 3 oscillatory region before reaching R ¼ . Thus we expect AdS3 × R . Moreover, the fermions lift to six-dimensional no singularity at ω ¼ Δ for this case; in fact we find no fermions as well; this lift identifies the charge q1 as Kaluza- Fermi surface at ω ¼ 0, either. Klein momentum in the sixth direction and places relations among the parameters q1, m2 and p1. VI. SIX-DIMENSIONAL LIFT The six-dimensional lift we demonstrate cannot be One concern about the 2QBH is its singularity at the considered the full picture, as several aspects of the fermion horizon. One may also wish to understand the geometrical are subleading (in the near-horizon expansion) and do not

046011-17 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) 1.5 2r2 L2 r2L2 Q2 dsˆ2 ¼ − dt2 þ dr2 þ dφ2 þ dx~2: ð120Þ L2 2r2 Q2 3 L2

One can calculate the sechsbeins and spin connections; we

k present them in Appendix C. The dimensional reduction of the Einstein-Hilbert action then gives   0.5 pffiffiffiffiffiffi ffiffiffiffiffiffi 1 1 −4ϕ p 2 pffiffi 2 L −ˆ ˆ − − ∂ϕ − 6 F EH ¼ gR ¼ g R 2 ð Þ 4 e ; ð121Þ 1.0

1.5 and the action for the Kaluza-Klein (KK) gauge field 1.5 1.0 0.5 matches that for the 1Q gauge field aμ (the one not turned on in the 2QBH background) if we define 1 ’ FIG. 9 (color online). Poles in the retarded Green s function aμ ≡ Aμ: ð122Þ from ω ¼ 0 to ω ¼Δ (red dashed lines) for the neutral fermion. 2

Thus the charge q1 is identified with Kaluza-Klein momentum. appear, in particular m1 and q2. Presumably including We may also obtain the leading term in the five- more subleading terms would provide a complete picture, dimensional potential as a reduction of a six-dimensional ultimately recovering the intermediate compactification 5 cosmological constant. Consider a cosmological term in between the (squashed) AdS5 × S lift and the five- 6D, dimensional reduction. The lift we display, however, elegantly resolves the singularity and establishes the pffiffiffiffiffiffi L −ˆΛˆ parameter relations, and so is sufficient for our purposes. Λ ¼ g ; ð123Þ and dimensionally reduce it, leading to A. Kaluza-Klein oxidation of metric ffiffiffiffiffiffi ϕ → 0 p pffiffi ˆ The near-horizon (r ) limit of the 5D extremal LΛ ¼ −ge 6Λ: ð124Þ 2QBH metric is The potential terms in the 5D theory are 2 8=3 2=3 4=3 2 2 − r 2 r Q 2 L 2   ds ¼ 2 2 3 dt þ 2 dx~ þ 4 3 2 3 dr ; ffiffiffiffiffiffi 8 ϕ 4 −2ϕ = 2 = = p pffiffi pffiffi L Q L r Q L − 6 6 pot ¼ g 2 e þ 2 e ; ð125Þ ð116Þ L L and the first term will dominate near the horizon. We can and the scalar diverges as match the 6D cosmological constant to the leading term if   we choose ϕ 1=3 pffiffi Q e2 6 ¼ : ð117Þ r ˆ 8 Λ ¼ ; ð126Þ L2 We now claim that this metric can be obtained as the dimensional reduction of a 6D metric. Using a hat for the and in 6D we have the term 6D metric, the Kaluza-Klein reduction ansatz is pffiffiffiffiffiffi 8 L −ˆ Λ ¼ g 2 : ð127Þ dsˆ2 ¼ e2αϕds2 þ e2βϕðdz þ AÞ2 L ϕ 3ϕ pffiffi 2 −pffiffi 2 Note that there is no obvious way to get the subleading term ¼ e 6ds þ e 6ðLdφ3 þ AÞ ; ð118Þ in this limit, since in 6D ϕ is part of the metric and one cannot simply write down terms with arbitrary powers of it. where we defined z ≡ Lφ3, and

1 3 B. Oxidation of gauge field α ¼ pffiffiffi ; β ¼ −3α ¼ − pffiffiffi : ð119Þ 2 6 2 6 The gauge field aμ is the 6D graviphoton; the other 5D gauge field Aμ must have another source in six dimensions. 3 The 6D metric is then AdS3 × R : In six dimensions, one can consider a one-form gauge field

046011-18 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) or a two-form gauge field. For the one-form case, the having used g ≡ 2=L. We see that the KK momentum n is reduction from six dimensions to five is also the charge q1. Moreover there is an effective mass

2ϕ 1 1 1 n pffiffi ˆ ˆ −2αϕ 6αϕ 6 − F2∧ ˆF2 − e F2∧ F2 − e F1∧ F1; m ¼ e : ð137Þ 2 ¼ 2 2 L ð128Þ This matches the leading term in the mass function (16) provided 2m2 ¼q1. This relation will hold for the where fermions as well. With the relation between m2 and the KK momentum n F2 ≡ dA1 − dA0∧A1;F1 ¼ dA0: ð129Þ in hand, we can provide a new interpretation of the gap Δ. Meanwhile, reducing a 6D two-form potential we have Consider a six-dimensional momentum vector in the KK ≡ 0 0 0 0 ω 0 0 0 0 and time directions kM ðkφ3 ;kt; ; ; ; Þ¼ðn; ; ; ; ; Þ. 1 1 1 The norm of this vector is − ˆ ∧ ˆ ˆ − −4αϕ ∧ − 4αϕ ∧ 2 H3 H3 ¼ 2 e H3 H3 2 e H2 H2; Q2 L2 L2 ð130Þ kMk ¼ n2 − ω2¼ ðΔ2 − ω2Þ; ð138Þ M r2L2 2r2 2r2 where 2 2 2 2 2 4 where we used n ¼ 2m2 and that Δ ¼ 8m2Q =L . Thus the gap represents the minimum energy required for the H3 ≡ dB2 − dB1∧A1;H2 ¼ dB1: ð131Þ six-dimensional momentum to be timelike; fluctuations of

Thus we see that, in order to match the gauge field Aμ in lesser energy, in the gapped region, correspond to spacelike five dimensions, we need to reduce a 6D two-form and 6D momenta due to the contribution of the momentum in identify the compact direction. The emergence of an AdS3 region suggests aspects of Bμ ∝ Aμ: ð132Þ the IR physics are controlled by a dual CFT2. A further understanding of the associated nonchiral Virasoro algebra Details of the coefficient of the kinetic term depend on could lead to greater insight concerning the nature of whether we take H3 to be self-dual or not; both matching the the gap. 5D Chern-Simons term, and the presumed ten-dimensional origin of this H3 in the five-form F5, suggest that it D. Fermion reduction should be. The normalization will not be needed for the arguments we make. Turn now to the fermionic case. The five-dimensional spinors in maximal gauged supergravity are four- Δ component symplectic Majorana pairs. These descend C. Probe scalar reduction and interpretation of gap from eight-component Majorana-Weyl fermions λa in six Although we are ultimately interested in fermions, it is dimensions: edifying to consider the simpler case of a 6D complex ˆ a a a b scalar η first. Starting with the six-dimensional action Γλ ¼ ελ ; λ ¼ B6Ωabðλ Þ : ð139Þ Z pffiffiffiffiffiffi 6 MN Here ε ¼1 gives the chirality, B6 is the six-dimensional Sη ≡ − d x −gˆgˆ ∂Mηˆ∂Nηˆ; ð133Þ conjugation matrix, and Ωab is an antisymmetric matrix in the space of spinors; details are relegated to Appendix B. we assume the KK ansatz The kinetic term in the 6D Lagrangian is

ηˆ ˆM inφ3 η μ pffiffiffiffiffiffi i ðx Þ¼e ðx Þð134Þ L −ˆ λ¯aΓM ˆN ∇ˆ λ ¼ g 2 e M N a; ð140Þ and arrive at the five-dimensional action   and a mass term is forbidden by the Weyl condition. The Z 2 ffiffiffiffiffiffi 4ϕ 5 p μν pffiffi n 2 covariant derivative is Sη → −2πL d x −g g DμηDνη þ e 6 jηj ; L2 ˆ 1 ∇ ∂ − ωˆ PQΓ : 141 ð135Þ N ¼ N 4 N PQ ð Þ where the covariant derivative is Let us reduce the kinetic term to five dimensions. Because of the symplectic Majorana condition, if we take one 6D n spinor to depend on φ3 ≡ z=L as a complex exponential, Dμ ≡∂μ − i Aμ ¼ ∂μ − ignaμ; ð136Þ L the other must depend on the conjugate. Furthermore, we

046011-19 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) will allow a rescaling by a power of the scalar ϕ. We choose and using that in the 2QBH we have masses of the form a basis for Γ matrices where the Weyl condition for one −ϕ 2ϕ 2 2ϕ chirality is solved by spinors of the form pffiffi pffiffi m2 pffiffi mðϕÞ¼gðm1e 6 þ m2e 6Þ ≈ e 6 þ; ð145Þ L  μ  ηffiffi ϕ χ1ðx Þ μ inφ3 p λ1ðz; x Þ ≡ e e 6 ; 0 we identify an effective mass, gauge coupling and Pauli  μ  coupling: ηffiffi ϕ χ2ðx Þ μ −inφ3 p λ2ðz; x Þ ≡ e e 6 ; ð142Þ ε ε 0 − n m2 ¼ 2 ;q1 ¼ n; p1 ¼ 4 : ð146Þ where the χa are four-component spinors that are sym- plectic Majorana in a 5D sense, and 0 stands for vanishing Thus these three five-dimensional quantities are determined four-component spinors. The other chirality would have the by two six-dimensional ones: the KK momentum n and the chirality ε. Since n is half-integer quantized, we expect q1 χi below the 0. Here n is half-integer quantized since the fields are fermionic. to be likewise and m2 to be quarter-integer quantized; we Performing the reduction (see Appendix C), we find we also find jp1j¼1=4. Since three quantities are determined can obtain a canonical 5D kinetic term with the choice from two underlying parameters, we have a relation, which η ¼ −1=4. Our total 5D Lagrangian becomes we can express as   1 ε 2ϕ −2 −1 i μ n pffiffi m2 ¼ q1p1; ð147Þ e L ¼ χγ¯ ∇μχ þ χ¯ e 6τ3χ 2 2 L     which is a result we advertised previously as Eq. (19). 1 2 ε −2ϕ n μ i pffiffi μν χ¯ γ τ χ χ¯ 6 γ χ Let us now compare these results to the fermions we þ 2 aμ 3 þ 2 4 e fμν ; ð143Þ L know in five dimensions. Below is the table from [59].5 One may verify that all the implied properties are present: where the τ3 Pauli matrix acts in the symplectic Majorana q1 is half-integer quantized (jq1j¼1=2 or jq1j¼3=2), m2 space (details in the Appendixes). Comparing to the is correspondingly quarter-integer quantized, jp1j¼1=4, canonical form of the Lagrangian and the relation (19) is satisfied. One can thus derive the 1 associated KK charge n and 6D chirality ε, and they are L χγ¯ μ∇ χ − χτ¯ χ χγ¯ μ τ χ χγ¯ μν χ ¼ 2 ði μ m 3 þ q aμ 3 þ ip fμν Þ added in the last two columns of the table. We find various modes involving the two lowest half-integer quantized ð144Þ charges, as well as both chiralities:

We note in passing that certain 5D spin-1=2 fields that mix We have not yet obtained the coupling of the fermion to with the gravitinos do not satisfy all the rules derived from the gauge field Aμ. One can consider a Pauli term in six this dimensional reduction; they do satisfy the relation dimensions, (147), but while they have jq1j¼1=2, they also have jm2j¼jp1j¼5=12. This indicates that those fields do not descend from a normal spin-1=2 kinetic term in 6D. The 5In [59], for all the fields whose near-boundary mass came out most natural explanation is that they are actually modes of negative (indicated as χ¯ in the first column), we flipped the signs the 6D gravitino, polarized in the z direction, though this of the gamma-matrix basis, thus flipping the signs of mi and pi. has not been checked. We have undone those flips here.

046011-20 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) i pffiffiffiffiffiffi 1 L −gˆλ¯aΓMNPH λ ; ν ≫ 1 → ∓ ≈ Pauli ¼ 2 MNP a ð149Þ ðjPj Þ jPj 2 jPj: ð152Þ and reduce this to five dimensions. Consider the case where Thus the six-dimensional lift is only fully self-consistent, one of the indices is polarized in the z direction. (If the field with no additional terms required, in this limit. However, strength is indeed self-dual, the other case will contribute even though this limit does not in general obtain for the similarly.) We then obtain schematically values of k we examine, the lift is nonetheless useful for the resolution of the singularity, for the proof of the relations ε ϕffiffi ffiffiffiffiffiffi i p p a μν between m2, q1 and p1, and for the intuition that excitations L ∼− e 6 −gχ¯ γ Fμνχ ; ð150Þ Pauli 2 a below the gap correspond to spacelike momenta. Subleading terms in r presumably can be introduced to resolve these 3 which is the right scaling of the scalar for the 5D Aμ Pauli issues, at the cost of giving up the simple AdS3 × R form. term. Thus one can obtain p2 simply by choosing the undetermined coefficient of the 6D Pauli term appropri- ACKNOWLEDGMENTS ately, since this is not fixed by anything else. We have enjoyed useful discussions and correspondence E. Validity of the six-dimensional lift with Mike Hermele, Shamit Kachru, Elias Kiritsis, Hong Liu, Leo Radzihovsky, and Subir Sachdev. The work This lift, while valuable, does not include all information. of O. D. was supported by the Department of Energy under In particular, the quantities m1 and q2 associated to each Grant No. DE-FG02-91-ER-40672. The work of S. S. G. fermion, which are subleading near the horizon, do not arise. was supported by the Department of Energy under Grant m1 is the subleading term in the potential and seems to be No. DE-FG02-91ER40671. The work of C. R. was sup- invisible. There also does not seem to be any 6D coupling ported in part by European Union’s Seventh Framework that generates q2, the ordinary gauge coupling in 5D: there Programme under Grant Agreements (FP7-REGPOT-2012- is no canonical coupling of a fermion to a two-form field 2013-1) No. 316165, PIF-GA-2011-300984, the EU that reduces to this. (It does come from the reduction of a program “Thales” MIS 375734 and was also cofinanced 6D one-form; but then one cannot produce the Pauli term, by the European Union (European Social Fund, ESF) and which is more important in the near-horizon limit.) Greek national funds through the Operational Program If one takes the 6D fermion or its 5D reduction and “Education and Lifelong Learning” of the National calculates the resulting Dirac equation, one gets a trunca- Strategic Reference Framework (NSRF) under “Funding tion of the equation analyzed in previous sections. In of proposals that have received a positive evaluation in the general, the first-order equations for r → 0 are 3rd and 4th Call of ERC Grant Schemes.”       a cω P APPENDIX A: A NOTE ON SECOND-ORDER ∂ þ þ b σ3 þ þ dω þ f iσ2 þ σ1 ψ ¼ 0; r r2 r2 r DECOUPLED DIRAC EQUATIONS ð151Þ In Secs. II–IV we use the second-order equations (27) associated to U, as described in [40]. In our previous work where the various constants are defined in (53); all these [59], as well as the numerical solutions in Sec. V, we use terms are required to obtain the correct limit of the Dirac ψ directly and the associated second-order equations equation. The 6D fermion, however, gives the equation with b, d and f dropped. While this might seem valid since ψ 00 − ψ 0 ∓ 0 − 2 2 − 2 ψ F þð X X þ Y Z XFÞ ; ðA1Þ they are all Oðr0Þ while the remaining terms are Oðr−1Þ or higher, it is actually not consistent; in generating the where second-order equations there are cross terms ab=r2 that 2 2 ≡∂ ∓ are no smaller than P =r . Thus we are not entitled to keep F r log ð Y þ ZÞ: ðA2Þ P if we drop b, d and f. The ultimate source of this subtlety is that the two components of ψ can be of different orders in The two formulations are rather similar, and both are the expansion in r. This is reflected in the fact that there is correct. The main difference, other than the introduction no rigorous scaling we can perform as r gets small that lets of a few factors of i, is that the formulation in terms of U us keep only P as a subleading term. involves X and Z in the coefficient of the first-order term ψ The effect of b, d and f in our analysis is to provide a (28), while the formulation in terms of has Y and Z constant term containing m1, m2 and q2 in the expression instead (A2). Because the energy ω is contained in Y, this for ν, which can play against the momentum-dependent P distinction is relevant when taking a small-ω limit. The ω term [see Eq. (59)]. We are thus only honestly entitled to definition of F puts in the denominator, and near- neglect b, d and f when they are much smaller than P. horizon expansions come with extra powers of ðr − rHÞ=ω. In the limit of large P we get for ν Such terms cannot be ignored; in the inner region they are

046011-21 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) of order one, while in the outer region they actually diverge. Define 8 × 8 gamma matrices in six dimensions via This ruins the near-horizon expansion for (2 þ 1)-charge ω ≠ 0 μ μ 5 black holes at . Thus it is preferred to work with the Γ ¼ γ ⊗ σ1; Γ ¼ I ⊗ iσ2; ðB5Þ quantities U. The analysis of [59] remains correct, however; the inner region was solved using the first-order and the six-dimensional chirality matrix is equations only and was matched to [4]. In the outer region, ω 0 0 1 2 3 4 5 ¼ was taken strictly before the near-horizon limit, Γ ≡ Γ Γ Γ Γ Γ Γ ¼ I ⊗ σ3; ðB6Þ which gives correct expressions. ψ One may ask whether the formulation affects the obeying as usual 2-charge black hole. For that case one finds Γ2 ¼ 1; fΓ; Γμg¼0: ðB7Þ 2 − ∓ P O F ¼ þ ðrÞ; ðA3Þ r cω We also have 6D conjugation and transpose matrices which implies that M −1 M M −1 M T B6Γ B6 ¼ðΓ Þ ;C6Γ C6 ¼ðΓ Þ ; ðB8Þ   aP 1 ∓X0 XF ¼ − þ O ; ðA4Þ defined as cωr2 r

B6 ¼ B ⊗ I2 2;C6 ¼ C ⊗ σ1; ðB9Þ and one ends up with Eq. (57) but with one term flipped: × ω which in the particular basis mentioned above take the c P aP 4 5 0 0 4 5 → : ðA5Þ form B6 ≡ −iΓ Γ Γ, C6 ¼ Γ B6 ¼ −iΓ Γ Γ Γ. Since σ1 is a cω μ real and symmetric, the Γ inherit the same reality and μ 0 1 2 3 Interestingly, this does not affect the near-horizon analysis. symmetry properties as the γ . Thus Γ , Γ , Γ and Γ 4 5 0 4 For solutions away from ω Δ, this term is dropped as are imaginary, while Γ and Γ are real, while Γ , Γ and j j¼ 5 1 2 3 subleading. For solutions at jωj¼Δ, the two terms (A5) Γ are antisymmetric while Γ , Γ and Γ are symmetric. are the same. In addition, the chirality matrix Γ is real and symmetric. The fermionic reduction is complicated by the presence of symplectic Majorana spinors in five and six dimensions. APPENDIX B: FIVE- AND SIX-DIMENSIONAL The index a; b; … runs over multiple spinors, and these will SPINOR CONVENTIONS be related in pairs by the symplectic Majorana condition. We use a mostly plus signature metric, but a “mostly We raise and lower symplectic Majorana spinor indices minus” type Clifford algebra, meaning that with the antisymmetric matrix Ωab:

γμ γν −2ημν b a ab f ; g¼ ; ðB1Þ χa ¼ Ωabχ ; χ ¼ Ω χb; ðB10Þ

ab ab a where we use the underline to denote flat-space indices. where Ω is the inverse of Ωab, Ω Ωbc ¼ δc;wemay 12 Five-dimensional gamma matrices are 4 × 4, and since we take conventions where Ω12 ¼ −Ω ¼ 1 for each pair of are in odd dimension there is a relation spinors. The symplectic Majorana condition is expressed in terms of the reality matrix B: γ4 ¼ γ0γ1γ2γ3: ðB2Þ a b χ ¼ BðχaÞ ¼ BΩabðχ Þ : ðB11Þ One can define conjugation and transpose matrices B and C obeying Defining the barred spinor as

γμ −1 γμ γμ −1 γμ T a † 0 B B ¼ð Þ ;CC ¼ð Þ ðB3Þ χ¯ ≡ ðχaÞ γ ; ðB12Þ and satisfying we may also express the symplectic Majorana condition as

−1 B ¼ B; BT ¼ B ¼ −B; χ¯a ¼ðχaÞTC: ðB13Þ C−1 ¼ CT ¼ −C; C ¼ C: ðB4Þ We can use the latter form to prove the bilinear “Majorana ” One choice is B ≡ iγ4, C ≡ iγ0γ4 which makes γ0, γ1, γ2 flip expression for symplectic Majorana spinors: γ3 γ4 γ0 γ4 and imaginary, while is real; and are antisym- μ μ μ μ χ¯aγ 1 …γ n ψ b ψ¯ bγ n …γ 1 χa metric while γ1, γ2 and γ3 are symmetric. ¼ : ðB14Þ

046011-22 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015) This constrains possible terms in the Lagrangian for dsˆ2 ¼ e2αϕds2 þ e2βϕðdz þ AÞ2; ðC1Þ symplectic Majorana spinors; some terms require a relative sign between the two spinors of the pair in order to end up the sechsbeins are nonzero. It is useful to use a Pauli matrix notation for the ˆμ αϕ μ ˆμ 0 z βϕ z βϕA symplectic Majorana space of each spinor pair to denote e ν ¼ e e ν; e z ¼ ;ez ¼ e ;eν ¼ e ν; χ¯ χ¯a χ χ this. If we agree that stands for and for b, the ðC2Þ allowed terms are μ where e ν is the 5D fünfbein. The inverse sechsbeins are 1 L χγ¯ μ∇ χ − χτ¯ χ χγ¯ μ τ χ χγ¯ μν χ ¼ ði μ m 3 þ q Aμ 3 þ ip Fμν Þ; ˆμ −αϕ μ ˆμ ˆz −βϕ 2 e ν ¼ e e ν; e z ¼ 0; e z ¼ e ; z −αϕ μ ðB15Þ eˆ ν ¼ −e e νAμ: ðC3Þ leading to the Dirac equation The spin connection is then

γμ∇ − τ γμ τ γμν χ 0 νρ 1 ði μ m 3 þ q Aμ 3 þ ip FμνÞ ¼ : ðB16Þ ωˆ ω νρ α ν ∂ρϕ − ρ ∂νϕ − 2ðβ−αÞϕA F νρ μ ¼ μ þ ðe μ e μ Þ 2 e μ ; Here we use τ to stand for Pauli matrices in the symplectic 1 ωˆ νz ðβ−αÞϕF ν − β ðβ−αÞϕA ∂νϕ Majorana space, to distinguish them from the σ which are μ ¼ 2 e μ e μ ; Pauli matrices in the Clifford algebra space. The factor of τ3 1 ωˆ νρ − 2ðβ−αÞϕF νρ in the mass and gauge coupling terms makes up for the z ¼ 2 e ; antisymmetry of the spinor contraction. The kinetic and ωˆ νz −β ðβ−αÞϕ∂νϕ Pauli terms do not need such a factor due to the minus signs z ¼ e ; ðC4Þ from integrating the derivative by parts and from the 6 where all flat indices on the right-hand side are obtained antisymmetry of Fμν, respectively. from the curved-space indices with the 5D fünfbein. We Completely analogous expressions to (B11) and (B13) notice that due to the origin of the spin connection terms hold for six-dimensional symplectic Majorana spinors. βϕ from eˆz ¼ e ðdz þ AÞ, we can write the last terms of the In that case we also have first and second lines as

½B6; Γ¼0; ðB17Þ νρ νρ v ρ ρ ν νρ ωˆ μ ¼ ωμ þ αðe μ∂ ϕ − eˆ μ∂ ϕÞþAμωˆ z 1 which along with Γ ¼ Γ implies that the symplectic νz ðβ−αÞϕ ν νz ωˆ μ ¼ e F μ þ Aμωˆ z : ðC5Þ Majorana condition is compatible with a Weyl condition 2 This will be useful in what follows. Γλ λ ¼ ; ðB18Þ Consider the reduction of the 6D fermion kinetic term (140) with the fermion ansatz (142). In all of our reductions allowing symplectic Majorana-Weyl spinors. In these con- I σ below, we will end up with 2×2 or 3 in the extra two- ventions, a 6D Weyl spinor is simply an eight-component component space, and the eight-component inner product spinor with only the top four or bottom four components between spinors will collapse to a four-component inner nonzero; if it is symplectic Majorana-Weyl, then thanks to product; because the fermion bilinears will be diagonal in (B9) the four nonzero components obey precisely the 5D the two-component space, the factors of einφ3 will cancel symplectic Majorana condition. Γ I ⊗ σ between the fermion and its conjugate. Since ¼ 4×4 3, the appearance of σ3 will give us a factor of the chirality ε. APPENDIX C: DETAILS OF FERMIONIC First consider the terms involving the partial derivative. KALUZA-KLEIN REDUCTION There are three such terms: Here we provide the details of the reduction of the 6D pffiffiffiffiffiffi i L −ˆ λ †Γ0 Γμ ν ∂ Γ5 ˆz ∂ Γμ ˆz ∂ λ Majorana-Weyl spinor to five dimensions discussed in ∂ ¼ g 2 ð aÞ ð e μ ν þ e z z þ e μ zÞ a; Sec. VI. Given the ansatz for the decomposition of the ðC6Þ 6D metric, ν where no fourth term appears since eˆ z ¼ 0. These three 6In the literature symplectic Majorana Lagrangians are typi- terms will become the 5D kinetic term, mass term and KK cally expressed in terms of a nondiagonalized basis for the gauge coupling term, respectively. Consider first the kinetic spinors, with τ2 instead of τ3. We are concerned with matching to term. We have our already-diagonalized eigenvectors, so we use the form above. The transformation from one basis to the other preserves the Γ0Γμ γ0γμ ⊗ I symplectic Majorana condition. ¼ 2×2; ðC7Þ

046011-23 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015)    ffiffiffiffiffiffi 1 ε 2ϕ and so the eight-component Weyl spinors reduce to four- p i μ n pffiffi φ L∂ ¼ −g χγ¯ ∂μχ þ χ¯ e 6 τ3χ component spinors as described above. The ein 3 factors 2 2 L cancel. We get one term where the partial derivative acts on    1 2n μ i μ the χ : þ χ¯ γ aμτ3χ − pffiffiffi χγ¯ ð∂μϕÞχ : ðC16Þ a 2 L 8 6 ϕ ffiffiffiffiffiffi 2ηϕ −ϕ pffiffi p i pffiffi † 0 μ pffiffi ν L 6 − 6 χ γ γ 2 6 ∂ χ kin ¼ e g 2 e ð aÞ e e μ ν a þ Consider now the spin connection terms. We will need to ffiffiffiffiffiffi ð1þ4ηÞϕ evaluate p pffiffi i μ − 2 6 χ¯aγ ∂ χ ¼ ge 2 μ a þ: ðC8Þ i pffiffiffiffiffiffi L − −ˆλ¯aΓM ˆN ωˆ PQΓ λ Thus to get the canonical kinetic term, we need to pick ω ¼ 8 g e M N PQ a: ðC17Þ η ¼ −1=4. Another term results when the derivative acts on the exponential before it cancels, so the total result coming We have from the first term in (C6) is   M N PQ μ ν PQ μ z PQ Γ eˆ ωˆ Γ ¼ Γ eˆ μωˆ ν Γ þ Γ eˆ μωˆ Γ ffiffiffiffiffiffi i i M N PQ PQ z PQ L p− χ¯aγμ∂ χ − ffiffiffi χ¯aγμ ∂ ϕ χ kin ¼ g μ a p ð μ Þ a : ðC9Þ z z PQ 2 8 6 þ Γ eˆ zωˆ z ΓPQ σ −αϕγμ ωˆ PQ − A ωˆ PQ Γ The latter part will be canceled by the spin connection. ¼ 1e ð μ μ z Þ PQ ∂ −βϕ PQ For the other two terms in (C6) we will need z þ iσ2e ωˆ z ΓPQ: ðC18Þ derivatives of the spinor: PQ Note that in the term in parentheses, the Aμωˆ z piece in in ωˆ PQ ∂zλ1 ¼ λ1; ∂zλ2 ¼ − λ2; ðC10Þ entirely cancels against terms in the μ piece due to the L L relations (C5). The first term in (C18) then gives us or in Pauli matrix notation −αϕ μ ˆ PQ ˆ PQ σ1e γ ðωμ − Aμωz ÞΓPQ in ∂ λ τ λ −αϕ μ νρ μ ðβ−αÞϕ μν z ¼ 3 : ðC11Þ ¼ σ1e ðγ ωμ γνρ − 8αγ ∂μϕ − σ3e γ F μνÞ; L ðC19Þ Using the gamma-matrix identity

0 5 0 where to get the coefficient in the middle term we used the Γ Γ ¼ −γ σ3; ðC12Þ 5D Clifford identity we obtain for the middle term in (C6) μ γ γμν ¼ −4γν: ðC20Þ ϕ ffiffiffiffiffiffi ϕ 3ϕ pffiffi p n − pffiffi † 0¯ pffiffi L 6 − 2 6 χ σ γ 2 6 τ bχ mass ¼ e g e ð aÞ 3 e ð 3Þa b 2L Meanwhile the second term in (C18) becomes 2ϕ pffiffi pffiffiffiffiffiffi nε 6 − χ¯a τ bχ ¼ e g ð 3Þa b; ðC13Þ 2L i σ −βϕωˆ PQΓ −2βσ −αϕγμ∂ ϕ − σ ðβ−2αÞϕγμνF i 2e z PQ ¼ 1e μ 2 2e μν; where the σ3 got us a factor of the chirality ε. Consider now the final term in (C6). This gives ðC21Þ

pffiffiffiffiffiffi i L −ˆ λ †Γ0Γμ ˆz ∂ λ and combining these and using β ¼ −3α we get KK gauge ¼ g 2 ð aÞ e μ z a ffiffiffiffiffiffi p n a μ b M N PQ ¼ −g χ¯ γ Aμðτ3Þ χ ; ðC14Þ Γ eˆ ωˆ Γ 2L a b M N  PQ  1 −αϕ μ νρ μ ðβ−αÞϕ μν σ1e γ ωμ γνρ − 2αγ ∂μϕ − σ3e γ F μν : or in terms of the 1Q gauge field aμ ¼ Aμ=2 ¼ 2

pffiffiffiffiffiffi 1 ð2n μ ðC22Þ L ¼ −g χ¯ γ aμτ3χ: ðC15Þ KK gauge 2 LÞ Now plug this in to the reduction of the Lagrangian (C17). Assembling these together, we get for the total terms (C6) The reduction of the metric determinant and the two λ’s into αϕ 0 ¯a coming from the partial derivative in the 6D spinor kinetic χ’s give a factor e . The Γ inside λ gives a factor σ1.We term thus find the terms

046011-24 FERMIONIC RESPONSE IN A ZERO ENTROPY STATE … PHYSICAL REVIEW D 91, 046011 (2015)  ffiffiffiffiffiffi p a i μ νρ i μ canonical kinetic term (C9). The last term provides a Pauli Lω ¼ −gχ¯ − γ ωˆ μ γνρ þ pffiffiffi γ ∂μϕ 8 8 6 coupling to the KK gauge field, which we rewrite using  A 2 ε aμ ¼ μ= . Our total 5D Lagrangian is now i ðβ−αÞϕ μν þ e γ F μν χ : ðC23Þ 16 a     1 ε 2ϕ 1 2 −1 i μ n pffiffi n μ e L ¼ χγ¯ ∇μχ þ χ¯ e 6τ3χ þ χ¯ γ aμτ3χ Finally we combine these with the rest of the terms we 2 2 L 2 L   already found from the partial derivative pieces (C16). The ε −2ϕ i pffiffi μν χ¯ 6 γ χ first new term simply gives the 5D spin connection, þ 2 4 e fμν ; ðC24Þ promoting the 5D partial derivative to a covariant deriva- μ tive. The second new term nicely cancels the extra γ ∂μϕ term generated by rescaling the spin-1=2 field to get a as given in (143).

[1] S.-S. Lee, A non-Fermi liquid from a charged black hole: A [16] E. Brynjolfsson, U. Danielsson, L. Thorlacius, and T. Zingg, critical Fermi ball, Phys. Rev. D 79, 086006 (2009). Black hole thermodynamics and heavy fermion metals, [2] H. Liu, J. McGreevy, and D. Vegh, Non-Fermi liquids from J. High Energy Phys. 08 (2010) 027. holography, Phys. Rev. D 83, 065029 (2011). [17] C. Charmousis, B. Gouteraux, B. Kim, E. Kiritsis, and R. [3] M. Cubrovic, J. Zaanen, and K. Schalm, String theory, Meyer, Effective holographic theories for low-temperature quantum phase transitions and the emergent Fermi-liquid, condensed matter systems, J. High Energy Phys. 11 (2010) Science 325, 439 (2009). 151. [4] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, Emergent [18] S. Sachdev, Holographic Metals and the Fractionalized quantum criticality, Fermi surfaces, and AdS2, Phys. Rev. D Fermi Liquid, Phys. Rev. Lett. 105, 151602 (2010). 83, 125002 (2011). [19] S. A. Hartnoll and A. Tavanfar, Electron stars for holo- [5] M. Berkooz, D. Reichmann, and J. Simon, A Fermi surface graphic metallic criticality, Phys. Rev. D 83, 046003 (2011). model for large supersymmetric AdS(5) black holes, J. High [20] S. A. Hartnoll, D. M. Hofman, and A. Tavanfar, Holo- Energy Phys. 01 (2007) 048. graphically smeared Fermi surface: Quantum oscillations [6] M. Berkooz and D. Reichmann, Weakly renormalized near and Luttinger count in electron stars, Europhys. Lett. 95, 1=16 SUSY Fermi liquid operators in N ¼ 4 SYM, J. High 31002 (2011). Energy Phys. 10 (2008) 084. [21] V. G. M. Puletti, S. Nowling, L. Thorlacius, and T. Zingg, [7] F. Denef, S. A. Hartnoll, and S. Sachdev, Quantum oscil- Holographic metals at finite temperature, J. High Energy lations and black hole ringing, Phys. Rev. D 80, 126016 Phys. 01 (2011) 117. (2009). [22] M. Edalati, R. G. Leigh, K. W. Lo, and P. W. Phillips, [8] S. S. Gubser and F. D. Rocha, Peculiar properties of a Dynamical gap and cuprate-like physics from holography, charged dilatonic black hole in AdS5, Phys. Rev. D 81, Phys. Rev. D 83, 046012 (2011). 046001 (2010). [23] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, [9] K. Goldstein, S. Kachru, S. Prakash, and S. P. Trivedi, Holographic non-Fermi liquid fixed points, Phil. Trans. R. Holography of charged dilaton black holes, J. High Energy Soc. A 369, 1640 (2011). Phys. 08 (2010) 078. [24] L. Huijse and S. Sachdev, Fermi surfaces and gauge-gravity [10] S. A. Hartnoll, J. Polchinski, E. Silverstein, and D. Tong, duality, Phys. Rev. D 84, 026001 (2011). Towards strange metallic holography, J. High Energy Phys. [25] K. Jensen, S. Kachru, A. Karch, J. Polchinski, and E. 04 (2010) 120. Silverstein, Towards a holographic marginal Fermi liquid, [11] T. Faulkner and J. Polchinski, Semi-holographic Fermi Phys. Rev. D 84, 126002 (2011). liquids, J. High Energy Phys. 06 (2011) 012. [26] S. A. Hartnoll, D. M. Hofman, and D. Vegh, Stellar spec- [12] M. Ammon, J. Erdmenger, M. Kaminski, and A. O’Bannon, troscopy: Fermions and holographic Lifshitz criticality, Fermionic operator mixing in holographic p-wave super- J. High Energy Phys. 08 (2011) 096. fluids, J. High Energy Phys. 05 (2010) 053. [27] N. Iqbal, H. Liu, and M. Mezei, Semi-local quantum liquids, [13] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, J. High Energy Phys. 04 (2012) 086. From black holes to strange metals, arXiv:1003.1728. [28] M. Edalati, K. W. Lo, and P. W. Phillips, Neutral order [14] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, parameters in metallic criticality in d ¼ 2 þ 1 from a hairy Strange metal transport realized by gauge/gravity duality, electron star, Phys. Rev. D 84, 066007 (2011). Science 329, 1043 (2010). [29] B. Gouteraux and E. Kiritsis, Generalized holographic [15] T. Hartman and S. A. Hartnoll, Cooper pairing near charged quantum criticality at finite density, J. High Energy Phys. black holes, J. High Energy Phys. 06 (2010) 005. 12 (2011) 036.

046011-25 DEWOLFE, GUBSER, AND ROSEN PHYSICAL REVIEW D 91, 046011 (2015) [30] S. Sachdev, A model of a Fermi liquid using gauge-gravity [51] R. A. Davison, K. Schalm, and J. Zaanen, Holographic duality, Phys. Rev. D 84, 066009 (2011). duality and the resistivity of strange metals, Phys. Rev. B 89, [31] N. Ogawa, T. Takayanagi, and T. Ugajin, Holographic Fermi 245116 (2014). surfaces and entanglement entropy, J. High Energy Phys. 01 [52] R. A. Davison, M. Goykhman, and A. Parnachev, AdS/CFT (2012) 125. and Landau Fermi liquids, J. High Energy Phys. 07 (2014) [32] L. Huijse, S. Sachdev, and B. Swingle, Hidden Fermi 109. surfaces in compressible states of gauge-gravity duality, [53] S. A. Hartnoll, Horizons, holography and condensed matter, Phys. Rev. B 85, 035121 (2012). arXiv:1106.4324. [33] E. Shaghoulian, Holographic entanglement entropy and [54] N. Iqbal, H. Liu, and M. Mezei, Lectures on holographic Fermi surfaces, J. High Energy Phys. 05 (2012) 065. non-Fermi liquids and quantum phase transitions, arXiv: [34] N. Iqbal and H. Liu, Luttinger’s theorem, superfluid 1110.3814. vortices, and holography, Classical Quantum Gravity 29, [55] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. 194004 (2012). Abrahams, and A. E. Ruckenstein, Phenomenology of the [35] A. Allais, J. McGreevy, and S. J. Suh, A Quantum Electron Normal State of Cu-O High-Temperature Superconductors, Star, Phys. Rev. Lett. 108, 231602 (2012). Phys. Rev. Lett. 63, 1996 (1989). [36] J. Polchinski and E. Silverstein, Large-density field theory, [56] P. W. Anderson, “Luttinger-Liquid” Behavior of the Normal viscosity, and “2kF” singularities from string duals, Metallic State of the 2D Hubbard Model, Phys. Rev. Lett. Classical Quantum Gravity 29, 194008 (2012). 64, 1839 (1990). [37] S. A. Hartnoll and E. Shaghoulian, Spectral weight in [57] P. Gegenwart, Q. Si, and F. Steglich, Quantum criticality in holographic scaling geometries, J. High Energy Phys. 07 heavy-fermion metals, Nat. Phys. 4, 186 (2008). (2012) 078. [58] O. DeWolfe, S. S. Gubser, and C. Rosen, Fermi Surfaces [38] K. Hashimoto and N. Iizuka, A comment on holographic in Maximal Gauged Supergravity, Phys. Rev. Lett. 108, Luttinger theorem, J. High Energy Phys. 07 (2012) 064. 251601 (2012). [39] M. Goykhman, A. Parnachev, and J. Zaanen, Fluctuations in [59] O. DeWolfe, S. S. Gubser, and C. Rosen, Fermi surfaces in finite density holographic quantum liquids, J. High Energy N ¼ 4 super-Yang-Mills theory, Phys. Rev. D 86, 106002 Phys. 10 (2012) 045. (2012). [40] S. S. Gubser and J. Ren, Analytic fermionic Green’s functions [60] J. P. Gauntlett, J. Sonner, and D. Waldram, Universal from holography, Phys. Rev. D 86, 046004 (2012). Fermionic Spectral Functions from String Theory, Phys. [41] M. Berkooz, A. Frishman, and A. Zait, Degenerate rotating Rev. Lett. 107, 241601 (2011). black holes, chiral CFTs and Fermi surfaces I—Analytic [61] R. Belliard, S. S. Gubser, and A. Yarom, Absence of a results for quasinormal modes, J. High Energy Phys. 08 Fermi surface in classical minimal four-dimensional gauged (2012) 109. supergravity, J. High Energy Phys. 10 (2011) 055. [42] R. Gopakumar, A. Hashimoto, I. R. Klebanov, S. Sachdev, [62] J. Gauntlett, J. Sonner, and D. Waldram, Spectral function of and K. Schoutens, Strange metals in one spatial dimension, the supersymmetry current (II), J. High Energy Phys. 11 Phys. Rev. D 86, 066003 (2012). (2011) 153. [43] M. Alishahiha, M. R. M. Mozaffar, and A. Mollabashi, [63] B. Pioline and J. Troost, Schwinger pair production in AdS Holographic aspects of two-charged dilatonic black hole (2), J. High Energy Phys. 03 (2005) 043. in AdS5, J. High Energy Phys. 10 (2012) 003. [64] K. Behrndt, M. Cvetic, and W. Sabra, Nonextreme black [44] M. Kulaxizi, A. Parnachev, and K. Schalm, On holographic holes of five-dimensional N ¼ 2 AdS supergravity, Nucl. entanglement entropy of charged matter, J. High Energy Phys. B553, 317 (1999). Phys. 10 (2012) 098. [65] M. Cvetic and S. S. Gubser, Thermodynamic stability and [45] A. Donos, J. P. Gauntlett, and C. Pantelidou, Semi-local phases of general spinning branes, J. High Energy Phys. 07 quantum criticality in string/M-theory, J. High Energy Phys. (1999) 010. 03 (2013) 103. [66] S. S. Gubser and I. Mitra, Instability of charged black holes [46] M. Berkooz, A. Frishman, and A. Zait, Stability of in Anti-de Sitter space, arXiv:hep-th/0009126. 5 rapidly-rotating charged black holes in AdS5 × S , arXiv: [67] S. S. Gubser and I. Mitra, The evolution of unstable black 1301.6524. holes in anti-de Sitter space, J. High Energy Phys. 08 (2001) [47] J. Erdmenger and S. Steinfurt, A universal fermionic 018. analogue of the shear viscosity, J. High Energy Phys. 07 [68] S. K. Domokos and J. A. Harvey, Baryon Number-Induced (2013) 018. Chern-Simons Couplings of Vector and Axial-Vector [48] M. V.Medvedyeva, E. Gubankova, M. Čubrović, K. Schalm, Mesons in Holographic QCD, Phys. Rev. Lett. 99, and J. Zaanen, Quantum corrected phase diagram of holo- 141602 (2007). graphic fermions, J. High Energy Phys. 12 (2013) 025. [69] S. Nakamura, H. Ooguri, and C.-S. Park, Gravity dual [49] L. Q. Fang, X.-H. Ge, and X.-M. Kuang, Holographic of spatially modulated phase, Phys. Rev. D 81, 044018 fermions with running chemical potential and dipole cou- (2010). pling, Nucl. Phys. B877, 807 (2013). [70] M. Rozali, D. Smyth, E. Sorkin, and J. B. Stang, Holo- [50] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, graphic Stripes, Phys. Rev. Lett. 110, 201603 (2013). Charge transport by holographic Fermi surfaces, Phys. Rev. [71] A. Donos and J. P. Gauntlett, Holographic charge density D 88, 045016 (2013). waves, Phys. Rev. D 87, 126008 (2013).

046011-26