Pomeranchuk Instability in a Non-Fermi Liquid from Holography

UTTG-24-11

Pomeranchuk Instability in a non-Fermi Liquid from Holography

Mohammad Edalati,1 Ka Wai Lo,2 and Philip W. Phillips2 1Weinberg Theory Group, University of Texas at Austin, Austin TX 78712, USA 2Department of Physics, University of Illinois at Urbana-Champaign, Urbana IL 61801, USA The Pomeranchuk instability, in which an isotropic Fermi surface distorts and becomes anisotropic due to strong interactions, is a possible mechanism for the growing number of experimental systems which display transport properties that differ along the x and y axes. We show here that the gauge- gravity duality can be used to describe such an instability in fermionic systems. Our holographic model consists of fermions in a background which describes the causal propagation of a massive neutral spin-two field in an asymptotically AdS spacetime. The Fermi surfaces in the boundary theory distort spontaneously and become anisotropic once the neutral massive spin-two field develops a normalizable mode in the bulk. Analysis of the fermionic correlators reveals that the low-lying fermionic excitations are non-Fermi liquid-like both before and after the Fermi surface shape distortion. Further, the spectral weight along the Fermi surface is angularly dependent and can be made to vanish along certain directions.

I. INTRODUCTION

To a surprising extent, the electronic properties of most metals are well described by Landau Fermi liquid theory. The key building blocks of this theory are quasi-particles and a Fermi surface, the surface in momentum space that demarcates filled from empty states. For a system of weakly interacting electrons, the shape (a) (b) (c) of the Fermi surface is isotropic. On the other hand, if the interactions among the quasi-particles are strong FIG. 1. (a) Examples of the Pomeranchuk instability in (a) l = 2, enough, other possibilities might emerge. Pomeranchuk (b) l = 3 and (c) l = 4 channels. The + and − signs on the [1] showed that forward scattering among quasi-particles anisotropic Fermi surfaces show the momentum gain and loss com- with non-trivial angular momentum and spin structure pared to the isotropic Fermi surfaces (which are shown above by can lead to an instability of the ground state towards the dashed lines). forming an anisotropic Fermi surface. For example, the simplest form of a two-dimensional nematic Fermi liquid Pomeranchuk mechanism is the only root cause2, it nev- can be thought of, in the continuum limit, as a realization ertheless offers a framework where such anisotropies can of the Pomeranchuk instability in the angular momentum be characterized in detail. So far, much of the work in l = 2 and spin s = 0 particle-hole channel whereby a cir- this direction has focused on reaching the nematic phases cular Fermi surface spontaneously distorts and becomes of correlated electron matter, via the Pomeranchuk insta- elliptical with a symmetry of a quadrupole (d-wave). bility, in a weakly interacting Fermi fluid [9, 10]. A nat- Such an instability, depicted in Figure1(a), is often re- ural question to ask is whether a nematic phase could be ferred to in the literature as the quadrupolar Pomer- approached, via a Pomeranchuk instability, from a non- 1 anchuk instability . In practice, however, the instability Fermi liquid phase. To answer the question, one has to breaks the discrete point group symmetry of the underly- address the origin of the Pomeranchuk instability beyond arXiv:1203.3205v2 [hep-th] 17 Sep 2012 ing lattice. As such, systems undergoing a Pomeranchuk the Fermi liquid framework. Unlike Fermi liquids, there is instability exhibit manifestly distinct transport proper- no notion of stable quasi-particles in non-Fermi liquids. ties along the different crystal axes. Experimentally sev- Nevertheless, non-Fermi liquids have sharp Fermi sur- eral instances of anisotropic transport in strongly cor- faces. Consequently, a Pomeranchuk instability, viewed related electronic systems have been reported [2–5] and simply as a spontaneous shape distortion of the underly- occupy much of the current focus in correlated electron ing Fermi surface, is still a theoretical possibility. matter. Although it is unclear at present whether the We explore here how such an instability can be studied using the gauge-gravity duality, hereafter referred to as holography. The duality, which maps certain d- dimensional strongly coupled quantum field theories to 1 The instabilities in other channels, including the antisymmetric spin channel, are also interesting. (Note that there is no Pomer- anchuk instability in the l = 1, s = 0 channel.) In this paper, however, we only focus on the quadrupolar Pomeranchuk insta- 2 Nematic phases could also be obtained by the melting of stripe bility and the resulting nematic phase. phases; see [7,8] and references therein. 2

(d + 1)-dimensional semiclassical gravitational theories III contains the fermonic degrees of freedom as well as the in asymptotically AdS spacetimes, is a promising tool results illustrating that the low-lying excitations are non- in modeling some features of strongly correlated con- Fermi liquid in nature as well as the shape distortion of densed matter systems. For reviews on the applications the underlying Fermi surface. of holography for condensed matter physics, see [11–17]. We take the system to be d = 2 + 1 dimensional, as this is the most interesting case experimentally [2–5]. II. BACKGROUND Also, we consider the case where the broken phase is a nematic (non-)Fermi liquid with the symmetry of a quadrupole. In other words, we only explore the possi- Since we want to explore the possibility of a transition bility of a quadrupolar Pomeranchuk instability. Such an from a non-Fermi liquid to a nematic Fermi liquid phase instability is characterized by an order parameter which in the boundary, the first step is to construct a back- is a neutral symmetric traceless tensor Oij where i and j ground which consistently incorporates a neutral mas- denote spatial indices. Thus, in order to use holography sive spin-two field ϕµν . This field is dual to the operator to realize the transition to such a phase in the boundary whose vacuum expectation value (vev) is the order pa- theory, one has to turn on a neutral massive spin-two rameter for the nematic phase. field ϕµν in the bulk and show that, under some circum- stances, it develops a non-trivial normalizable profile in the bulk radial direction. From the point of view of the A. BGP Lagrangian physics in the bulk, there is a serious hurdle that one needs to overcome in order to successfully describe the In flat (d + 1)-dimensional Minkowski spacetime, Fierz propagation of a massive spin-two field, which has to do and Pauli suggested [18] a Lagrangian for a free neutral with the existence of extra unwanted degrees of freedom massive spin-two field which is quadratic in derivatives (ghosts) and superluminal modes. Eliminating ghosts and correctly describes the propagating degrees of free- and superluminal modes within a theory that also allows dom for such a field. Generalizing to curved spacetimes, for the condensation of the operator dual to the massive Buchbinder, Gitman and Pershin [19], following [20], pro- spin-two field is the first step. In addition, a coupling of posed a simple Lagrangian (quadratic in the massive the correct type must be introduced in the bulk between spin-two field, and also quadratic in derivatives) which the massive spin-two field and the fermions in order to describes the causal propagation of the correct number bring about the shape distortion of the Fermi surface in of degrees of freedom of a neutral massive spin-two field the boundary. We present here a minimal model which in a fixed Einstein background in arbitrary dimensions. can address each of these problem, and show that the In four bulk dimensions, their Lagrangian for ϕ , de- isotropic Fermi surface of the boundary non-Fermi liquid µν noted by L , reads gets spontaneously distorted and becomes elliptical. We BGP also find that the broken (nematic) phase is a non-Fermi LBGP 1n µ νρ µ √ = − ∇µϕνρ∇ ϕ + ∇µϕ∇ ϕ liquid. We interpret this transition as a holographic re- −g 4 alization of the quadrupolar Pomeranchuk instability in ρ νµ µ ν non-Fermi liquids. Nonetheless, a disclaimer is neces- + 2∇µϕνρ∇ ϕ − 2∇ ϕµν ∇ ϕ (1) sary here. The background we study cannot be extended 2 µν 2 R µν R 2o − m ϕµν ϕ − ϕ + ϕµν ϕ − ϕ . consistently to T = 0 on the count that so doing would 2 4 require the back reaction of the gauge field. Such a back- µ reacted metric would be non-Einstein and there is no Here R is the Ricci scalar. Also, we have defined ϕ = ϕµ. causal Lagrangian for a spin-two field in a curved back- Several comments are warranted here. First, it is crucial ground that is not Einstein. Within the background we that the fixed background satisfies the Einstein relation consider, the vaccuum expectation value of the spin-two (which, in four bulk dimensions, reads) Rµν = Λgµν , with field actually grows as the temperature is increased. Con- Λ being the cosmological constant, otherwise some consequently, there is strictly no instability in the traditional straint equations become dynamical which would give sense in the context of spontaneous symmetry breaking. rise to extra propagating degrees of freedom for ϕµν . Hence, our usage of the term instability is not entirely In our context this restriction simply implies that one warranted. Nonetheless, the program we outline here should ignore the backreaction of ϕµν , as well as other does provide a consistent holographic framework to ad- bulk fields that will be introduced later, on the metric of dress how to engineer shape distortions of the underlying the fixed Einstein background. Second, in the flat space- Fermi surface. Hence, it serves as a valid first step in time limit, the above Lagrangian becomes the Fierz-Pauli realizing the physics of the Pomeranchuk instability. Lagrangian [18]. Third, the massless limit of the BGP Lagrangian, i.e. m2 = 0, has an emergent gauge sym- The outline of this paper is as follows. Section II sets metry (diffeomorphism) which is unique to the graviton: up the Lagrangian for the neutral massive spin-two field. ϕµν → ϕµν +∇(µξν) with ξµ being an infinitesimal vector. Included here will be an explicit mechanism for obtaining Indeed, as pointed out in [19], this symmetry has been a normalizable profile for the bulk spin-two field. Section used to settle the ambiguity associated with the definition 3

2 of the mass of ϕµν . Fourth, for m = R/6 (here d+1 = 4) Suppose now there exists a (2+1)-dimensional field the above Lagrangian has another emergent gauge sym- theory dual to the gravitational setup under consider- metry [21]: ϕµν → ϕµν +(∇µ∇ν +gµν R/12) with being ation here. Focusing for the moment on the bulk neutral an infinitesimal scalar gauge parameter. In the following, massive spin-two field ϕµν , holography tells us that it we always assume that m2 =6 0 and m2 =6 R/6. Exclud- should be dual to some operator in the boundary the- ing these two special values for the mass, the equations ory. Since ϕµν is a bulk field with spin greater than zero, of motion and constraints obtained from (1) are given by its number of components does not naively match the [19] number of components of the dual operator. But, as ex- plained in [23], since ϕµν can be put in the form (8), it is ∇2 − 2 ρ σ 0 = m ϕµν + 2R µ ν ϕρσ, (2) ϕij which sources a neutral symmetric traceless operator µ 0 =∇ ϕµν , (3) Oij in the boundary theory. (The vev of the operator O 0 =ϕ, (4) ij determines the nematicity of the (2+1)-dimensional boundary theory.) More concretely, equation (2) implies 0 =ϕ, ˙ (5) that ϕij(r) takes the following form near the boundary 00 i 0i 0 0i 0 0 =g ∇0∇iϕ ν − g ∇0∇iϕ ν − g ∇i∇0ϕ ν as r → ∞ ij 0 ρ 0 σ 2 0 − g ∇i∇jϕ − 2R ϕρσ + m ϕ . (6) ∆−1 2−∆ ν ν ν ϕij(r) = Aij r 1 + ··· + Bij r 1 + ··· , (10) The above expressions give the correct number of propa- with A and B being symmetric traceless tensors. A gating degrees of freedom for a massive spin-two field in ij ij ij is the source for the boundary theory operator O while (3+1)-dimensions. Such a field transforms in the five- ij B is proportional to its vev, hO i. In (10), ∆ denotes dimensional irreducible representation of SO(3), with ij ij the UV scaling dimension of O , which is related to the SO(3) being the little group of the Lorentz group ij mass of the bulk field ϕ via SO(3,1). µν r As we alluded to earlier, in order to satisfy the con- 3 9 ∆ = + + m2. (11) straint Rµν = Λgµν , we will work in a regime where mat- 2 4 ter fields do not backreact on the bulk geometry. So, we take the Schwarzschild AdS4 black hole as our back- Note that for m = 0, i.e. when ϕµν is the graviton, equa- ground geometry (Λ = −3) tion (11) yields ∆ = 3 which is the dimension of Tµν , the energy-momentum tensor operator, of the (2+1)- dimensional boundary theory. There is a Breitenlohner- 2 µ ν ds = gµν dx dx Freedman bound [24] for the propagation of ϕµν in an 2 dr2 asymptotically AdS4 geometry which reads m ≥ 0 2 − 2 2 2 = r f(r) dt + dx + dy + 2 , (7) [23, 25, 26]. Equation (11) then implies that ∆ ≥ 3. r f(r) In our discussions in this paper, we will always take the 3 3 mass squared of ϕµν in the asymptotic AdS4 region of where f(r) = 1−(r /r ) with r0 being the horizon radius, 0 the geometry to satisfy the condition m2 > 0. Note that given by the largest real root of f(r0) = 0. Note that we 2 are working in units where we set the curvature radius L the condition m =6 R/6 = −2 is then trivially satisfied. of AdS4 equal to unity. Also, note that in the coordinates Given the ansatz (8), we are interested in solutions in we have chosen above, the asymptotic boundary of the which ϕµν (r), or rather ϕij(r), is regular near the horizon spacetime is at r → ∞. The temperature of the black and normalizable as r → ∞. More concretely, near the hole is given by T = 3r0/(4π). horizon, we look for a solution of the equation (2) which is regular as r → r0, namely To solve for ϕµν , we only consider the configuration where ϕ = ϕ (r) for all µ, ν ∈ {t, r, x, y}. Analyzing µν µν ϕ (r) = a + b (r − r ) + c (r − r )2 + ··· , (12) the equations (2)–(6), one can show [22, 23] that it is ij ij ij 0 ij 0 consistent to put ϕµν (r) in the form where the coefficient tensors bij, cij, ··· are all deter- → ∞ 0 0 mined in terms of aij. In the boundary as r , we ϕµν (r) = , (8) demand ϕij(r) to be normalizable 0 ϕij(r) 2−∆ ϕij(r) = Bij r 1 + ··· . (13) where ϕij(r), with i, j = x, y, is a symmetric traceless two-by-two matrix. Defining the “director” ~ϕ(r) = If a solution with the above two boundary conditions ex- ϕ (r) + iϕ (r), under a rotation θ in the (x,y)-plane, xx xy ists, then the boundary theory would be in a phase where ~ϕ(r) transforms as the neutral symmetric traceless operator Oij (which is ~ϕ(r) → e2iθ ~ϕ(r). (9) the operator dual to ϕij) spontaneously condenses. It is easy to show that given the set of equations (2)–(6), such One can then choose a particular value of θ to set either a solution does not exist. As we will explain in the next ϕxx(r) or ϕxy(r) equal to zero, or make them equal. section, one way forward is to minimally modify the BGP 4

Lagrangian (1) such that the new action yields the cor- 100 rect number of causal propagating degrees of freedom for ϕµν , and also permits the aforementioned spontaneous 50 condensation to occur.

B 0 B. Modifying the BGP Lagrangian

!50 Undoubtedly, there are many ways to modify the BGP Lagrangian to facilitate spontaneous condensation of the !100 operator dual to ϕµν . Instead of categorizing all such !1.0 !0.5 0.0 0.5 1.0 modifications, we take perhaps the simplest possibility. A Gubser [27] has shown that coupling a neutral scalar field 2 2 to the square of the Weyl tensor leads to a scalar hair FIG. 2. A plot of Bij verse Aij for ` = 1.0866 and m = 1/100. The values of B and A have been rescaled so that they are on an asymptotically flat Schwarzschild black hole. This ij ij dimensionless. The gauge in (9) has been used to set ϕxx = ϕxy. coupling is of interest here because the square of the Weyl tensor vanishes at the boundary (r → ∞) and hence does not affect the scaling dimension of the dual boundary bound as before. Thus, regardless of the value of the cou- operator. To this end, we modify the BGP Lagrangian 2 pling ` , the asymptotic AdS4 region of the background (1) as follows remains stable.

2 As Figure2 illustrates, there exists a normalizable so- ` √ µνρσ γδ 2 Lϕ = LBGP + −g CµνρσC ϕγδϕ − ϕ , (14) lution with A = 0 and B =6 0. (We have used the 4 ij ij gauge in (9) to set ϕxx = ϕxy.) The holographic inter- with Cµνρσ the Weyl tensor. In the Appendix, we show pretation of this normalizable solution is that Oij has that the above modification to the BGP Lagrangian, spontaneously condensed in the boundary theory. While though not unique, offers a valid description of the neu- it may be possible to obtain a normalizable solution for tral massive spin-two field ϕµν , meaning that ϕµν still ϕµν using an alternative mechanism, we believe that the propagates causally in a fixed Einstein spacetime with the details of how the dual operator condenses in the bound- correct number of degrees of freedom. In what follows, we ary theory should be irrelevant to the Pomeranchuk in- show that the equations of motion for ϕµν obtained from stability as we discuss in the next section. the new Lagrangian now admits non-trivial normalizable solutions (whose near horizon and asymptotic boundary behaviors are given by (12) and (13), respectively). D. Maxwell’s Lagrangian

Since we will be considering the boundary theory at a C. Normalizable Solution ﬁnite chemical potential µ for a U(1) charge, we need to introduce in the bulk an Abelian gauge ﬁeld Aµ with the From the Appendix, the relevant equation for deter- Lagrangian mining whether there exists a normalizable solution for 1√ µν ϕ is L = − −g Fµν F . (16) µν M 4 2 2 2 γδρσ ρ σ ∇ − m + ` CγδρσC ϕµν + 2R µ ν ϕρσ = 0. (15) The equations of motion for Aµ is then

µ Note that the background metric must again satisfy the ∇ Fµν = 0. (17) Einstein relation. Since we have taken the spacetime to We will only be interested in the case where A is non-zero be the Schwarzschild AdS4 black hole, the square of the t µνρσ 6 and, moreover, depends only on the radial coordinate Weyl tensor is CµνρσC = 12(r0/r) . One can easily show that, given the constraints presented in the Ap- r. So, we take the ansatz Aµ = At(r)δµ0. Given our pendix and the equation (15), it is again consistent to assumption that Aµ does not back react on the metric put ϕµν in the form given in (8). Notice that as shown (7), the solution to (17) is easily obtained to be above, the square of the Weyl tensor evaluated on the r0 background vanishes as r → ∞. This then implies that A = µ 1 − dt, (18) r the relationship between the asymptotic mass m of ϕij and the UV dimension ∆ of the dual operator Oij is still where the boundary conditions at the horizon and the given by (11). Also, the mass of ϕij in the asymptotic asymptotic boundary have been ﬁxed by demanding AdS4 region satisﬁes the same Breitenlohner-Freedman At(r0) = 0 and At(r → ∞) = µ. 5

III. FERMIONS ing aside those terms which would vanish once evaluated on the background, we focus on the leading-order3 non- trivial coupling (which can potentially contribute asym- In this section, we study the shape distortion of the metric features to the fermionic spectral function) Fermi surface as a result of the condensation of the √ boundary theory operator Oij. We start by introducing µ ν Lint = −iλ −g ϕµν ψ¯ Γ D ψ, (20) a bulk spinor field ψ with mass mψ and charge q which is dual to a boundary theory fermionic operator Oψ with with λ real. Thus, the bulk fermion Lagrangian that we a UV scaling dimension ∆ψ and charge q. Note that the study is Lψ = LDirac + Lint. bulk spinor ψ is a four-component Dirac spinor while the The Dirac equation following from Lψ takes the form dual operator Oψ is a two-component Dirac spinor. In 1 this paper, we take mψ ∈ [0, ) and choose the so-called µ ν 2 (D/ − m + λϕµν Γ D )ψ = 0. (21) “conventional quantization” where the scaling dimension of the fermionic operator is related to the mass of the To solve the above equation, we go to momentum space 3 µ ik.x µ bulk spinor field through ∆ψ = 2 + mψ. by Fourier transforming ψ(r, x ) ∼ e ψ(r, k ), where kµ = (ω,~k). It is expedient to choose a basis for the Dirac matrices as follows A. Fermion Lagrangian and Equation of Motion −σ 0 iσ 0 Γr − − = 3 , Γt = 1 , 0 −σ3 0 iσ1 In our discussions below, the bulk spinor field ψ is −σ 0 0 σ treated as a probe where its backreaction on the back- Γx = 2 , Γy = 2 . (22) ground metric, gauge field and neutral massive spin-two 0 σ2 σ2 0 field is ignored. The Dirac Lagrangian, It is also convenient to remove the spin connection √ from the Dirac operator D/ by rescaling ψ according LDirac = − −g iψ¯(D/ − m)ψ, (19) to ψ → (−ggrr)−1/4ψ. Splitting the new (rescaled) † t T ~ describes the bulk spinor field, where ψ¯ = ψ Γ and spinor ψ according to ψ = (ψ1, ψ2), where ψ1(r; ω, k) / µ c 1 ab µ ab ~ D = ec Γ ∂µ + 4 ωµ Γab − iqAµ and ea and ωµ being and ψ2(r; ω, k) are two-component spinors, and given respectively the vielbein and the spin connection. Also, the ansatz for the background form of ϕµν where the a, b, ··· = {t, x, y, r} denote the tangent space indices, only non-vanishing components are ϕxx = −ϕyy and t x y r Γ , Γ , Γ , Γ are the Dirac matrices satisfying the Clif- ϕxy = ϕyx, the equation (21) results in the following a b ab 1 two coupled differential equations for ψ and ψ ford algebra {Γ , Γ } = 2η and Γab = 2 [Γa, Γb]. The 1 2 overall negative sign in (19) ensures that the bulk ac- √ √ xx xx tion, once holographically renormalized, gives rise to a D˜ r − iσ2 g k˜x ψ1 + iσ2 g k˜yψ2 = 0, (23) positive spectral density for the fermionic operator Oψ. √ √ ˜ xx ˜ xx ˜ In (19), gµν and Aµ are the background metric and the Dr + iσ2 g kx ψ2 + iσ2 g kyψ1 = 0, (24) background gauge field whose expressions are given in (7) and (18), respectively. where we have defined In the probe limit that we are considering here the √ p ˜ − rr − tt − Dirac Lagrangian (19) is not capable of holographically Dr = σ3 g ∂r + σ1 g (ω + qA2) mψ, (25) xx xx realizing how the spectral function of the boundary the- k˜x = kx (1 + λg ϕxx) + λg ϕxyky, (26) ory fermionic operator Oψ is affected in the presence the xx xx k˜y = ky (1 − λg ϕxx) + λg ϕxykx. (27) order parameter hOiji. To do so, one has to explicitly introduce bulk couplings between the spinor field ψ and For λ = 0, or when ϕxx = ϕxy = 0, the Dirac equa- the neutral massive spin-two field ϕµν . A similar problem tion (21) has been studied extensively in the literature arises in the context of holographic d-wave superconduc- (although mainly on the Reissner-NordströmAdS4 black tivity [22, 23] although in that case one has to deal with hole background), following the work of [28–31]. These coupling the bulk spinor to a charged massive spin-two studies show that in these cases there are symmetrical field. Since our goal here is to capture the leading order Fermi surfaces in the boundary theory whose underlying effects on the retarded two-point function of the fermionic excitations can either be Fermi or non-Fermi liquid-like. operator due to the presence of the order parameter, we On the other hand, when ϕxx develops a normalizable consider those couplings in the bulk which are quadratic in the spinor field and have relatively low mass dimension. Furthermore, among the aforementioned couplings, 3 √ 5 µ ν we will only be interested in those which will potentially There is also a coupling of the form λ5 −g ϕµν ψ¯ Γ Γ D ψ that give rise to anisotropic features in the fermion spectral breaks parity in the boundary theory when the operator dual to function. In other words, we ignore the terms which ϕµν condenses. We will not consider such a coupling in this would just modify the mass of the spinor field. Leav- paper. 6 mode in the bulk, k˜x and k˜y will differ thereby giving rise to asymmetrical features in the fermionic correlators of the boundary theory. This can be shown explicitly. To compute the retarded correlator of the boundary theory operator Oψ, one needs to solve the equations (23) and (24) with in-falling boundary condition at the horizon [32] and read off the source and the expectation value of Oψ from the asymptotic expansion of ψα (α = 1, 2) following the prescription of [33]. Indeed, choosing in- falling boundary conditions for ψα near the horizon, the leading-order asymptotic (r → ∞) behavior of ψα takes −m m T the form ψα(r → ∞) = (Bα r ψ ,Aαr ψ ) , where the T two-component spinor A = (A1,A2) sources the bound- T ary theory fermionic operator Oψ, while B = (B1,B2) gives the vev of the operator. The spinor B is related to the spinor A through B = SA from which the retarded Green function of the operator Oψ take the form [33]

t GR(ω,~k) = −iSγ . (28)

t Note that in our basis of gamma matrices γ = iσ1. Up to a numerical constant, the spectral function of the operator Oψ is given by ρ(ω,~k) = Tr ImGR(ω,~k). FIG. 3. Density plots of the spectral function of the fermionic operator Oψ in the boundary field theory. a) Spectral density when the boundary field theory is in the unbroken phase b) Spectral B. Fermi Surfaces and Low-Energy Excitations density in the broken phase showing an elliptic Fermi surface where we set λ = −0.4, c) spectral density in the broken phase where λ = −0.95 showing a suppression of the spectral weight and d) To obtain the retarded Green function (28) of the spectral density, also in the broken phase, when λ = −1.4. fermionic operator Oψ, one should numerically solve the equations (23) and (24), or the flow equations obtained from them, with in-falling boundary condition at the open circles) reveals that in the unbroken phase of the horizon. We do this computation at finite temperature boundary theory where the Fermi surface is circular, the mainly because our background, in which there is a nor- dynamical exponent z = 0.64 which indicates that the malizable solution for the neutral massive spin-two field, low-lying excitations form a non-Fermi liquid. Impor- is not, strictly speaking, valid at zero temperature. The ~ ~ tantly then, any subsequent breaking of rotational sym- Femi momentum kF appears as a pole in Re GR(ω, k), al- metry of the Fermi surface will be from a non-Fermi liq- though at finite temperature such poles, instead of being uid state, one of the key hurdles in describing the strong the delta function, become broadened. correlations in the experimental systems. Carrying out the numerical computation, we find mul- Focusing now on the interesting phase where the neu- tiple Fermi surfaces for the parameters chosen. To tral symmetric traceless operator Oij has spontaneously demonstrate proof of concept, it is sufficient to focus on condensed, we find three significant features in the spec- the first Fermi surface, where |~k | (measured compared F tral function of the fermionic operator Oψ. First, the to the effective chemical potential qµ) has the smallest rotational symmetry of the Fermi surface is broken. As value. In generating the plots shown in this section, we we alluded to earlier, such behavior is expected given the first used the gauge (9) to set ϕxx = ϕxy and then nu- form of k˜x and k˜y at each r-slicing in the bulk. Second, merically solved the Dirac equations for a very small but the spectral weight is angularly dependent. For interme- non-zero frequency such that the delta functions at the diate values of the coupling λ, a gap-like feature opens Fermi surface are broadened. at the two end points of the major axis. This highly First, consider the case where the background value of non-trivial behavior is not a generic feature of traditional ϕµν (r) identically vanishes in the bulk. In other words treatments of the Pomeranchuk instability [8] but can be the boundary theory is in the symmetry-unbroken phase understood within a WKB approximation to the spec- hOiji = 0. Not surprisingly, the Fermi surface obtained tral function as outlined in the next subsection. The in this phase is isotropic, as is evident from Figure3(a). diminished spectral density reappears once λ exceeds a To investigate the nature of the excitations, we com- critical value as shown in Figure3(d). Finally, the excita- puted the quasiparticle dispersion relation by focusing tions near the Fermi surface remain non-Fermi liquid-like. on the peak in the spectral function. In a Fermi liquid, The solid circles and triangles in Figure4 show that the z ω(k) = k⊥ with k⊥ = k − kF and z = 1. Figure4 (see dispersions along kx and ky deviate from linearity with 7

! (a) (b) ! 0.00002 ! !

0.000015 ! !Ω

! 0.00001 ! (c) (d) 0.00002 0.00003 0.00005 0.00007

!k"

z FIG. 4. The dispersion relation, ω(k) ∝ k⊥, of quasi-particles around kF plotted on a log-log scale. The three cases correspond to a) open circles for the unbroken phase of the boundary theory with a circular Fermi surface where we find z = 0.64, b) solid circles for the dispersion relation (in the broken phase) along x- axis where we find z = 0.74 with λ = −0.4 and c) solid triangles for the dispersion relation along y-axis (in the broken phase) where ˜0 z = 0.84 with λ = −0.4. FIG. 5. Contour plot of kx for a) λ = 0, b) λ = −0.4, c) λ = −0.95 and d) λ = −1.4. The plots have been generated with m = 0, ω = 0 and qµ = 17. exponents of z = 0.74 and z = 0.84, respectively. These values are, of course, parameter dependent. analysis of the Dirac equation. In order to make the ˜0 notation less cluttered, we will replace kx by k in what follows. In order to apply the WKB approximation, we C. WKB Analysis of Spectral Function will take the limit where k, m, ω and q are large, but their ratio remains constant. To this end, it is convenient to In this section we employ a WKB approximation to rescale all of these quantities with a factor γ such that uncloak uncloak the origin of the momentum-dependent k = γkˆ, mψ = γmˆ , q = γqˆ and ω = γωˆ, where γ is taken spectral weight in Figure3. Such an approximation has to be large (compared to one). Wednesday, July 4, 2012 been used recently in [34] for the semi-analytic analysis Central to the WKB approximation is the construction of spectral functions in the electron star background [35]. of a Schrödinger-like equation for each of the components Setting kx = k cos θ and ky = k sin θ, we define new of the two-component spinor momentum coordinates q 0 0 Φ1 k˜ = k˜2 + k˜2 k˜ = 0, (29) ψ1 = . x x y y Φ2 which are written in the so-called “lab” frame. Note that The same treatment applies equally to ψ with the re- the expressions for k˜ and k˜ have been defined in (26) 2 x y placement of k → −k. To leading order in γ, the second- and (27), respectively. With these new momenta and for order differential equation for Φ takes on the form a particular θ and k, the original Dirac equations (23) 1 and (24) give rise to two decoupled equations 00 2 n 2 tt 2 Φ1 = γ grr mˆ + g [ˆω + µqˆ(1 − r0/r)] √ ˜ xx ˜0 Dr − iσ2 g kx ψ1 = 0, (30) xx 2o + g kˆ Φ1, (32) √ ˜ xx ˜0 Dr + iσ2 g kx ψ2 = 0. (31) with √ Any anisotropy that arises now in the spectral function rr ˜0 (− g ∂r − mψ)Φ1 must be tied to kx. To confirm this, we construct a con- Φ2 = √ . (33) p tt xx ˜0 − −g (ω + qAt) + g k tour plot of kx at the horizon where the anisotropy is largest. The plots in Figure5 show clearly that for a ˜0 In the context of a Schrödingerequation, the right-hand fixed value of k, kx is minimum at θ = π/8 and 9π/8 which is consistent with the values of θ where the spec- side of equation (32) can be regarded as the zero-energy tral density is a minimum. potential n o Indeed, one can develop a better understanding on how 2 2 tt 2 iiˆ2 ˜0 V = γ grr mˆ + g [ˆω + µqˆ(1 − r0/r)] + g k . (34) kx affects the spectral function by performing a WKB 8

The poles of the retarded Green function only come from V (r) G and are given by Y = πn − ie−2X + ··· where we have included the imaginary part up to the leading order only. Since we are interested in the non-analytic dependence of the Green function on frequency and momentum, we will r3 r2 r1 r focus on the behavior in the vicinity of the pole where G can be written as X i G = . (41) − −2X(ω,k) n Y (ω, k) πn + ie

horizon (n) 4 Close to ω = 0 and k = kF , we can expand Y (ω, k) as (n) (n) (n) (n) Potential V in the Schr¨odingerequation as a function of FIG. 6. Y 0, kF +ω∂ωY 0, kF +(k −kF )∂kY 0, kF +··· . the radial coordinate r. The classical turning points as measured As a result of this expansion, the retarded Green function from the horizon are denoted by r3, r2, and r1. takes the form

−2an X −cne G (ω, k) ∝ (42) It is the turning points of this potential that governs the R (n) ω + v (k − k ) + ic e−2X(ω,k) physics of the WKB approximation. In the near-horizon n n F n (n) −2an 2 −2an −2X region, the potential is approximated by X − ω + vn(k − k ) cne + icne e ∝ F , − (n) 2 2 −4X 2 2 h 2 i n ω + vn(k kF ) + cne L γ 2ˆ2 2 2 2 L r0 V (r → r0) = 3 L k +m ˆ r0 − ωˆ 3r0(r − r0) 3 (r − r0) where + ··· (35) (n) ∂kY 0, k v = F , Ifω ˆ is non-zero, the near horizon potential is dominated n (n) ∂ωY 0, k by the term proportional toω ˆ2 and is negative for real F 1 positiveω ˆ. Close to the boundary, the leading order term c = , (43) n (n) of the potential ∂ωY 0, kF Z ∞ q 2 2 2 (n) γ L mˆ an = dr V 0, k , V (r → ∞) = + ··· , (36) F r2 r1 is always positive. Between these two regions, there can which is essentially the equation (48) of [34]. be at most three turning points which we denote by r1, Close to the Fermi surface, where the ﬁrst term in r2 and r3, with r1 being the turning point closest to the the denominator of the second equation in (42)) can be boundary as depicted in Figure6. For the parameter dropped5, the spectral function A(ω, k) can be extracted range of interest here, all three turning points will enter from the imaginary part of the retarded Green function. the matching conditions as we describe below. One then obtains What remains is a simple matching procedure to de- A(ω, k) ∝ e−2an+2X . (44) termine the form of Φ1 in each of the regions that bracket the turning points. This procedure, detailed previously We can now understand the anisotropy of spectral func- in [34], utilizes the functions tion from the dependence of an and X on the area en-

Z r2 √ closed by the potential integrated between the turning 6 X = dr V + log 2, (37) points r2 and r3 and between r1 and ∞ which will de- r3 termine an. In fact, for m = 0, r1 → ∞, an therefore Z r1 √ π vanishes and the angular dependence of the spectral func- − Y = dr V + , (38) tion will be dominated by X alone. Close to the horizon, r2 2 which directly enter the retarded Green function

Z r √ 4 iG 2mL 0 Note that in equation (34) we have written the potential in terms GR(ω, k) ∝ lim r exp −2 dr V (39) of the hatted quantitiesω ˆ, kˆ, however, in doing the expansion of 2 r→∞ r 1 Y (ω, k), the factors of γ cancel and we can drop the hat on ω and k. where 5 We have assumed that in the limit ω → 0, ω vanishes faster than e−2X(ω,k) which is true in the range of parameters in which we cosh(X + iY ) + sinh(X − iY ) G = . (40) are interested. cosh(X − iY ) − sinh(X + iY ) 6 This will deﬁne the quantity X. 9

120 desirable since they would not only enable an analysis at zero-temperature but would also cure at finite tempera- 100 ture some of the peculiar thermodynamical properties of 80 the conduced phase in the probe limit. In addition, is the angular dependent spectral weight a generic feature V 60 of holographic non-Fermi liquids with (partially) broken 40 rotational symmetry? If so, then holography would have provided a key ingredient missing from most condensed 20 matter analyses of the Pomeranchuk instability, namely 0 a vanishing spectral weight for a finite range of momenta and hence a pseudogap as a function of frequency. Also, 1.00 1.02 1.04 1.06 1.08 1.10 how does the resistivity tensor behave in the two different rr0 directions? In our probe analysis, where the backreaction of the massive spin-two field on the metric is ignored, FIG. 7. Effective WKB potential for λ = −0.4, m = 0 at the first the resistivity tensor is not sensitive to the anisotropy Fermi surface. The solid line corresponds to θ = 0 and the dashed line to θ = π/8. of the system when the boundary theory is in the broken phase. Nevertheless, one might be able to see the anisotropic contribution of the fermions to the resistivity through a fermion loop computation in the bulk similar the first turning point of the potential, r3, is roughly given by to the analysis of [36].

ω2L2r r ≈ r + 0 . (45) 3 0 2 2 2 2 3 m r0 + k(r0) L ACKNOWLEDGMENTS

From this we can see that in the limit ω → 0, that is, close to the Fermi surface, r3 → r0. We would like to thank R. Leigh, C. Herzog, and E. Fradkin for several important discussions at the outset In our analysis of the spectral function, where we of this work. P. W. P. would like to thank S. Hartnoll have set m = 0, only X determines the spectral func- for comments on the manuscript. M. E. is supported by tion. To illustrate the angular dependence, we plot the the NSF under Grant Number PHY-0969020 while K. near-horizon effective WKB Schrodinger potential for L. and P. W. P. acknowledge financial support from the λ = −0.4, m = 0 at first Fermi surface (n = 0) in Figure NSF-DMR-0940992 and DMR-1104909. 7 (corresponding to Figure 3(b)). The dashed line corresponds to the potential along θ = π/8, while the solid line corresponds to θ = 0. It is now obvious that r2 is larger for θ = 0 than for θ = π/8 and the potential is APPENDIX always larger for θ = 0 than θ = π/8 in the range of r shown. Hence X and spectral weight are larger along θ = 0 than θ = π/8, thereby explaining our key finding In this appendix, we demonstrate that adding a cou- 2 γδρσ µν 2 that the spectral weight is angularly dependent. In prin- pling of the form ` CγδρσC (ϕµν ϕ −ϕ ) to the BGP ciple, the WKB approximation works only for large n, Lagrangian, given that the background spacetime is an nevertheless, we see that the result qualitatively agrees Einstein manifold, does not induce any extra degrees of with our numerical results even for small n. freedom for the neutral massive spin-two field ϕµν nor vi- olate its causal propagation. Our discussion below closely follows that of [19], where it has been shown that neutral massive spin-two field in an Einstein spacetime, such IV. DISCUSSION as the AdSd+1 Schwarzschild black hole, has the correct number of degrees of freedom and propagates causally. In We have shown here how holography can be used to our discussion below, we keep the number of dimensions model a shape distortion of the underlying Fermi surface of the bulk spacetime (denoted by d + 1) arbitrary. in a non-Fermi liquid through the condensation of a neu- Consider the following Lagrangian [20] tral symmetric traceless operator which is dual to a neutral massive spin-two field in the bulk. An open challeng- 1n µ νρ µ µ ν L = − ∇µϕνρ∇ ϕ + ∇µϕ∇ ϕ − 2∇ ϕµν ∇ ϕ ing problem remains: Is the condensation of the bound- 4 ary theory operator Oij possible starting from a bulk ρ νµ 2 µν 2 + 2∇µϕνρ∇ ϕ − m (ϕµν ϕ − ϕ ) geometry which is not an Einstein manifold? In other 0 µν 2 µλνρ words, what is the backreacted background that allows + 2a1R ϕµν ϕ + 2a2R ϕ + 2a3R ϕµν ϕλρ o for the ghost-free and causal propagation of a neutral + 2a Rµν ϕ ϕλ + 2a Rµν ϕ ϕ , (46) massive spin-two field. These extensions are extremely 4 µλ ν 5 µν 10

µ where ϕ = ϕ µ. The above Lagrangian is the most gen- the covariant derivative of the equations of motion eral two-derivative action for a neutral massive spin-two µ ˜ field ϕµν (up to the quadratic order in ϕµν , though) in a ∇ Eµν = 0. (50) curved background. The coefficients in the first two rows µ µ 2 µν 2 of (46) are fixed because in the flat spacetime limit, one Again, neither ∇ Eµν nor ∇ C (ϕµν ϕ − ϕ ) in- demands the Lagrangian to go over to the Fierz-Pauli La- volves two time-derivatives of the spin-two field. Thus, grangian [18], which is known to have the correct number the equations (50) give us d + 1 secondary constraints. of causally-propagating degrees of freedom for a neutral Following [19], the equations (50) should containϕ ˙ µ0 massive spin-two field. Indeed, in order for a massive through a rank d matrix in order for their derivatives to define d accelerationsϕ ¨ : spin-two field (represented by a symmetric tensor ϕµν ) i0 to have the correct number of propagating degrees of E µ = A ϕ˙ + Bjϕ˙ + ··· , freedom in a (d + 1)-dimensional spacetime, one needs µ0, 00 j0 µ j 2(d + 2) constraint equations to kill the extra degrees of Eµi, = Ci ϕ˙00 + Di ϕ˙j0 + ··· , (51) freedom. j ˆ ν AB rank Φµ ≡ rank j = d. Now, add to the above Lagrangian an additional cou- Ci Di 2 2 µν 2 pling of the form ` C (ϕµν ϕ − ϕ ) and define the La- ˆ ν grangian The explicit form for Φµ when ` = 0 appears in [19]. The j additional non-vanishing contributions to A, B , Ci and i Dj are given by 1 2 2 µν 2 Lϕ = L + ` C ϕµν ϕ − ϕ , (47) 4 j 2 2 j0 i 2 2 00 i δB = −` C g , δDj = ` C g δj. (52)

ˆ ν where, in order to make the expressions less cluttered, Hence, in order for the rank of Φµ to be d, we require 2 γδρσ 2 2 2 we defined C = CγδρσC . For ` = 0, the analy- m0 − ` C + 2(1 − ξ)R/(d + 1) =6 0, where ξ is arbitary sis has been performed in [19], where, in order to get parameter. We now have 2d + 2 constraints. rid of the extra degrees of freedom of ϕµν , one obtains One of the remaining constraints can be obtained from a one-parameter family of solutions for the coefficients a linear combination of the equations of motion and the ··· a1, , a5 (labeled by an arbitrary real parameter ξ) primary and secondary constraints: which can be represented as 2 − 2 2 − m0 ` C ˜µ µ ν ˜ 2(1 ξ) ˜µ Eµ + ∇ ∇ Eµν + REµ ξ 1 − 2ξ d − 1 (d + 1)(d − 1) a1 = , a2 = , a3 = a4 = a5 = 0, (48) ϕ 2(1 − ξ) d + 1 2d + 2 = R + m2 − `2C2 d − 1 d + 1 0 d + 1 − 2ξd 2 2 2 provided that the background metric gµν satisfies the × R + (m0 − ` C )d (53) 2 d + 1 Einstein relation Rµν = 2Λgµν /(d − 1), and m0 =6 2 2 2 ν 2 µ 2(ξ − 1)/(d + 1). Defining m = m0 + 2(1 − ξ)R/(d + 1) + 2` (∇ C )(∇ ϕµν − ∇ν ϕ) and given the above solution for a ’s, the Lagrangian (46) µ ν i + `2(∇ ∇ C2)(ϕ − g ϕ) ≈ 0. takes the form given in the expression (1), the so-called µν µν BGP Lagrangian. This equation can be used as a constraint since it does For ` =6 0, one can easily obtain the constraint equa- not containϕ ¨µν . It containsϕ ˙ µν which can be removed µ tions following [19]. Basically, constraints come from the by the equations ∇ E˜µν = 0. After such a removal, the equations of motion and their derivatives which do not time-derivative of (53) does not containϕ ¨µν and can be contain two time derivatives of spin-two fieldϕ ¨µν . The used as the last constraint. We then have in total 2d + 4 equations of motion read constraints and hence the correct number of propagating degrees of freedom. Following the discussion in [19], it can be seen easily E˜ = E + `2C2 (ϕ − g ϕ) = 0, (49) µν µν µν µν that the characteristic matrix remains unchanged for ` =6 0, due to the fact that the Weyl squared coupling does not affect any of the terms with two time-derivatives. So, where Eµν denotes the equations of motion for the ` = 0 the equations remain hyperbolic and causal even for the case. The explicit form of Eµν is given in [19]. Since ` =6 0 case. The resulting equations of motion are Eµ0 does not involve second time-derivatives of ϕµν , and obviously the new term in (49) does not involve any sec- 2 2 2 2 ρ σ ∇ − m + ` C ϕµν + 2R µ ν ϕρσ = 0. (54) ond time-derivative of ϕµν either, the equations E˜µ0 = 0 actually give us d + 1 (primary) constraints. 2 2 where, as we have defined before, m = m0 + 2(1 − In order to obtain the secondary constraints, we take ξ)R/(d + 1). 11

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