Charge transport by holographic Fermi surfaces
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Citation Faulkner, Thomas, Nabil Iqbal, Hong Liu, John McGreevy, and David Vegh. “Charge transport by holographic Fermi surfaces.” Physical Review D 88, no. 4 (August 2013). © 2013 American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevD.88.045016
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/81381
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 88, 045016 (2013) Charge transport by holographic Fermi surfaces
Thomas Faulkner,1 Nabil Iqbal,2 Hong Liu,3 John McGreevy,4,* and David Vegh5 1Institute for Advanced Study, Princeton, New Jersey 08540, USA 2KITP, Santa Barbara, California 93106, USA 3Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Department of Physics, University of California at San Diego, La Jolla, California 92093, USA 5Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland (Received 1 July 2013; published 21 August 2013) We compute the contribution to the conductivity from holographic Fermi surfaces obtained from probe fermions in an AdS charged black hole. This requires calculating a certain part of the one-loop correction to a vector propagator on the charged black hole geometry. We find that the current dissipation is as efficient as possible and the transport lifetime coincides with the single-particle lifetime. In particular, in the case where the spectral density is that of a marginal Fermi liquid, the resistivity is linear in temperature.
DOI: 10.1103/PhysRevD.88.045016 PACS numbers: 11.25.Tq, 04.50.Gh, 71.10.Hf
low energy excitations near the Fermi surface [4–6]. Other I. INTRODUCTION non-Fermi liquids include heavy fermion systems near a For over fifty years our understanding of the low- quantum phase transition [7,8], where similar anomalous temperature properties of metals has been based on behavior to the strange metal has also been observed. Laudau’s theory of Fermi liquids. In Fermi liquid theory, The strange metal behavior of high Tc cuprates the ground state of an interacting fermionic system is and heavy fermion systems challenges us to formulate characterized by a Fermi surface in momentum space, a low energy theory of an interacting fermionic system and the low energy excitations are weakly interacting fer- with a sharp Fermi surface but without quasiparticles mionic quasiparticles near the Fermi surface. This picture (see also [9,10]). of well-defined quasiparticles close to the Fermi surface Recently, techniques from the AdS/CFT correspondence provides a powerful tool for obtaining low temperature [11] have been used to find a class of non-Fermi liquids properties of the system and has been very successful [12–17] (for a review see [18]). The low energy behavior of in explaining most metallic states observed in nature, these non-Fermi liquids was shown to be governed by a from liquid 3He to heavy fermion behavior in rare earth nontrivial infrared (IR) fixed point which exhibits non- compounds. analytic scaling behavior only in the time direction. In Since the mid 1980s, however, there has been an accu- particular, the nature of low energy excitations around mulation of metallic materials whose thermodynamic and the Fermi surface is found to be governed by the scaling transport properties differ significantly from those pre- dimension of the fermionic operator in the IR fixed point. 1 dicted by Fermi liquid theory [1,2]. A prime example of For >2 one finds a Fermi surface with long-lived quasi- these so-called non-Fermi liquids is the strange metal particles while the scaling of the self-energy is in general 1 phase of the high Tc cuprates, a funnel-shaped region different from that of the Fermi liquid. For 2 one in the phase diagram emanating from optimal doping instead finds a Fermi surface without quasiparticles. At T ¼ 1 at 0, the understanding of which is believed to be ¼ 2 one recovers the ‘‘marginal Fermi liquid’’ (MFL) essential for deciphering the mechanism for high Tc super- which has been used to describe the strange metal phase conductivity. The anomalous behavior of the strange of cuprates. metal—perhaps most prominently the simple and robust In this paper we extend the analysis of [12–15]to linear temperature dependence of the resistivity—has address the question of charge transport. We compute the resisted a satisfactory theoretical explanation for more contribution to low temperature optical and DC conductiv- than 20 years (see [3] for a recent attempt). While photo- ities from such a non-Fermi liquid. We find that the optical emission experiments in the strange metal phase do reveal and DC conductivities have a scaling form which is again a sharp Fermi surface in momentum space, various anoma- characterized by the scaling dimension of the fermionic lous behaviors, including the ‘‘marginal Fermi liquid’’ [4] operators in the IR. The behavior of optical conductivity form of the spectral function and the linear-T resistivity, gives an independent confirmation of the absence of qua- imply that quasiparticle description breaks down for the siparticles near the Fermi surface. In particular we find for 1 ¼ 2 , which corresponds to MFL, the linear-T resistivity *On leave from Department of Physics, MIT, Cambridge, is recovered. A summary of the qualitative scaling behav- Massachusetts 02139, USA. ior has been presented earlier in [16]. Here we provide a
1550-7998=2013=88(4)=045016(34) 045016-1 Ó 2013 American Physical Society FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) systematic exposition of the rather intricate calculation described by a charged black hole in d þ 1-dimensional behind them and also give the numerical prefactors. anti-de Sitter spacetime (AdSdþ1)[26,27]. The conserved There is one surprise in the numerical results for the current J of the boundary global Uð1Þ is mapped to a bulk prefactors: for certain parameters of the bulk model Uð1Þ gauge field AM. The black hole is charged under this (co-dimension 1 in parameter space), the leading contribu- gauge field, resulting in a nonzero classical background tion to the DC and optical conductivities vanishes; i.e. the for the electrostatic potential AtðrÞ. As we review further in actual conductivities are higher order in temperature than the next subsection, in the limit of zero temperature, the that presented [16]. This happens because the effective near-horizon geometry of this black hole is of the form Rd 1 vertex determining the coupling between the fermionic AdS2 , exhibiting an emergent scaling symmetry operator and the external DC gauge field vanishes at lead- which acts on time but not on space. ing order. The calculation of the leading nonvanishing The non-Fermi liquids discovered in [13,14] were found order for that parameter subspace is complicated and will by calculating the fermionic response functions in the finite not be attempted here. density state. This is done by solving Dirac equation for a While the underlying UV theories in which our models probe fermionic field in the black hole geometry (2.2). The are embedded most likely have no relation with the UV Fermi surface has a size of order OðN0Þ and various argu- description of the electronic system underlying the strange ments in [13,14] indicate that the fermionic charge density metal behavior of cuprates or a heavy fermion system, it is associated with a Fermi surface should also be of order tantalizing that they share striking similarities in terms of OðN0Þ. In contrast, the charge density carried by the black infrared phenomena associated with a Fermi surface with- hole is given by the classical geometry, giving rise to a 1 2 out quasiparticles. This points to a certain ‘‘universality’’ boundary theory density of order 0 OðGN Þ OðN Þ. underlying the low energy behavior of these systems. The Thus in the large-N limit, we will be studying a small part emergence of an infrared fixed point and the associated of a large system, with the background OðN2Þ charge scaling phenomena, which dictate the electron scattering density essentially providing a bath for the much smaller rates and transport, could provide important hints in for- OðN0Þ fermionic system we are interested in. This ensures mulating a low energy theory describing interacting fermi- that we will obtain a well-defined conductivity despite onic systems with a sharp surface but no quasiparticles. the translation symmetry. See Appendix A for further elaboration on this point. A. Setup of the calculation The optical conductivity of the system can be obtained from the Kubo formula In the rest of this introduction, we describe the setup of our calculation. Consider a d-dimensional conformal field 1 1 yy ð Þ¼ hJ ð ÞJ ð Þi G (1.1) theory (CFT) with a global Uð1Þ symmetry that has an AdS i y y retarded i R gravity dual (for reviews of applied holography see e.g. where J is the current density for the global Uð1Þ in y [19–22]). Examples of such theories include the N ¼ 4 y direction at zero spatial momentum. The DC conductivity super-Yang-Mills theory in d ¼ 4, maximally supercon- is given by1 formal gauge theory in d ¼ 3 [23–25], and many others with less supersymmetry. These theories essentially consist DC ¼ lim ð Þ: (1.2) of elementary bosons and fermions interacting with non- !0 Abelian gauge fields. The rank N of the gauge group is The right-hand side of (1.1) can be computed on the gravity G mapped to the gravitational constant N of the bulk gravity side from the propagator of the gauge field Ay with end such that 1 / N2. A typical theory may also contain 2 GN points on the boundary, as in Fig. 1.Ina1=N expansion, another coupling constant which is related to the ratio of the leading contribution—of OðN2Þ—comes from the the curvature radius and the string scale. The classical background black hole geometry. This reflects the presence gravity approximation in the bulk corresponds to the of the charged bath, and not the fermionic subsystem in large-N and strong coupling limits in the boundary theory. which we are interested. Since these fermions have a The spirit of the discussion of this paper will be similar to density of OðN0Þ, they will give a contribution to of that of [13,14]; we will not restrict to any specific theory. order OðN0Þ. Thus to isolate their contribution we must Since Einstein gravity coupled to matter fields captures perform a one-loop calculation on the gravity side as universal features of a large class of field theories with a indicated in Fig. 1. Higher loop diagrams can be ignored gravity dual, we will simply work with this universal sector, since they are suppressed by higher powers in 1=N2. essentially scanning many possible CFTs. This analysis The one-loop contribution to (1.1) from gravity contains does involve an important assumption about the spectrum many contributions and we are only interested in the part of charged operators in these CFTs, as we elaborate below. One can put such a system at a finite density by turning 1Note that we are considering a time-reversal invariant system on a chemical potential for the Uð1Þ global symmetry. which satisfies ð Þ¼ ð Þ. Thus the definition below is On the gravity side, such a finite density system is guaranteed to be real.
045016-2 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013)
FIG. 1 (color online). Conductivity from gravity. The horizon- tal solid line denotes the boundary spacetime, and the vertical FIG. 2 (color online). The imaginary part of the current- axis denotes the radial direction of the black hole, which is the current correlator (1.1) receives its dominant contribution from direction extra to the boundary spacetime. The dashed line diagrams in which the fermion loop goes into the horizon. This denotes the black hole horizon. The black hole spacetime also gives an intuitive picture that the dissipation of current is asymptotes to that of AdS4 near the boundary and factorizes controlled by the decay of the particles running in the loop, R2 into AdS2 near the horizon. The current-current correlator which in the bulk occurs by falling into the black hole. in (1.1) can be obtained from the propagator of the gauge field Ay with end points on the boundary. Wavy lines correspond to gauge field propagators and the dark line denotes the bulk propagator hybrid approach. We first write down an integral expres- for the probe fermionic field. The left diagram is the tree-level sion for the two-point current correlation function propagator for Ay, while the right diagram includes the contri- yy bution from a loop of fermionic quanta. The contribution from GE ði lÞ in Euclidean signature. We then perform analytic the Fermi surface associated with corresponding boundary fer- continuation (1.3) to Lorentzian signature inside the mionic operator can be extracted from the diagram on the right. integral. This gives an intrinsic Lorentzian expression for the conductivity. The procedure can be considered as the generalization of the procedure discussed in [28] for coming from the Fermi surface, which can be unambigu- tree-level amplitudes to one-loop. After analytic continu- ously extracted. This is possible because conductivity from ation, the kind of diagrams indicated in Fig. 2 are included independent channels is additive, and, as we will see, the unambiguously. In fact they are the dominant contribution contribution of the Fermi surface is nonanalytic in tem- to the dissipative part of the current correlation function, perature T as T ! 0. We emphasize that the behavior of which gives resistivity. interest to us is not the conductivity that one could measure Typical one-loop processes in gravity also contain UV most easily if one had an experimental instantiation of this divergences, which in Fig. 1 happen when the two bulk system and could hook up a battery to it. The bit of interest vertices come together. We are, however, interested in the is swamped by the contribution from the OðN2Þ charge leading contributions to the conductivity from excitations density, which however depends analytically on tempera- around the Fermi surface, which are insensitive to short ture. In the large-N limit which is well described by distance physics in the bulk.2 Thus we are computing a classical gravity, these contributions appear at different orders in N and can be clearly distinguished. In cases UV-safe quantity, and short distance issues will not be where there are multiple Fermi surfaces, we will see that relevant. This aspect is similar to other one-loop applied the ‘‘primary’’ Fermi surface (this term was used in [14]to holography calculations [29,30]. denote the one with the largest k ) makes the dominant The plan of the paper is follows. In Sec. II, we first F briefly review the physics of the finite density state and contribution to the conductivity. 2 Calculations of one-loop Lorentzian processes in a black discuss the leading OðN Þ conductivity which represents a hole geometry are notoriously subtle. Should one integrate foreground to our quantity of interest. Section III outlines the interaction vertices over the full black hole geometry or the structure of the one-loop calculation. Section IV only the region outside the black hole? How should one derives a general formula for the DC and optical conduc- treat the horizon? How should one treat diagrams (such as tivity respectively in terms of the boundary fermionic that in Fig. 2) in which a loop is cut in half by the horizon? spectral function and an effective vertex which can One standard strategy is to compute the corresponding in turn be obtained from an integral over bulk on-shell correlation function in Euclidean signature, where these quantities. Section V discusses in detail the leading low issues do not arise, and then obtain the Lorentzian expres- temperature behavior of the effective vertices for the DC and optical conductivities. In Sec. VI we first derive the sion using analytic continuation (for l > 0) scaling behavior of DC and optical conductivities and then yy yy present numerical results for the prefactors. Section VII GR ð Þ¼GE ði l ¼ þ i Þ: (1.3) concludes with some further discussion of the main results. This, however, requires precise knowledge of the Euclidean correlation function which is not available, 2For such a contribution the two vertices are always far apart given the complexity of the problem. We will adopt a along boundary directions.
045016-3 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) A number of appendices contain additional background and material and fine details. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gFQ 2ðd 2Þ 2 d 2 ;cd : (2.5) II. BLACK HOLE GEOMETRY AND cdR r0 d 1 O N2 ð Þ CONDUCTIVITY It is useful to parametrize the charge of the black hole by a In this section we first give a quick review of the AdS length scale r , defined by charged black hole geometry and then consider the leading sffiffiffiffiffiffiffiffiffiffiffiffi OðN2Þ 3 d contribution to the conductivity. Our motivation is Q rd 1: (2.6) twofold. Firstly, this represents a ‘‘background’’ from d 2 which we need to extract the Fermi surface contribution; we will show that this contribution to the conductivity In terms of r , the density, chemical potential and tem- perature of the boundary theory are is analytic in T, unlike the Fermi surface contribution. Secondly, the bulk-to-boundary propagators which deter- 1 r d 1 1 ¼ ; mine this answer are building blocks of the one-loop 2 (2.7) R ed calculation required for the Fermi surface contribution. dðd 1Þ r r d 2 A. Geometry of a charged black hole ¼ 2 ed; (2.8) d 2 R r0 We consider a finite density state for CFT by turning on d a chemical potential for a Uð1Þ global symmetry. On the dr r2d 2 T ¼ 0 1 (2.9) gravity side, in the absence of other bulk matter fields to 4 R2 r2d 2 take up the charge density, such a finite density system is 0 described by a charged black hole in d þ 1-dimensional where we have introduced 4 anti-de Sitter spacetime (AdSdþ1)[26,27]. The conserved g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF current J of the boundary global Uð1Þ is mapped to a bulk ed : (2.10) 2dðd 1Þ Uð1Þ gauge field AM. The black hole is charged under this gauge field, resulting in a nonzero classical background for In the extremal limit T ¼ 0, r0 ¼ r and the geometry the electrostatic potential A ðrÞ. For definiteness we take r r Rd 1 t near the horizon (i.e. for r 1)isAdS2 : the charge of the black hole to be positive. The action for a vector field A coupled to AdS gravity can be written R2 r2 M dþ1 ds2 ¼ 2 ð dt2 þ d 2Þþ dx~2 (2.11) Z 2 2 1 pffiffiffiffiffiffiffi dðd 1Þ R2 R dþ1 R S ¼ 2 d x g þ 2 2 F F 2 R gF where R2 is the curvature of AdS2, and (2.1) 2 1 R2 ed R2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR; ¼ ;At ¼ : (2.12) with the black hole geometry given by [26,27] dðd 1Þ r r 2 2 2 2 ds gttdt þ grrdr þ giidx~ In the extremal limit the chemical potential and energy density are given by r2 R2 dr2 2 2 ¼ 2 ð hdt þ dx~ Þþ 2 (2.2) d 1 2 2 R r h dðd 1Þ r R ðd 1Þ R d ¼ 2 ed; ¼ 2 (2.13) with d 2 R d 2 r Q2 M rd 2 with charge density still given by (2.7). 0 r0 r h ¼ 1 þ 2d 2 d ;At ¼ 1 d 2 : (2.3) At a finite temperature T , 1 and the r r r r near-horizon metric becomes that of a black hole in Note that here we define g (and gtt) with a positive sign. Rd 1 tt AdS2 times r0 is the horizon radius determined by the largest positive 0 ! 1 root of the redshift factor R2 2 d 2 r2 ds2 ¼ 2 @ 1 dt2 þ A þ dx~2 (2.14) 2 2 2 2 R2 d Q 0 1 2 hðr Þ¼0; ! M ¼ r þ (2.4) 0 0 0 rd 2 0 with 3 2 This was also considered in [31–33]. ed R2 4 At ¼ 1 ; 0 (2.15) For other finite density states (with various bulk matter 0 r0 r contents, corresponding to various operator contents of the dual QFT), see e.g. [34–49] for an overview. and the temperature
045016-4 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) 1 The small frequency and small temperature limit of the ¼ T : (2.16) solution to Eq. (2.22) can be obtained by the matching 2 0 technique of [14]; the calculation parallels closely that of In this paper we will be interested in extracting the [14], and was also performed in [31]. The idea is to divide leading temperature dependence in the limit T ! 0 (but the geometry into two regions in each of which the equa- with T 0) of various physical quantities. For this purpose tion can be solved approximately; at small frequency, these it will be convenient to introduce dimensionless variables regions overlap and the approximate solutions can be 2 matched. For this purpose, we introduce a crossover radius TR2 1 T ¼ ; 0 T 0 ¼ ; Tt (2.17) rc, which satisfies r r 2 2 rc r TR2 after which (2.14) becomes 1; c 1; (2.24) 0 ! 1 r rc r R2 2 d 2 r2 ds2 ¼ 2 @ 1 d 2 þ A þ dx~2 (2.18) and will refer to the region r>rc as the outer (or UV) 2 2 2 2 0 1 2 R region and the region r0 2 2 Note that in this limit the system is still at a nonzero m R ¼ 2;r! r : (2.26) 1 eff 2 temperature as 0 ¼ 2 remains finite. An advantage of (2.18) and (2.19) is that in terms of these dimensionless As a result the IR region differential equation becomes the variables, T completely drops out of the metric. same as that of a neutral scalar field in AdS2 with this mass. Thus at k ¼ 0, the CFT mode Jy to which the gauge field couples flows to a scalar operator in the IR. It then follows B. Vector boundary to bulk propagator that the IR scaling dimension of Jy is given by [see (56) We now calculate the conductivity of the finite density 1 of [14]] IR ¼ 2 þ ¼ 2 as state described by (2.2) using the Kubo formula (1.1), to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi leading order in T in the limit 1 2 2 3 ¼ þ m R ðr Þ ¼ : (2.27) 4 eff 2 2 T ! 0;s ¼ fixed: (2.21) T Using current conservation this translates into that t t To calculate the two-point function of the boundary IRðJ Þ¼1; i.e. J is a marginal operator in the IR, which is expected as we are considering a compressible system.5 current Jy, we need to consider small fluctuations of the Near the boundary of the inner region (i.e. ! 0), the gauge field Ay ay which is dual to Jy with a nonzero 1 1 3 a 2 2 2 frequency and k ¼ 0. In the background of a charged solutions of (2.22) behave as y . We will black hole, such fluctuations of a mix with the vector choose a basis of solutions which are specified as (which y also fixes their normalization) fluctuations hty of the metric, as we discuss in detail in 1 3 Appendix B. This mixing has a simple boundary interpre- r r 2 2 1 3 ð ; sÞ! ¼ 2 2; ! 0: (2.28) tation; acting on a system with net charges with an electric I 2 TR2 field causes momentum flows in addition to charge flows. After eliminating h from the equations for a , we find Note that since the metric (2.18) has no explicit T depen- ty y ð Þ that (see Appendix B for details) dence, as a function of s (2.21), I s; also have no ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi p rr yy p yy 2 tt 2 @rð gg g @rayÞ gg ðmeff g Þay ¼ 0 5Note that here the IR marginality of Jt only applies to zero t (2.22) momentum. That J must be marginal in the IR for any com- pressible system leads to general statements of the two-point y where a acquires an r-dependent mass given by function of J in the low frequency limit [50]. For example for a y system whose IR limit is characterized by a dynamical exponent z, then Jy must have IR dimension 2 þ d 2 .Ford ¼ 3 this r 2d 2 z 2 2 further implies for any z the two-point function of Jy must scale meffR2 ¼ 2 : (2.23) r like 3 at zero momentum. 045016-5 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) explicit T dependence. This will be important for our Note that in the above expression all the T dependence is discussion in Sec. V. made manifest. a ð0Þ The retarded solution (i.e. y is infalling at the horizon) The leading order solutions in the outer region can for the inner region can be written as [14] be determined analytically [31]; with d 2 ðretÞ þ ð0Þ r r ay ð ; sÞ¼ þ G ðsÞ ; (2.29) I y I þ ðrÞ¼ 2 1 (2.36) ðd 2ÞR2 r G where y is the retarded function for ay in the AdS2 region, 3 and thus which can be obtained by setting ¼ 2 and q ¼ 0 in Eq. (D27) of Appendix D of [14]6 r bð0Þ að0Þ ¼ ; þ ¼ rd 2: (2.37) þ ðd 2ÞR2 að0Þ is 2 þ G ðsÞ¼ ðs2 þð2 Þ2Þ: (2.31) y 3 að0Þ bð0Þ Useful relations among , can be obtained from In the outer region we can expand the solutions to (2.22) the constancy in r of the Wronskian ffiffiffi ffiffiffi in terms of analytic series in and T. In particular, p yy rr p yy rr W½a ;a a gg g @ra a gg g @ra (2.38) the zeroth order equation is obtained by setting in (2.22) 1 2 1 2 2 1 ¼ 0 and T ¼ 0 (i.e. the background metric becomes where a1;2 are solutions to (2.22). In particular, equating ð0Þ ð0Þ that of the extremal black hole). Examining the behavior W½ þ ; at boundary and horizon gives of the resulting equation near r ¼ r , one finds that 1 3 ð0Þ ð0Þ 2 3 2 2 bð0Þa að0Þb d 3 2 d 3 2 ay ðr r Þ , which matches with those of the inner þ þ ¼ r R ¼ r R : (2.39) region in the crossover region (2.25). It is convenient to use d 2 d 2 ð0Þ Note that this equation assumes the normalization of the the basis of the zeroth order solutions ðrÞ which are specified by the boundary condition gauge field specified in (2.32). 1 3 ð0Þ r r 2 2 C. Tree-level conductivity ðrÞ! 2 ;r! r : (2.32) R2 We now proceed to study the low frequency and low Note that in this normalization in the overlapping region temperature conductivity at tree level in the charged black we have the matching hole, using the boundary to bulk propagator just discussed. The OðN2Þ conductivity is given by the boundary value ð0Þ þ ð0Þ 2 of the canonical momentum conjugate to a (in terms of r þ $ T I ; $ T I : (2.33) y foliation) evaluated at the solution (2.35) Near the AdS boundary, ð0Þ can be expanded as pffiffiffiffiffiffiffi dþ1 2R2 ggyygrr@ K ð Þ¼ lim r A (2.40) ð0Þ r!1 að0Þ bð0Þ 2 d r!1 g2 2 i þ r (2.34) F að0Þ bð0Þ which gives with , some functions of k (and does not depend , T). We can now construct the bulk-to-boundary (retarded) 2R3 d 1 bð0Þ þ G T3bð0Þ ð Þ¼ðd 2Þ þ y : (2.41) propagator to leading order in the limit (2.21), which 2 2 i að0Þ G 3að0Þ gF þ þ yT we will denote as KAðr; Þ with boundary condition KAðr; Þ!1 at the AdSdþ1 boundary (r !1). From We thus find (2.29), the matching (2.33) and (2.34), we thus find the 3 d full bulk-to-boundary propagator is then 2R 1 ð Þ¼ðd 2Þ 2 2 8 gF i ð0Þ G 3 ð0Þ > þ ðrÞþ yðsÞT ðrÞ ! <> outer region ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ að0ÞþG ðsÞT3að0Þ b b a a b þ y þ G 3 þ þ KAðr; Þ¼ : (2.35) þ yT þ (2.42) > þþG ðsÞ að0Þ að0Þ 2 :T I y I inner region þ ð þ Þ að0ÞþG ðsÞT3að0Þ þ y r d 2 i ¼ Kðd 2Þ 6 2 We copy it here for convenience: R d 2 d 3 d 5 ð 2 Þ ð1 þ is þ iqe Þ ð1 þ iqe Þ þ K ð 2 þð2 TÞ2Þþ G 2 2 2 d 2 d dðd 1Þ e RðsÞ¼ð4 Þ 1 is 1 : d ð2 Þ ð þ iqedÞ ð iqedÞ 2 2 2 (2.43) (2.30) Note that the above equation differs from that of Appendix D of where we have used (2.37) and (2.39) as well as (2.13) 2 K 2Rd 1 K [14] by a factor of T due to normalization difference in (2.28). and introduced 2 2 . , which also appears in the gF 045016-6 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) vacuum two-point function, specifies the normalization of the boundary current and scales as OðN2Þ. Using (2.13) and (2.7), the first term in (2.43) can also be written as 2 i ð Þ¼ þ (2.44) þ P where P ¼ d 1 is the pressure. When supplied with the standard i prescription, this term gives rise to a contribu- tion proportional to ð Þ in the real part of ð Þ. This delta function follows entirely from kinematics and repre- FIG. 3 (color online). The bulk Feynman diagram by which sents a ballistic contribution to the conductivity for a clean the spinor contributes to the conductivity. charged system with translational and boost invariance, as we review in Appendix A. It is also interesting to note that theory retarded Green’s function of a fermionic operator from the bulk perspective the delta function in the con- has a Fermi surface-like pole at ! ¼ 0 and k ¼ kF of ductivity is a direct result of the fact that the fluctuations of the kind described in [13,14]. We will neglect many the bulk field ay are massive, as is clear from the perturba- ‘‘complications,’’ including spinor indices, matrix struc- tion equation (2.22). This is not a breakdown of gauge tures, gauge-graviton mixing, and a host of other important invariance; rather the gauge field acquired a mass through details, which turn out to be inessential in understanding its mixing with the graviton. In the absence of such a mass the structure of the calculation. term the radial equation of motion is trivial in the hydro- The important bulk Feynman diagram is depicted in dynamic limit (as was shown in [51]) and there is no such Fig. 3. Note that it is structurally very similar to the delta function. particle-hole bubble which contributes to the Fermi liquid In (2.43), the important dynamical part is the second conductivity (see e.g. [52,53]): an external current source term, which gives the dissipative part of the conductivity. creates a fermion-antifermion pair, which then recombines. This part, being proportional to 2 þð2 TÞ2, is analytic The calculation differs from the standard Fermi liquid in both T and and the DC conductivity goes to zero in calculation in two important ways. The first, obvious dif- the T ! 0 limit. This has a simple physical interpretation; ference is that the gravity amplitude involves integrals over the dissipation of the current arises from the neutral com- the extra radial dimension of the bulk geometry. It turns ponent of the system, whose density goes to zero in the out, however, that these integrals can be packaged into T ! 0 limit, leaving us with only the ballistic part (2.44) factors in the amplitude (called below) that play the for a clean charged system. We also note that for a clean role of an effective vertex. The second main difference system such as this, there is no nontrivial heat conductivity is that actual vertex correction diagrams in the bulk are or thermoelectric coefficient at k ¼ 0, independent of the suppressed by further powers of N 2 and are therefore charge conduction. This is because momentum conserva- negligible, at least in the large-N limit in which we work. tion is exact, and just as in the discussion of Appendix A, We now proceed to outline the computation. While it is there can therefore be no dissipative part of these transport more convenient to perform the tree-level calculation of coefficients. the last section in the Lorentzian signature, for the one- loop calculation it is far simpler to work in Euclidean III. OUTLINE OF COMPUTATION OF signature, where one avoids thorny conceptual issues Oð1Þ CONDUCTIVITY regarding the choice of vacuum and whether interaction vertices should be integrated through the horizon or not. As explained in the introduction, in order to obtain the Our strategy is to first write down an integral expression 7 yy contribution of a Fermi surface to the conductivity, we for the Euclidean two-point function GE ði lÞ and then need to extend the tree-level gravity calculation of the analytically continue to Lorentzian signature inside the previous section to the one-loop level with the correspond- integral (for l > 0) ing bulk spinor field running in the loop. This one-loop calculation is rather complicated and is spelled out in detail yy yy G ð Þ¼G ði l ¼ þ i Þ (3.1) in the next section. In this section we outline the main R E ingredients of the computation suppressing the technical details. which will then give us the conductivity via the Kubo formula (1.1) and (1.2). A. Cartoon description 7We put the i in the argument of all Euclidean correlation In this subsection we will describe a toy version of the functions to eventually make the analytic continuation to one-loop conductivity. We will assume that the boundary Lorentzian signature more natural. 045016-7 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) We now turn to the evaluation of the diagram in Fig. 3, B. Performing radial integrals which works out to have the structure Inserting (3.3) into (3.2) we can now use standard X Z manipulations from finite-temperature field theory to yy ~ ~ GE ði lÞ T dkdr2dr2DEðr1;r2; i l þ i!n; kÞ rewrite the Matsubara sum in (3.2) in terms of an integral i!n over Lorentzian spectral densities. The key identity is ~ (see Appendix G for a discussion) KAðr1; i lÞDEðr2;r1; i!n; kÞKAðr2; i lÞ: X (3.2) 1 1 fð! Þ fð! Þ T ¼ 1 2 ið!m þ lÞ !1 i!m !2 !1 i l !2 The ingredients here require further explanation. !m DEðr1;r2; i!n;kÞ is the spinor propagator in Euclidean (3.5) space. KAðr; i lÞ is the boundary-to-bulk propagator for the gauge field [i.e. Euclidean analytic continuation of with (2.35)]; it takes a gauge field source localized at the bound- 1 fð!Þ¼ (3.6) ary and propagates it inward, computing its strength at a e ! 1 bulk radius r. The vertices have a great deal of matrix structure that we have suppressed, and the actual derivation where the upper (lower) sign is for fermion (boson). Using of this expression from the fundamental formulas of the above identity (3.2) can now be written as AdS/CFT requires a little bit of manipulation that is dis- Z yy fð!1Þ fð!2Þ cussed in the next section, but its structure should appear G ði Þ d! d! dkdr~ dr E l 1 2 1 2 ! ! i plausible. The radial integrals dr should be understood as 1 2 l ~ ~ including the relevant metric factors to make the expres- ðr1;r2; !; kÞKAðr1; i lÞ ðr2;r1; i l; kÞ sion covariant, and dk~ denotes integration over spatial KAðr2; i lÞ: (3.7) momentum along boundary directions. We would now like to perform the Euclidean frequency We then analytically continue the above expression to the sum. This is conveniently done using the spectral repre- Lorentzian signature by setting i l ¼ þ i .Wenow sentation of the Euclidean Green’s function for the spinor, realize the true power of the spectral decomposition (3.3) and (3.4); the bulk-to-bulk propagator factorizes into a Z ~ ~ d! ðr1;r2; !; kÞ product of spinor wave functions, allowing us to do each DEðr1;r2; i!m; kÞ¼ ; (3.3) ð2 Þ i!m ! radial integral independently. In this way all of the radial integrals can be repackaged into an effective vertex ~ where ðr1;r2; !; kÞ is the bulk-to-bulk spectral density. Z ~ ~ c ~ c ~ As we discuss in detail in Appendix C, ðr1;r2; !; kÞ can ð!1;!2; ; kÞ¼ dr ðr; !1; kÞKAðr; Þ ðr; !2; kÞ; be further written in terms of boundary theory spectral density Bð!; kÞ as (again schematically, suppressing all (3.8) indices) where the propagator KAðr; Þ has now become a ðr; r0; !; k~Þ¼c ðr; !; k~Þ ð!; kÞc ðr0; !; k~Þ (3.4) Lorentzian object that propagates infalling waves, and we B are left with the formula where c ðrÞ is the normalizable spinor wave function8 to Z yy ~ fð!1Þ fð!2Þ ~ the Dirac equation in the Lorentzian black hole geometry. GR ð Þ d!1d!2dk Bð!1; kÞ Equation (3.4) can be somewhat surprising to some readers !1 !2 i ~ ~ ~ and we now pause to discuss it. The bulk-to-bulk spectral ð!1;!2; ; kÞ Bð!2; kÞ ð!2;!1; ; kÞ: density factorizes in the radial direction; thus in some sense the density of states is largely determined by the analytic (3.9) ~ structure of the boundary theory spectral density Bð!; kÞ. This formula involves only integrals over the boundary We will see that this means that despite the presence of the theory spectral densities; the radial integral over spinor and extra radial direction in the bulk, the essential form of one- gauge field wave functions simply provides an exact loop calculations in this framework will be determined by derivation of the effective vertex that determines how the boundary theory excitation spectrum, with all radial strongly these fluctuations couple to the external field integrals simply determining the structure of interaction theory current, as shown in Fig. 4. vertices that appear very similar to those in field theory. Computing the effective vertex requires a complete We now turn to the evaluation of the expression (3.2). solution to the bulk wave equations; however, we will show that in the low temperature and low frequency limit, 8As discussed in Appendix C, up to normalization, there is a the conductivity (3.9) is dominated by the contribution near unique normalizable solution with given !, k~. the Fermi surface, where we can simply replace by a 045016-8 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) Appendix E). We will show in Appendix F that such contributions give only subleading corrections in the low temperature limit and will be omitted. (3) The careful treatment of spinor fields and associated indices will require some care. FIG. 4 (color online). Final formula for conductivity; radial A. A general formula integrals only determine the effective vertex , with exact We consider a free spinor field in (2.2) with an action propagator for boundary theory fermion running in loop. Z pffiffiffiffiffiffiffi dþ1 M S ¼ d x giðc DM c mccÞ (4.1) constant. Thus if one is interested in extracting low temperature DC and optical conductivities in the low fre- where c ¼ c y t and quency regime, the evaluation of (3.9) reduces to a familiar 1 one as that in a Fermi liquid (without vertex corrections). D ¼ @ þ ! ab iqA : (4.2) M M 4 abM M IV. CONDUCTIVITY FROM A SPINOR FIELD The abstract spacetime indices are M; N ... and the In this section we describe in detail the calculation abstract tangent space indices are a; b; .... The index with an underline denotes that in tangent space. Thus a leading to (3.9), paying attention to all subtleties. For denotes gamma matrices in the tangent frame and M those readers who want to skip the detailed derivation, the final in curved coordinates. According to this convention, for results for the optical and DC conductivities are given example, by (4.49) and (4.50), with the relevant vertices given by (for d ¼ 3)(4.53)–(4.56). pffiffiffiffiffiffiffi M ¼ aeM; r ¼ grr r: (4.3) Before going into details, it is worth mentioning some a important complications which we ignored in the last We will make our discussion slightly more general, appli- section: cable to a background metric given by the first line of (2.2) (1) As already discussed in Sec. II B, the gauge field with gMN depending on r only. Also for notational sim- perturbations on the black hole geometry mix with plicity we will denote the graviton perturbations; a boundary source for the bulk gauge field will also lead to perturbations in Z Z dd 1k ¼ metric, and as a result the propagator K in (3.2) d 1 : (4.4) A k ð2 Þ should be supplemented by a graviton component. Thus the effective vertex is rather more involved We are interested in computing the one-loop correction than the schematic form given in (3.8). to the retarded two-point function of the boundary vector (2) Another side effect of the mixing with graviton is current due to a bulk spinor field. In the presence of a that, in addition to Fig. 3, the conductivity also background gauge field profile, the fluctuations of the bulk receives a contribution from the ‘‘seagull’’ diagram gauge field mix with those of the metric. Consider small j j of Fig. 5, coming from quartic vertices involving perturbations in aj Aj and ht ¼ gt . In Appendix E terms quadratic in metric perturbations (given in we find that the corrections to the Dirac action are given at cubic order by XZ pffiffiffiffiffiffiffi j c dþ1 c S3½aj;ht ; ¼ i d x g j 1 hj t@ þ g @ hj rtj iqa j c t j 8 jj r t j (4.5) where rtj t r j. Note that in the above equation and below the summations over boundary spatial indices will be indicated explicitly (with no summation associated with repeated indices). There are also quartic corrections FIG. 5 (color online). The ‘‘seagull’’ diagram coming from (involving terms quadratic in bosonic fluctuations) which quartic vertices involving terms quadratic in bosonic perturba- are given in Appendix E. These terms give only subleading tions. The boundary is represented by a circle. corrections, as we discuss in Appendix F. 045016-9 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) Now we go to Euclidean signature via where the boundary spectral function B is Hermitian and c is the normalizable Lorentzian wave function for the t ! itE ! ! i!E At ! iA iS ! SE: 9 free Dirac equation in the black hole geometry. Bð!; kÞ is (4.6) the boundary theory spectral density of the holographic It is helpful to keep in mind that c and c y do not change non-Fermi liquid, and was discussed in detail in [14] (see t Appendix D for a review). Note that in (4.12) bulk spinor under the continuation, and we do not change . The c Euclidean spinor action can then be written as indices are suppressed and , in label independent Z normalizable solutions and as discussed in Appendix C can pffiffiffi ð0Þ be interpreted as the boundary spinor indices. S ¼ i ddþ1x gc ð MD mÞc þ S þ (4.7) E M 3 Now we introduce various momentum space Euclidean boundary-to-bulk propagators for the gauge field and where S3 can be written in momentum space as graviton: XXZ Z ffiffiffiffiffiffiffi 2 p S3 ¼ T dr g ajðr; i lÞ¼KAðr; i lÞAjði lÞ; k !m (4.13) l hjðr; i Þ¼K ðr; i ÞA ði Þ ~ ~ ~ t l h l j l c ðr;i!m þi l;kÞBðr;i l;kÞc ðr;i!m;kÞ: (4.8) where Ajði lÞ is the source for the boundary conserved ~ Here the kernel Bðr; i l; kÞ contains all dependence on the current in Euclidean signature. These are objects which gauge and metric fluctuations, which we have also Fourier propagate a gauge field source at the boundary to a gauge expanded: field or a metric fluctuation in the interior, and so should A A X g perhaps be called KA and Kh ; we drop the second A label ~ j t jj j rtj Bðr;i l;kÞ¼ i ikjht ðr;i lÞ þ @rht ðr;i lÞ since we will never insert metric sources in this paper. j 8 KAðr; i lÞ and Khðr; i lÞ go to zero (for any nonzero j l) at the horizon and they do not depend on the index j iqajðr;i lÞ : (4.9) due to rotational symmetry. Kh and KA are not indepen- dent; in Appendix B we show that Note that since we are only interested in calculating the pffiffiffiffiffiffiffiffiffiffiffiffi dþ1 conductivity at zero spatial momentum, we have taken a 2 j @rKh ¼ C grrgttg KA; (4.14) j ii and ht to have zero spatial momentum. We now evaluate the one-loop determinant by integrat- with ing out the fermion field. We seek the quadratic depen- C ¼ 2 2 (4.15) dence on the gauge and graviton fields; the relevant term in the effective action is given by the Feynman diagram in where is the background charge density (2.7). Fig. 3 and is Now using the definition of the propagators (4.13) we can 2 X Z Z qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi write the kernel Bðr; i ; k~Þ in (4.9) in terms of a new object j T l ½aj;ht ¼ dr1 gðr1Þdr2 gðr2Þ Qjðr ;i ; k~Þ 2 k ; l that has the source explicitly extracted: !m; l X ~ j ~ ~ ~ Bðr; i l; kÞ¼ Q ðr; i l; kÞAjði lÞ (4.16) trðDEðr ;r ; i!m þ i l; kÞBðr ; l; kÞ 1 2 2 j D ðr ;r ; i! ; k~ÞBðr ; ; k~ÞÞ (4.10) E 2 1 m 1 l with where the tr denotes the trace in the bulk spinor indices and g Qjðr;i ;k~Þ¼ i ik K ðr;i Þ t þ jj @ K ðr;i Þ rtj D ðr ;r ; i! ; k~Þ denotes the bulk spinor propagator in the l j h l r h l E 1 2 m 8 Euclidean signature. As always we suppress bulk spinor j indices. The bulk spinor propagator is discussed in some iqKAðr;i lÞ : (4.17) detail in Appendix C. The single most important property that we use is its spectral decomposition Plugging (4.16) into (4.10), we can now express the entire expression in terms of the boundary gauge field Z ~ ~ d! ðr1;r2; !; kÞ source Ajði lÞ. Taking two functional derivatives of this DEðr1;r2; i!m; kÞ¼ ; (4.11) 2 i!m ! expression with respect to Aj, we find that the boundary Euclidean current correlator can now be written as ~ where ðr1;r2; !; kÞ is the bulk spectral function. As is shown in (C53) of Appendix C, the bulk spectral function 9 can be written in terms of that of the boundary theory as For a precise definition of the normalizable Lorentzian wave function, see again Appendix C. In this section, to avoid clutter, 0 ~ c ~ c 0 we will use the boldface font to denote the normalizable solu- ðr; r ; !; kÞ¼ ðrÞ B ð!; kÞ ðr Þ (4.12) tion. The non-normalizable solution will not appear. 045016-10 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) X Z Z qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ij ~ i ~ ~ j ~ GE ði lÞ¼ T dr1 gðr1Þdr2 gðr2Þ trðDEðr1;r2; i!m þ i l; kÞQ ðr2; i l; kÞDEðr2;r1; i!m; kÞQ ðr1; i l; kÞÞ: k !m (4.18) Note that all objects in here are entirely well defined and self-contained; we now need only evaluate this expression. To begin this process we first plug (4.11) into (4.18) and then perform the sum over !m using the techniques that were outlined earlier in (3.5)–(3.7). We find that Eq. (4.18) can be further written as Z Z Z qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ij d!1 d!2 fð!1Þ fð!2Þ GE ði lÞ¼ dr1 gðr1Þdr2 gðr2Þ k 2 2 !1 i l !2 ~ i ~ ~ j ~ trð ðr1;r2; !1; kÞQ ðr2; i l; kÞ ðr2;r1; !2; kÞQ ðr1; i l; kÞÞ; (4.19) Now plugging (4.12) into (4.19) we find that Z Z d! d! fð! Þ fð! Þ ij 1 2 1 2 ~ i ~ ~ j ~ GE ði lÞ¼ B ð!1; kÞ ð!1;!2;i l; kÞ B ð!2; kÞ ð!2;!1; i l; kÞ (4.20) k 2 2 !1 i l !2 with We thus find that the retarded Green’s function for the currents can be written as i ~ ð!1;!2;i l;kÞ Z Z ð Þ ð Þ Z pffiffiffi ij d!1 d!2 f !1 f !2 c ~ i ~ c ~ GR ð Þ¼ ¼ dr g ðr;!1;kÞQ ðr;i l;kÞ ðr;!2;kÞ: (4.21) k 2 2 !1 !2 i ð ~Þ i ð ~Þ ð ~Þ Let us now pause for a moment to examine this expression. B !1; k !1;!2; ; k B !2; k In the last series of manipulations we replaced the interior j ~ ð!2;!1; ; kÞ (4.25) frequency sum with an integral over real Lorentzian fre- quencies; however by doing this we exploited the fact that where the spectral density factorizes in r. This allowed us to Z pffiffiffi i ~ c i c absorb all radial integrals into , which should be thought ð!1;!2; ; kÞ¼ dr g Q (4.26) of as an effective vertex for the virtual spinor fluctuations. Now only the boundary theory spectral density B appears with explicitly in the expression. This form for the expression is g j ~ t jj rtj perhaps not surprising from a field-theoretical point of Q ðr; ; kÞ¼ i ikjKhðr; Þ þ @rKhðr; Þ view; however it is interesting that we have an exact 8 expression for the vertex, found by evaluating radial inte- iqK ðr; Þ j : (4.27) grals over normalizable wave functions. A We now obtain the retarded Green’s function for the Note that in (4.20) both ið! ;! ; i ; k~Þ analytically Gijði Þ > 1 2 l currents by starting with E l for l 0 and analyti- i 10 ij ð ~Þ cally continuing G ði Þ to Lorentzian signature via continue to !1;!2; ; k . E l The complex, frequency-dependent conductivity is ij ij G ð Þ¼G ði ¼ þ i Þ: (4.22) ij R E l G ð Þ ijð Þ R (4.28) We will suppress the i in equations below for notational i simplicity but it is crucial to keep it in mind. For simplicity which through (4.25) is expressed in terms of intrinsic of notations we will denote the Lorentzian analytic con- boundary quantities; can be interpreted as an effective tinuation of various quantities only by their argument, e.g. vertex. Note that from (4.24) one can readily check that K ðr; i Þj ! K ðr; Þ: (4.23) i y t i t A l i l¼ A ðQ ðr; ; k~ÞÞ ¼ Q ðr; ; k~Þ (4.29) i We also make the analogous replacements for Kh, Q and which implies that . Note that KAðr; Þ and Khðr; Þ have now become i ~ i ~ retarded functions which are infalling at the horizon and ð!1;!2; ; kÞ¼ ð!2;!1; ; kÞ: (4.30) satisfy K ðr; Þ¼KAðr; Þ;Kðr; Þ¼Khðr; Þ: (4.24) 10 A h This is due to the fact that KAð i lÞ¼KAði lÞ. 045016-11 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) We now make a further manipulation on the expression For this purpose we choose a reference direction, say, with ~ ~ for the effective vertex (4.26) to rewrite the first term there kx ¼ k jkj and all other spatial components of k vanish- in terms of @rKh, which can then be related simply to KA ing. We will denote this direction symbolically as ¼ 0 via (4.14). To proceed note that the wave function c below. Then from the transformation properties of spinors satisfies the Dirac equation (C2), which implies it is easy to see that that11 (see Appendix C1for details) ~ y c t c BðkÞ¼Uð Þ Bðk; ¼ 0ÞU ð Þ; ðr;!1Þ ðr;!2Þ (4.35) ~ y i 1 pffiffiffi iðkÞ¼Rijð ÞUð Þ jðk; ¼ 0ÞU ð Þ c r c ¼ pffiffiffi@rð g ðr;!1Þ ðr;!2ÞÞ: (4.31) !1 !2 g where Rijð Þ is the orthogonal matrix which rotates a j We now use this identity in the first term of Q in (4.27) and vector k~ to ¼ 0 and U is the unitary matrix which does integrate by parts. We can drop both boundary terms: the the same rotation on a spinor (i.e. in , space). The term at infinity vanishes since the c are normalizable, and angular integral in (4.25) is reduced to the term at the horizon vanishes because the graviton wave Z function hi (and thus K ) vanishes there.12 We then find 1 t h dd 2 R ð ÞR ð Þ¼C (4.36) that (4.27) can be rewritten as ð2 Þd 1 ik jl ij kl k j ~ j r d 2 Q ðr; ; kÞ¼ i @rKhðr; Þ where C is a normalization constant and d denotes the !1 !2 1 measure for angular integration. Note that C ¼ 4 for 1 gii d ¼ 3 and C ¼ for d ¼ 4. The conductivity can now þ @ K ðr; Þ rtj iqK ðr; Þ j ; 12 2 8 r h A be written as (4.32) ijð Þ¼ ij ð Þ (4.37) where it is important to note that this expression makes sense only when sandwiched between the two on-shell with spinors in . This manipulation replaced the K with h Z Z its radial derivative @ K , and one can now use the relation C 1 d! d! fð! Þ fð! Þ r h ð Þ¼ dkkd 2 1 2 1 2 between gauge and graviton propagators (4.14) to elimi- i 2 2 ! ! i X 0 1 2 nate @rKh in favor of KA, leaving us with i B ð!1;kÞ ð!1;!2; ;kÞ B ð!2;kÞ iY k i j ~ j 2 j r r t j Q ðr; ; kÞ¼KAðr; Þ Y1 þ þ iY3 i !1 !2 ð!2;!1; ;kÞ (4.38) (4.33) i where Bð!; kÞ and ð!2;!1; ;kÞ in (4.38) and in all where expressions below should be understood as the correspond- 1 dþ1pffiffiffiffiffiffi 1 d ing quantities evaluated at ¼ 0 as in (4.35). Y ¼ qg 2;Y¼ Cg 2 g ;Y¼ g 2C: 1 jj 2 jj tt 3 8 jj Equation (4.38) can now be further simplified in a basis (4.34) in which B is diagonal, i.e. B ¼ B , leading to j Z 1 Z This is the form of Q that will be used in the remainder of C d 2 d!1 d!2 fð!1Þ fð!2Þ C C ð Þ¼ dkk this calculation. In (4.33), the -dependent terms [ was i 0 2 2 !1 !2 i given in (4.15)] can be interpreted as giving a ‘‘charge ð! ;kÞM ð! ;! ; ;kÞ ð! ;kÞ (4.39) renormalization’’ resulting from mixing between the gauge B 1 1 2 B 2 field and graviton. where (there is no summation over , below) X B. Angular integration i i M ð!1;!2; ;kÞ¼ ð!1;!2; ;kÞ ð!2;!1; ;kÞ We will now use the spherical symmetry of the under- i lying system to perform the angular integration in (4.25). (4.40) 11 i For !1 ¼ !2 the equation below reduces to the conservation with given by (4.26). From (4.30) we also have of fermionic number. 12 This is analogous to the well-known statement that At van- M ð! ;! ; ;kÞ¼M ð! ;! ; ;kÞ: (4.41) ishes at black hole horizons, and is similarly most transparent in 1 2 1 2 i Euclidean signature, where a nonzero ht at the shrinking time cycle indicates a delta-function contribution to the Einstein The DC conductivity can now be obtained by taking the tensor. ! 0 limit in (4.39), which can be written as 045016-12 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) X Z 1 Z Z 1 Z C d 2 d! @fð!Þ C d 2 d!1 d!2 fð!1Þ fð!2Þ DC ¼ dkk ð Þ¼ dkk 2 ; 0 2 @! i 0 2 2 !1 !2 i 1 1 M S Bð!1;kÞM11ð!1;!2; ;kÞ Bð!2;kÞ (4.49) Bð!; kÞ ð!; kÞ Bð!; kÞþ (4.42) and we will only need to calculate M11. Similarly, for the where one-loop DC conductivity, M Z 1 Z ð!; kÞ lim M ð! þ ;!; ;kÞ (4.43) C d 2 d! @fð!Þ !0 DC ¼ dkk 2 0 2 @! 1 M 1 is real [from (4.41)]. Note that the term written explicitly in Bð!; kÞ 11ð!; kÞ Bð!; kÞ: (4.50) (4.42) is obtained by taking the imaginary part of 1 in (4.39). The rest, i.e. the part proportional !1 !2 i to the real part of 1 , is collectively denoted as S. C. M11 in d ¼ 3 !1 !2 i We will see in Sec. VI B that such contribution vanishes For definiteness, let us now focus on d ¼ 3 and choose in the low temperature limit, so we will neglect it from the following basis of gamma matrices: now on. ! ! 3 0 i 1 0 Let us now look at the ! 0 limit of (4.43), for which r ¼ ; t ¼ ; 1 3 i 1 the ! ! term in (4.33) has to be treated with some care. 0 0 1 2 ! !(4.51) Naively, it appears divergent; however, note that 2 0 0 2 x ¼ ; y ¼ : 1 0 2 2 0 c ~ r c ~ lim ðr; ! þ ; kÞ ðr; !; kÞ !0 Now we write c ~ r c ~ ! ! ¼ ðr; !; kÞ @! ðr; !; kÞ (4.44) rr 1 1 rr 1 0 c ¼ð gg Þ 4 ; c ¼ð gg Þ 4 (4.52) 1 2 is finite because [see Eq. (C39) of Appendix C] 0 2 c ðr; !; k~Þ r c ðr; !; k~Þ¼0: (4.45) where 1;2 are two-component bulk spinors. As discussed in Appendix D, in the basis (4.51), the fermion spectral Introducing function is diagonal and the subscripts 1, 2 in (4.52) can be interpreted as the boundary spinor indices. Also note that i i ð!; kÞ lim ð! ;!; ;kÞ (4.46) in this basis the Dirac equation is real in momentum space !0 and 1;2 can be chosen to be real. x we thus have It then can be checked that (evaluated at ¼ 0) only y X has diagonal components while only has off-diagonal i i M ð!; kÞ¼ ð!; kÞ ð!; kÞ (4.47) components. From (4.40), we then find that i x x M11 ¼ 11ð!1;!2; ;kÞ 11ð!2;!1; ;kÞ; (4.53) and from (4.44) and (4.43) where Z ffiffiffiffiffiffiffi Z j ~ p c ~ pffiffiffiffiffiffiffi ð!;kÞ¼ dr gKAðr; ¼0Þ ðr;!;kÞ x T 11ð!1;!2; ;kÞ¼ dr grrKAð Þ 1 ð!1;kÞ ðY j þiY rtj ik Y r@ Þc ðr;!;k~Þ: 1 3 j 2 ! 3 iY2k 2 1 Y1 þ Y3 (4.48) !1 !2 1ð!2;kÞ: (4.54) The above expressions (4.39) and (4.42) are very gen- eral, but we can simplify them slightly by using some Similarly, for the DC conductivity, explicit properties of the expression for . We seek B M x x singular low-temperature behavior in the conductivity, 11 ¼ 11ð!; kÞ 11ð!; kÞ (4.55) which will essentially arise from low frequency singular- with ities in B. At the holographic Fermi surfaces described Z pffiffiffiffiffiffiffi in [14], at discrete momenta k ¼ kF, only one of the x T 1 11ð!; kÞ¼ dr grrKAðr; ¼ 0Þ 1 ðr; !; kÞ eigenvalues of B, say B, develops singular behavior. We can extract the leading singularities in the T ! 0 limit ðY 3 þ Y þ ikY 2@ Þ ðr; !; kÞ: 1 2 1 3 1 2 ! 1 by simply taking the term in (4.39) proportional to ð BÞ . Thus (4.39) simplifies to (4.56) 045016-13 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) Equations (4.53)–(4.56) are a set of complete and as rc ! r . Below we will see in some parameter range this self-contained expressions that can be evaluated numeri- is not so and the limit should be treated with care. cally if the wave functions 1;2 are known. Y1;2;3 were Now let us look at the integrand of the vertex (4.54)in introduced here in (4.34). the limit (5.3). For this purpose let us first review the As this was a somewhat lengthy exposition, let us briefly behavior of the vector propagator KA and spinor wave recap: after a great deal of calculation, we find the optical function in the IR and UV regions, which are discussed and DC conductivities are given by (4.49) and (4.50), with respectively in some detail in Sec. II B and Appendix D M (for d ¼ 3) M11 given by (4.53) and (4.54) and 11 given (please refer to these sections for definitions of various by (4.55) and (4.56). notations below): (1) From Eq. (2.35), we find in the outer region V. EFFECTIVE VERTICES ð0Þ K ð Þ¼ þ þ iOðT3Þ In this section we study in detail the analytic properties A að0Þ of the effective vertices (4.53)–(4.56) appearing respec- þ r r tively in the expressions for optical and DC conductivities ¼ þ OðTÞþiOðT3Þ (5.8) (4.49) and (4.50) in the regime of low frequencies and r d ¼ temperatures. We will restrict to 3. with the leading term independent of and T For simplicity of notations, from now on we will sup- and real. In the above we have also indicated the M x press various superscripts and subscripts in 11, 11, leading temperature dependence of the imaginary M x M 11, 11 and 1, and denote them simply as M, , , part [from (2.31)]. In the inner region from the and . Recall that under complex conjugation second line of Eq. (2.35) we have ð!1;!2; ;kÞ¼ ð!2;!1; ;kÞ (5.1) 4 KAð Þ¼TKAðs; ÞþOðT Þ (5.9) M ð!1;!2; ;kÞ¼Mð!1;!2; ;kÞ (5.2) where and both ð!; kÞ and Mð!; kÞ are real. Introducing scaling 1 K ðs; Þ¼ ð þ þ G ðsÞ Þ (5.10) variables A að0Þ I y I þ ! ! w 1 ;w 2 ;s (5.3) 1 T 2 T T has no explicit T dependence. (2) In the outer region, to lowest order in T, the normal- we will be interested in the regime izable spinor wave function can be expanded in ! as w1;w2;s¼ fixed;T! 0: (5.4) ¼ ð0Þ þ ! ð1Þ þ (5.11) A. Some preparations where ð0Þ and ð1Þ are defined respectively in As in the discussion of Sec. II B it is convenient to (D32) and (D33) and are T-independent. separate the radial integral in (4.54) into two parts, coming In the inner region, to leading order, can be from IR and UV regions respectively, i.e. written as [from (D35)] ¼ þ (5.5) IR UV ð0Þ aþ ð ; wÞ¼ T k þ (5.12) with W I Z 1 ffiffiffiffiffiffiffi Z r ffiffiffiffiffiffiffi p c p ð0Þ UV ¼ dr grr ...; IR ¼ dr grr ... (5.6) where aþ and W0 are some k-dependent constants rc r0 (but independent of ! and T), and I ð ; wÞ do not where r0 is the horizon at a finite temperature and rc is the have any explicit dependence on T. The above crossover radius specified in (2.24) and (2.25). In the inner expression, however, does not apply near a Fermi ð0Þ (IR) region it is convenient to use coordinate introduced surface k ¼ kF where aþ ðkFÞ is zero. Near a Fermi in (2.17), and then surface we have instead [see discussion in Z Appendix D around (D38)] 0 pffiffiffiffiffiffiffiffi IR ¼ d g ... (5.7) c 1 k ð ; w; TÞ¼ ½aþðk; !; TÞT F ð ; wÞ In the limit (5.3), as discussed around (2.25), we can take W I ð0Þ k þ rc ! r and c ! 0 in the integrations of (5.6) and (5.7). a ðkFÞT F I ð ; wÞ (5.13) Note, however, this limit can only be straightforwardly taken provided that the integrals of (5.6) are convergent where 045016-14 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) aþðk; !; TÞ¼c1ðk kFÞ c2! þ c3T þ 1 1 g ;Y1;3 Oð1Þ;Y2 ð 0 Þ2: 0 ¼ c1ðk kFð!; TÞÞ (5.14) (5.23) with real coefficients c1, c2, c3. Again in (5.13) all the T dependence has been made manifest. Here One can then check that the integrals for the we have also introduced a ‘‘generalized’’ Fermi IR part (5.7) are always convergent near the horizon . momentum kFð!; TÞ defined by 0 1 c k ð!; TÞ¼k þ ! 3 T þ (5.15) B. Low temperature behavior F F v c F 1 With the preparations of the last subsection, we will now with v ¼ c1 . proceed to work out the low temperature behavior of the F c2 (3) We now collect the asymptotic behavior of various effective vertices (4.54) and (4.56), which in turn will play functions appearing in the effective vertices (4.54) an essential role in our discussion of the low temperature and (4.56): behavior of the DC and optical conductivities in Sec. VI. (a) For r !1, The stories for (4.54) and (4.56) are rather similar. For illustration we will mainly focus on (4.56) and only point ffiffiffiffiffiffiffi mR p 1 out the differences for (4.54). The qualitative behavior of r ;KAð Þ Oð1Þ grr ; r the vertices will turn out to depend on the value of k.We 1 1 will thus treat different cases separately. Y ;Y (5.16) 1 r 2;3 rd < 1 and thus the UV part of the integrals are always 1. k 2 convergent as r !1. Note that in our conven- Let us first look at the inner region contribution. 1 tion mR > 2 with the negative mass corre- Equations (5.9) and (5.12) give the leading order tempera- sponding to the alternative quantization. ture dependence as (b) For r ! r , in the outer region, ð0Þ /ða Þ2T1 2 k þ (5.24) pffiffiffiffiffiffiffi 1 þ grr ;Y1;Y3 Oð1Þ;Y2 r r : pffiffiffiffiffiffi r r where we have also used that gtt / T. The outer region 0 (5.17) contribution UV starts with order OðT Þ and we can thus ignore (5.24) at leading order. The full vertex can be From (5.8), K ð Þ r r . From (D32) and A written as (D33), 1 2 k 1 ð!; kÞ¼ 0ðkÞþOðT Þ (5.25) ð0Þ ¼ ðað0Þ ð0Þ að0Þ ð0ÞÞ; (5.18) W þ þ where 0ðkÞ is given by the zeroth order term of UV and and can be written as 1 ð1Þ ð1Þ ð0Þ ð0Þ ð1Þ Z 3 ¼ ðaþ þ aþ 1 pffiffiffiffiffiffiffir r qR CR W ð0ÞT 3 ð0Þ 0ðkÞ¼ dr grr þ 3 1 ð1Þ ð0Þ ð0Þ ð1Þ r r r 8r a þ a þ Þ (5.19) pffiffiffiR3 ikC h ð0ÞT 2 ð1Þ (5.26) where r3 ðnÞ k n ðr r Þ ;r! r : (5.20) where we have taken rc ! r in the lower limit of the integral [as commented below (5.7)], and have plugged in (c) Near the event horizon ! 0, the explicit form of Y1;2;3. From (5.16)–(5.20) it can also be readily checked that the integral is convergent on both ends. i s K ðs; Þ¼ð Þ 4 ð1 þ Þ; (5.21) A 0 Note that 0ðkÞ is independent of both T and ! and real. Similarly for (4.54), one has and i w i w ð! ;! ; ; kÞ¼ ðw ;w ;s; kÞþOðT1 2 k Þ (5.27) I ðw; Þ c ð 0 Þ 4 þ c ð 0 Þ 4 1 2 0 1 2 (5.22) with the leading term 0 given by where c are some -independent constant spinors (which depend on w and k). Also note 0ðw1;w2;s; kÞ¼ 0ðkÞ: (5.28) 045016-15 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) 1 2. k 2 where kFð!; TÞ was the ‘‘generalized Fermi momentum’’ 1 introduced in (5.15). In (5.32) we have also used When k 2 , the inner region contribution (5.24)is no longer negligible for generic momentum. Closely ðk Þ¼ ðk Þ (5.33) related to this, the leading outer region contribution, which 0;finite F 0 F is given by (5.26), now becomes divergent at the lower as from (5.29) 0 is finite at kF. Note that expression (5.32) end (near r ). More explicitly, from Eqs. (5.17)–(5.20) < 1 applies to all kF including kF 2 . we find that as r ! r , the integrand of (5.26) behaves From (5.32), note that at the Fermi surface k ¼ kF, schematically as ( OðT0Þ < 3 ð0Þ 2 2 k 0 kF 2 ðaþ Þ ðr r Þ þ Oððr r Þ Þþ : (5.29) ð!; k Þ (5.34) F 3 2 3 OðT kF Þ The divergence is of course due to our artificial separation kF 2 of the whole integral into the IR and UV regions and there and the vertex develops singular temperature dependence should be a corresponding divergence in IR in the limit for 3 . However, at the generalized Fermi momentum kF 2 c ! 0 to cancel the one from (5.26). What the divergence kFð!; TÞ, the singular contribution is suppressed. This signals is that the leading contribution to the full effective structure will be important below in understanding the vertex now comes from the IR region, as the UV low temperature behavior of the DC and optical region integral is also dominated by the IR end. Thus for conductivities. a generic momentum k, the effective vertex has the Similarly the vertex (4.54) can be written for k kF & leading behavior OðTÞ as ð0Þ 2 1 2 k ð!; kÞ ðaþ Þ T þ : (5.30) ~ 2 ð!1;!2; ;kÞ¼Bð!1;!2; ;kFÞc1ðk kFð!1;TÞÞ 1 A slightly tricky case is ¼ , for which has a 1 2 k 2 0 ðk k ð! ;TÞÞT kF þ ðk Þþ logarithmic divergence and could lead to a log T contribu- F 2 0 F tion once the divergence is canceled. We have not checked (5.35) its existence carefully, as it will not affect the leading ~ behavior of the DC and optical conductivities (as will be where Bð!1;!2; ;kÞ is a smooth function of k, which 0 clear in the discussion of the next section). Thus in what scales with temperature as OðT Þ, and we have used (5.28). 1 0 To summarize the main results of this section: follows, it should be understood that for k ¼ , the OðT Þ 2 1 ð 0Þ behavior in (5.30) could be log T. (1) For k < 2 , the effective vertices are O T for all It can also be readily checked that for generic k, the momenta. For both the DC and optical conductiv- effective vertex has the same temperature scaling as . ities they are given by 0ðkÞ of (5.26), which is a ð0Þ smooth function of k. At a Fermi surface k ¼ kF, aþ ðkFÞ¼0 [14], for which 1 (2) For k 2 , the vertices develop singular tempera- the leading order term in (5.30) vanishes. Thus near a 1 2 Fermi surface, which is the main interest of this paper, ture dependence for generic momenta as T k . But we need also to examine subleading terms. Plugging (5.13) near the Fermi surface [more precisely at the gen- into the expression (4.56) for the vertex, we find that near eralized Fermi momentum kFð!; TÞ] the singular contribution is suppressed. kF the temperature dependence of (including both IR and 13 (3) For all values of , the vertices for the DC and UV contributions) can be written as kF optical conductivities are given by (5.32) and (5.35) 2 1 2 k ð!; kÞ¼Bð!; kÞðaþðk; !; TÞÞ T þ 0;finite þ OðTÞ respectively. (5.31) where Bðk; !Þ is a smooth function of k and nonvanishing VI. EVALUATION OF CONDUCTIVITIES near kF. At low temperatures it scales with temperature as With the behavior of the effective vertices in hand we ð 0Þ O T . 0;finite denotes the finite part of (5.26), which is can now finally turn to the main goal of the paper: the again ! and T-independent, and a smooth function of k & leading low temperature behavior of the DC and optical (also at kF). For k kF OðTÞ, using (5.14) we can conductivities. We will first present the leading tempera- further write (5.31)as ture scaling and then calculate the numerical prefactors in 2 2 1 2 k the last subsection. In the discussion below we will only ð!; kÞ¼Bð!; kFÞc1ðk kFð!; TÞÞ T F consider a real k. Depending on the values of q and m, þ ðkFÞþ (5.32) 0 there could be regions in momentum space where k is imaginary, referred to as oscillatory regions in [13,14]. We 13 consider the contribution from an oscillatory region in There is also a term proportional to aþðk; !; TÞ@! 1 2 k aþðk; !; TÞT F from the last term in (4.56), but its coeffi- Appendix F2. We will continue to follow the notations T 2 ! 1 cient, being proportional to I I is zero. introduced in (5.3) with below w ¼ T and fðwÞ¼ewþ1 . 045016-16 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) A. DC conductivity where we have used (5.32). Now the key is that since 2 2 Let us first consider the leading low temperature depen- T kF , by scaling y ! T kF y, the term proportional 1þ2 dence of the DC conductivity (4.50) which we copy here to A in the last parenthesis becomes proportional to T kF for convenience and can be ignored. Now the integral can be straightfor- wardly evaluated and we find that Z dw @fðwÞ DC ¼ Iðw; TÞ (6.1) C0 2 @w 2 k Iðw; TÞjFS ¼ T F (6.9) ! 2ImgðwÞ with w ¼ T and (dropping all the super- and subscripts) Z where C 1 ð Þ d 2 2 ð Þ 2ð Þ I w; T dkk B w; k; T w; k; T : (6.2) 0 2 d 2 2 2 0 C ¼ 2 Ch1kF 0ðkFÞ: (6.10) Note that as a function w, the Fermi function fðwÞ is Clearly (6.9) dominates over the contribution from regions independent of T; thus all the T dependence of DC is of momentum space away from a Fermi surface which contained in the momentum integral Iðw; TÞ. from (6.4) can at most be OðT0Þ.14 As discussed in [14] (and reviewed in Appendix D), for Plugging (6.9) into (6.1), we then find that for all kF the generic momentum, the spinor spectral function B has DC conductivity has the following leading low temperature leading low temperature dependence behavior: T2 k : (6.3) B 2 k DC ¼ T F (6.11) Using (6.3), (5.25), and (5.30), we find that for a generic momentum (up to possible logarithmic corrections) where is a numerical prefactor given by ( 0 Z 4 1 C dw @fðwÞ 1 T k k < ¼ : (6.12) 2 2 2 (6.4) 2 2 @w ImgðwÞ B 2 1 T k 2 We will discuss the numerical evaluation of in Sec. VI C. where the leading contribution of the first line (for < 1 ) @f k 2 Note that since both @w and ImgðwÞ are positive and comes from the UV part of the vertex, while for the second even, the integral in the above expression is manifestly line the leading contribution comes from the IR part of the positive. vertex. We emphasize that in the above derivation it is crucial Near a Fermi surface as reviewed at the end of that the same k kFð!; TÞ appears in both the spectral Appendix D function (6.5) and the effective vertex (5.32). As a result the leading contribution to the DC conductivity is dominated 2h1Im by the UV part of the effective vertex due to suppression at B ¼ 2 2 ; (6.5) ðk kFð!; TÞ Re Þ þðIm Þ 1 kFð!; TÞ, despite the fact that for k > 2 the vertex is generically dominated by the IR part. where h is a positive constant, k ð!; TÞ is given by (5.15) 1 F Finally note that in writing down (6.11) we have and assumed there is a single Fermi surface. In the presence 2 ¼ T kF gðwÞ: (6.6) of multiple Fermi surfaces, the contribution from each of them can be simply added together and the one with the gðwÞ is a T-independent scaling function (depending on largest kF dominates. kF= ) which can be obtained from the retarded function in AdS evaluated at k [see (D43)–(D47) for explicit 2 F B. Optical conductivity expressions]. Now let us consider the momentum integral (6.2) near a Let us now look at the optical conductivity, which from Fermi surface. For this purpose it is convenient to intro- (4.49) can be written as duce a new integration variable C Z d! d! ð Þ¼ 1 2 y ¼ k kFð!; TÞ (6.7) i 2 2 in terms of which (6.2) can be written to leading order as fð!1Þ fð!2Þ Ið!1;!2; ;TÞ (6.13) !1 !2 i Z 1 2 2 d 2 Im Iðw; TÞjFS ¼ 2Ch1kF dy 2 2 1 ðy Re Þ þðIm Þ where 2 2 1 2 k 2 ðBðkFÞc y T F þ 0ðkFÞÞ þ 1 14See Appendix F2for a discussion of the contribution from (6.8) oscillatory regions which is again at most of order OðT0Þ. 045016-17 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) Z 1 2 k d 2I comes from the region y OðT F Þ. One then finds from a Ið!1;!2; ;TÞ dkk ð!1;!2; ;kÞ (6.14) 2 0 simple scaling that the term proportional to 0ðkFÞ (i.e. the UV part of the vertex) is dominating. The corresponding with temperature scaling of (6.19) depends on the range of . 2 I kF ð!1;!2; ;kÞ¼ Bð!1;kÞ ð!1;!2; ;kÞ For OðT Þ, one has 2 k 2 k ð!2;!1; ;kÞ Bð!2;kÞ: (6.15) Ið!1;!2; ;TÞjFS T F ; OðT F Þ (6.21) Recall that the vertex satisfies (5.1) which implies that while for OðTÞ, one finds ( 2 2 k 1 ð Þ¼ ð Þ (6.16) T F k < Ið! ;! ; ;TÞj 0 F 2 : (6.22) 1 2 FS 2 2 2 1 T kF as one would expect since the system has time-reversal 0 kF 2 symmetry. To summarize, the contribution from near the Fermi surface is given by 1. Temperature scaling Z Z C 2ðk Þ d! d! All the temperature dependence of (6.13)isin ð Þ¼ 0 F dkkd 2 1 2 i 2 2 Ið!1;!2; ;TÞ which we examine first. As in (6.3) and (6.4) one finds that away from a Fermi surface I has the fð!1Þ fð!2Þ Bð!1;kÞ Bð!2;kÞþ following leading T dependence: !1 !2 i ( 4 k 1 (6.23) T k < Ið! ;! ; ;kÞ 2 (6.17) 1 2 2 1 which is of the form of that for a Fermi liquid in the T k 2 absence of vortex corrections. 1 where for k < 2 the leading contribution comes from the UV part of the vertex, while for the second line the leading 2. Contribution from Fermi surface contribution comes from the IR part of the vertex. Thus one could at most get Now let us look at the contribution from the Fermi surface in detail and work out the explicit frequency 0 Ið!1;!2; ;TÞ OðT Þ (6.18) dependence. As discussed above we only need include the UV part of the effective vertex, which gives from regions of momentum space away from a Fermi Z surface. IðFSÞð! ;! ; ;TÞ¼ 2ðk Þkd 2 dk ð! ;kÞ ð! ;kÞ: Near a Fermi surface has the low temperature 1 2 0 F F B 1 B 2 expansion (5.35) and B is given by (6.5) and (6.6). (6.24) Introducing y ¼ k k ð! ;TÞ, then the integral has the F 1 The latter integral can be done straightforwardly (see following structure Appendix G2for details) and gives Z Im 1 ðFSÞð Þ¼ 2ð Þ d 2 ð Þ Ið!1;!2; ;TÞjFS dy 2 2 I !1;!2; ;T 2 h1 0 kF kF B !2;K2 ðy Re 1Þ þðIm 1Þ 2 d 2 Im ¼ 2 h1 0ðkFÞkF Bð!1;K1Þ 2 2 2 (6.25) ðy þ Re 2Þ þðIm 2Þ 1 2 k ðb1yðy þ ÞT F þ 0ðkFÞÞ where K2 kFð!1;TÞþ ð!1Þ and K1 kFð!2;TÞþ ð! Þ. We now plug (6.25) into (6.13) and evaluate one 1 2 k 2 ðb2yðy þ ÞT F þ 0ðkFÞÞ of the frequency integral as follows. Split the integrand into (6.19) two terms; in the one with the fð!2Þ, we use the second line of (6.25) and do the !1 integral, which can be written as where Z d!1 Bð!1;K1Þ R 1 ¼ G ð!2 þ ;K1Þ ð! ! Þ; (6.20) 2 !1 !2 i v 1 2 F h ¼ 1 ð! Þ, and b are some y-independent functions 1;2 1;2 1;2 þ ð!2Þ ð!2 þ Þ 0 vF of !1, !2, which scale with temperature as OðT Þ.For y OðT0Þ (i.e. away from the Fermi surface) the integrand (6.26) scales as (6.17). Near the Fermi surface, i.e. in the range where in the first line we used the spectral decomposition y & OðTÞ, as in the analysis of (6.8), due to the fact that of the boundary fermionic retarded function GR and the 2 k 1;2 T F , the dominant contribution in the y-integral second line used (D39). Similarly for the term with fð!1Þ 045016-18 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) we can use the first line of (6.25) and do the !2 integral, behavior (6.29) and (6.31) are indicative of a system which gives without a scale and with no quasiparticles. > 1 Z (2) kF 2 : In this case there are two regimes: d!2 Bð!2;K2Þ h1 2 kF 1 ¼ : (a) with u ¼ 2 ¼ fixed and s ¼ uT ! 0, T kF 2 !1 !2 i þ ð!1Þ ð!1 Þ vF we find (6.28) becomes (6.27) 2 k ð Þ¼T F F2ðuÞ (6.33) Combining them together we thus find that with Z C0 d! fð!Þ fð! þ Þ ð Þ¼ 0 Z ð Þ (6.28) C @f w 1 i 2 þ ð!Þ ð! þ Þ F2ðuÞ¼ dw u : vF 2 i @w þ 2iImgðwÞ vF where C0 was introduced before in (6.10). It is now mani- (6.34) fest from the above equation that in the ! 0, we recover @f (6.11). This confirms the claim below (4.43) that S in (4.42) Since @w is peaked around w ¼ 0, we can ap- vanishes at leading order at low temperatures. proximate the above expression by setting gðwÞ We now work out the qualitative dependence of (6.28) to its value at w ¼ 0, leading to a Drude form which has a rich structure depending on the value of k . 0 2 k 2 F iC T F 1 !p As stated earlier we work in the low temperature limit ð Þ ¼ 2 u þ 2iImgð0Þ 1 i vF T ! 0 with s ¼ T fixed. < 1 (6.35) (1) kF 2 : In this case given (6.6), to leading order we can ignore the term proportional to in the down- with stairs of the integrand of (6.28). Then ð Þ can be 0 written in a scaling form 2 vFC 1 2 ! ; 2Imgð0Þv T kF : p 2 F 2 k ð Þ¼T F F1ð =TÞ (6.29) (6.36) with F1ðsÞ a universal scaling function given by This behavior is consistent with charge transport Z from quasiparticles with a transport scattering dw fðw þ sÞ fðwÞ 0 2 k F1ðsÞ¼C rate given by / T F . Furthermore we could 2 is interpret C0 as proportional to the quasiparticle 1 density. Indeed from (6.10) it is proportional to (6.30) gðw þ sÞ g ðwÞ the area of the Fermi surface. Note that 0ðkFÞ in C0 can be interpreted as the effective charge with g given by (D44). In the s ! 0 limit we recover of the quasiparticles. the DC conductivity (6.11). In the limit s !1, (b) For s ¼ ¼ fixed, the two terms in the which corresponds to the regime T , T downstairs of the integrand of (6.28) are much using (D46) we find that15 smaller than the term, and we can then 00 2 ð Þ¼C ð i Þ kF (6.31) expand in power series of , with the lowest two terms given by where C00 is a real constant given by 2 i!p 2 1 C0 ð Þ¼ ð1 þ T kF kðsÞþ Þ (6.37) C00 ¼ 4 ih2 Z with 1 dy Z 2 2 vF 1 cðk Þð1 þ yÞ kF c ðk Þð1 yÞ kF kðsÞ¼ dwðfðwÞ fðw þ sÞÞ F F s2 (6.32) ðg ðwÞ gðw þ sÞÞ: (6.38) with cðkFÞ given by (D45). In obtaining (6.31)we s In the large s limit using (D46) we find have made a change of variable w ¼ 2 ðy 1Þ in (6.30) and taken the large s limit. Note that 2 1 4vFh2ImcðkFÞ kðsÞ! að 2isÞ kF ;a¼ the falloff in (6.31) is much slower than the 2 þ1 Lorentzian form familiar from Drude theory. The kF (6.39) 15Note that in our setup is a UV cutoff scale; thus we always in which case ð Þ (i.e. for T ) can assume . be written as 045016-19 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) Z 2 C0 d! fð!Þ fð!þ Þ i!p 2 2 2 ð Þ¼ 2a! ð 2i Þ kF þ : ð Þ¼ (6.44) p i 2 þ ð!Þ ð!þ Þ vF (6.40) 0 2 d 2 2 where C ¼ 2 Ch1kF 0ðkFÞ. The formula implicitly The leading 1= piece in (6.37) gives rise to a depends on the bulk fermion mass m and charge q.We term proportional to ð Þ with a weight are using this version since it only contains one ! integral consistent with (6.35). The subleading scaling and it is easier to evaluate than (4.39). In order to compute behavior may be interpreted as contribution ð Þ for a fixed m and q, we need the following quantities: from the leading irrelevant operator. (i) Fermi momentum: kFðm; qÞ. Note that in both regimes discussed above the At T ¼ 0, ReG 1ðk; ! ¼ 0Þ changes its sign at the temperature scalings are consistent with those Fermi momentum. We determine the location of this identified earlier in (6.21) and (6.22). For real sign change using the Newton method (up to 40 part of ð Þ we have / !1 !2 / as iterations). The algorithm needs an initial k value constrained by the delta function resulting where the search starts. This initial value was set by from the imaginary part of 1 in !1 !2 i empirical linear fits on kFðm; qÞ. When there were (6.23), while for the imaginary part of ð Þ multiple Fermi surfaces, we picked the primary the dominant term (i.e. the term proportional Fermi surface (the one with the largest kF). i 1 to ) comes from the region !1 !2 Computing G involves solving the Dirac 2 OðT kF Þ. equation in the bulk. We used Mathematica’s (3) ¼ 1 : For the marginal Fermi liquid, the term NDSolve to solve the differential equation kF 2 in the downstairs of (6.28) is of the same order as , using AccuracyGoal=PrecisionGoal¼12...22, and and we have WorkingPrecision ¼ 70. Typical IR and UV cutoffs are 10 12 ...10 20 and 10 25 ...10 40, respectively. 1 T The resulting k values are typically accurate to the ð Þ¼T F3 ; log (6.41) F T 10th digit. (ii) Numerator of the Green’s function: h ðm; qÞ. where F can be written as 1 2 The numerator of the fermionic Green’s function is Z 0 dw fðw þ sÞ fðwÞ determined by fitting a parabola on six data points F ¼ C 1 3 2 is of G ðk; ! ¼ 0Þ near the Fermi surface (i.e. ¼ 5 þ 5 1 k kF 10 ...kF 10 ), and then taking the (6.42) derivative of the parabola at k ¼ k . The computa- s þ gðw þ sÞ g ðwÞ F vF tion of redundant data points makes the resulting h1 with gðwÞ now given by (D47). Due to time reversal value somewhat more accurate, but its main func- tion is to monitor the stability of the numerics: symmetry the real part 1ð Þ of ð Þ is an even function in and thus for =T < 1, one can again whenever the six points are not forming an approxi- k approximate ð Þ by a Drude form with the trans- mately straight line, we know that the F finding port scattering time / 1 .For T, using (D48) algorithm has failed. In this case, we need to go T back and ‘‘manually’’ obtain the value of the Fermi we find that momentum. 1 C0 1 1 1þi 1 (iii) Self-energies: ð!; m; q; TÞ. ð Þ¼ þ þ ic 2 v c Let ~ð!Þ denote the self-energy at the Fermi sur- 2 1 log T ðlog T Þ 2 F 1 face with the linear ! term included. Then, þ (6.43) vF h which is analogous to (6.31), but with logarithmic G ð!; kÞ¼ 1 : (6.45) R ~ modifications. Recall that there are no logarithmic ðk kFÞ ð!Þ corrections for the DC conductivity (6.11). ~ Since we already know h1 and kF, we determine by computing the fermionic Green’s function C. Numerical coefficients ~ at k ¼ kF. Computing both ð!; m; q; TÞ and In this section, we discuss the numerical computation of ~ð! þ ; m; q; TÞ then gives the denominator þ d ¼ the conductivities in 2 1 boundary dimension ( 3). of (6.44). (iv) Effective vertex: ð!1;!2; ;k; m; qÞ. 1. Optical conductivity The numerical code computed the frequency- i The optical conductivity is given by (6.28) that we copy dependent ð!1;!2; ;k; m; qÞ [see (4.26) for here for convenience: an explicit formula] instead of the simpler 0ðkFÞ. 045016-20 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) 500 2.0 400 1.5 300 1.0 200 Re I, Im I Re I, Im I 0.5 100 0 0.0 0.0010 0.0005 0.0000 0.0005 0.0010 0.012 0.010 0.008 0.006 0.004 0.002 0.000 R 1 FIG. 6 (color online). Typical functions whose integral gives the conductivity: ð Þ¼ 1 Ið!Þd!. The two figures correspond to T and T. The real and imaginary parts are indicated by blue (dark) and orange (light) curves, respectively. We determine ð!1;!2; ;kÞ by first numerically computing KAðr; Þ, which is the bulk-to-boundary 0.008 gauge field propagator with ingoing boundary con- 1 ditions at the horizon. (Note that at ¼ 0 this may F 0.006 be done analytically.) We then compute the spinor ,Im 1 0.004 propagator with normalizable UV boundary condi- F tions at both !1 and !2 and also compute the Re 0.002 integral using a single NDSolve call. The integra- i 2 tion proceeds towards the horizon where it oscil- 0.000 lates somewhat before converging. 6 4 2 0 2 4 6 8 By using the above quantities, we compute the conduc- log T tivity at a fixed and T by performing the integral over ! in (6.44). The Fermi functions suppress the integral expo- 0.015 nentially outside a certain window set by the parameters; 2 F see Fig. 6. The size of this window can be determined and 0 is used to automatically set the integration limits. The 0.010 1 i ,Im 2 integrand is computed at 15...30 points. We used F Mathematica’s parallel computing capabilities in order to Re 0.005 compute three data points at the same time. i 1 Figure 7 show the scaling functions F1;2 defined in the 0.000 1 1 6 4 2 0 2 4 previous section for k < and k > . The behavior in F 2 F 2 log T 2 limiting cases agrees with the analysis above. FIG. 7 (color online). These figures show the scaling functions 2. DC conductivity for the optical conductivity. Top: Real and imaginary parts of the The DC conductivity can be computed using (6.11): scaling function F1 for m ¼ 0, q ¼ 1, where the IR fermion exponent is kF 0:24. Bottom: Real and imaginary parts of the Z k C @fð!Þ h2 2ð!; TÞ scaling function F2 for m ¼ 0, q ¼ 2, where the IR fermion ðTÞ¼ F d! T 1 : (6.46) exponent is 0:73 and hence we are in the regime with a DC ð Þ kF 2 @! Im !; kF;T stable quasiparticle. As indicated in the figure, both real and imaginary parts resemble the Drude behavior. The computation of kF, h1 and was detailed in the previous subsection. The ! derivative inside ð!; TÞ is computed by taking the difference of the wave functions The numerically unstable areas (with error larger than 3%) ¼ 6 8 with ! 10 ...10 . The numerical AC conductivity are colored gray in the figure. For kF > 1:1 (in the G2 in the zero frequency limit matched the output of the DC component) the numerical inaccuracies became too large. conductivity code. For q<0:3, the automated kF finding algorithm typically For a given pair ðm; qÞ, the DC conductivity is computed failed and we had to determine kF manually. 4 8 at different temperatures between T ¼ 10 ...10 . Note the deep blue line in the G1 spinor component. At Then, the temperature-independent coefficient is com- these points, the effective vertex 0ðkFÞ changes sign and 2 k puted by a fit using ðm; qÞ¼ DCðm; q; TÞT F . In the therefore the leading contribution to the DC conductivity m-q space, the resolution was 45 45 with a computation vanishes (so does the leading contribution to the optical time of approximately 22 hours. The results are seen conductivity since it is also proportional to 0). Since in Fig. 8. The plot shows log ðm; qÞ using a color code. the DC effective vertex is real, this happens along a 045016-21 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) FIG. 8 (color online). The plots show the coefficient log ðm; qÞ for the primary Fermi surface for both spinor components. ð Þ In the white regions there is no Fermi surface. Black lines indicate half-integer kF m; q . In the two gray regions in G2 the numerical computations were unreliable (with error greater than 3%). codimension 1 line in the m-q plane. This ‘‘bad metal’’ line ¼ crosses the kF 1=2 line at around m 0:18 and q 3:4 (not in the figure). VII. DISCUSSION AND CONCLUSIONS FIG. 9 (color online). The system can be described by a low Despite the complexity of the intermediate steps, the energy effective action where fermionic excitations around a result that we find for the DC conductivity is very simple. free fermion Fermi surface hybridize with those in a strongly One can package all radial integrals into effective vertices coupled sector (labeled as SLQL in the figure) described on the in a way that makes it manifest that the actual conductivity gravity side by the AdS2 region [14,71,72] (see [18] for a more extensive review). is completely determined by the lifetime of the one-particle excitations, as is clear from the formula (6.23). We should stress that from a field-theoretical point of effective action [14,71,72] in which fermionic excitations view this conclusion is not a priori obvious, as the single- around a free fermion Fermi surface hybridize with particle lifetime measures the time needed for the particle those of a strongly coupled sector, which can be considered to decay, whereas the conductivity is sensitive to the way in as the field theory dual of the AdS2 region and was referred which it decays. For example, if the Fermi quasiparticles to as a semilocal quantum liquid (SLQL) in [73]. See are coupled to a gapless boson (as is the case in many Fig. 9. The SLQL provides a set of fermionic gapless field-theoretical constructions of non-Fermi liquids; see modes to which the excitations around the Fermi surface e.g. [1,54–70] and references therein), small-momentum can decay. In our bulk treatment this process has a nice scattering is strongly preferred because of the larger phase geometric interpretation in terms of the fermion falling into space available to the gapless boson at smaller momenta. the black hole, as in Fig. 2. The crucial point is that because However, this small-momentum scattering does not of the semilocal nature of the SLQL—as exhibited by the degrade the current and so contributes differently to the self-energy (6.6)—there are gapless fermionic modes for conductivity than it does to the single-particle lifetime, any momentum.16 Thus the phase space for scattering is not meaning that the resistivity grows with temperature with sensitive to the momentum transfer, and the conductivity is a higher power than the single-particle scattering rate [64]. determined by the one-particle lifetime. Such systems are therefore better metals than one would We find that the conductivity at the marginal Fermi liquid point ¼ 1 (when the single-particle spectral have guessed from the single-particle lifetime. kF 2 In our calculation, the current dissipation is more effi- cient. To understand why, note that in our gravity treatment 16This is similar to that postulated for the bosonic fluctuation the role played by the gapless boson in the above example spectrum in the MFL description of the cuprates [4]. But an is instead filled by the AdS2 region. From a field theoretical important distinction is that here the gapless modes are point of view, our system can be described by a low energy fermionic. 045016-22 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) function takes the MFL form) is consistent with a linear ji ¼ ui ; i ¼ uið þ PÞ: (A2) resistivity, just as is observed in the strange metals. The correlation between the single-particle spectral function This gives and the collective behavior and transport properties is a ji ¼ i; (A3) strong and robust prediction of our framework. While it is þ P fascinating that this set of results is self-consistent, we do stress that the marginal ¼ 1 point is not special from which is effectively a constitutive relation. In a nonrelativ- kF 2 istic system, the enthalpy þ P reduces to the mass of the our gravity treatment, and more work needs to be done to particles. Combining this with conservation of momentum understand if there is a way to single it out in holography.17 (Newton’s law) i i ACKNOWLEDGMENTS @t ¼ E (A4) We thank M. Barkeshli, E. Fradkin, T. Grover, and Fourier transforming gives S. Hartnoll, G. Kotliar, P. Lee, J. Maldacena, W. Metzner, S. Sachdev, T. Senthil, and B. Swingle for valuable dis- i 2 jið Þ¼ Eið Þ (A5) cussions and encouragement. During its long gestation, ð þ PÞ this work was supported in part by funds provided by the U.S. Department of Energy (DOE) under cooperative and hence research agreements DE-FG0205ER41360, DE-FG02- 2 92ER40697, DE-FG0205ER41360, and DE-SC0009919, Re ð Þ¼ ð Þ (A6) þ P and the OJI program, in part by the Alfred P. Sloan Foundation, and in part by the National Science plus, in general, dissipative contributions. Note that in ~ Foundation under Grants No. NSF PHY05-51164 and systems where the relation J ¼ þP ~ is an operator equa- No. PHY11-25915. All the authors acknowledge the hos- tion, momentum conservation implies that there are no pitality of the KITP, Santa Barbara, at various times, in dissipative contributions, and the conductivity is exactly particular the miniprogram on Quantum Criticality and the given by (A6). This is indeed consistent with the leading AdS/CFT Correspondence in July, 2009, and the program term we obtained in (2.44). Holographic Duality and Condensed Matter Physics in fall The Fermi surface contribution which is the main result 2011. T.F was also supported by NSF Grants No. PHY of the paper does not contain a delta function in , as the 0969448, No. PHY05-51164, the UCSB physics depart- Fermi surface current can dissipate via interactions with ment and the Stanford Institute for Theoretical Physics. the OðN2Þ bath. Although the total momentum (of the D. V. was also supported by the Simons Institute for Fermi surface plus bath) is conserved, the time it takes Geometry and Physics and by a COFUND Marie Curie the bath to return momentum given to it by the Fermi Fellowship, and acknowledges the Galileo Galilei Institute surface degrees of freedom is parametrically large in N, for Theoretical Physics in Florence for hospitality during as in probe-brane conductivity calculations [75]. The DC part of this work. conductivity we obtain is averaged over a long time that is of OðN0Þ. APPENDIX A: RESISTIVITY IN CLEAN SYSTEMS Our discussion here is somewhat heuristic, but a more careful hydrodynamic analysis that also takes into account In a translation-invariant and boost-invariant system at the leading frequency dependence was performed e.g. in finite charge density (without disorder or any other mecha- [76], where it was explicitly shown that the presence of nism by which the charge carriers can give away their impurities broadens this delta function into a Drude peak. momentum), the DC resistivity is zero. An applied electric field will accelerate the charges. This statement is well known but we feel that some clarification will be useful. It APPENDIX B: MIXING BETWEEN GRAVITON can be understood as follows. Start with a uniform charge AND VECTOR FIELD density at rest and in equilibrium, in a frame where In this section we construct the tree-level equations of jt 0;Ttt ¼ 0;Tii ¼ P; motion for the coupled vector-graviton fluctuations about (A1) the charged black brane background. The action can be i i ti j ¼ 0; T ¼ 0; written in the usual form, 18 Z is the energy density and P is the pressure. Boost by a 1 pffiffiffiffiffiffiffi R2 i dþ1 R velocity u to a frame where S ¼ 2 d x g 2 2 F F ; (B1) 2 gF 17See [50,74] for recent work in this direction. with background metric given by 18 i We perform a small boost u c, so we can use the Galilean 2 2 2 2 transformation, even if the system is relativistic. ds ¼ gttdt þ grrdr þ giidxi (B2) 045016-23 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) ffiffiffiffiffiffiffi y p rr yy 0 y and a nonzero background profile A0ðrÞ. It is convenient to ¼ gg g ay Qht (B13) work with the radial background electric field Er ¼ @rA0, pffiffiffiffiffiffiffi which satisfies the equation of motion y yy tt 2 ¼ ffiffiffiffiffiffiffi @r gg g ! ay 0 (B14) p tt rr @rðEr gg g Þ¼0: (B3) pffiffiffiffiffiffiffi 2 2C Qa ¼ gg grrgtt@ hy: (B15) We will denote the r-independent quantity 1 y yy r t Taking a derivative of the first equation with respect to r ffiffiffiffiffiffiffi 2 p tt rr gF Q E gg g ¼ 2 (B4) one can derive an equation for ay alone r 2R2 pffiffiffiffiffiffiffi grrgtt where we have used (2.7). @ ð ggrrgyya0 Þþ 2 2C Q2 pffiffiffiffiffiffiffi r y 1 gg Now consider small fluctuations yy pffiffiffiffiffiffiffi ggyygtt!2 a ¼ 0: (B16) AM ! A0 M0 þ aM;gMN ! gMN þ hMN: (B5) y We seek to determine the equations of motion for these Note now that using fluctuations; we first use our gauge freedom to set 2 1 2 gtt ¼ fr ;grr ¼ ;gii ¼ r (B17) hrM ¼ ar ¼ 0: (B6) r2f At quadratic level in fluctuations the Maxwell action can we find that (B16) becomes now be written as !2rd 5 Z d 1 0 CQ d 1 1pffiffiffiffiffiffiffi @rðr fayÞþ r ay ¼ 0: (B18) S ¼ C ddþ1x gf fMN f EM 1 4 MN In the last expression we introduced the constant 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi E ð ggttgrr þ ggtrgtrÞ E r ð2Þ r C 2 2C Q ¼ 2 2 : (B19) 2 1 pffiffiffiffiffiffiffi pffiffi tt rr Q i i 4 3 r 2 Erðg g gÞð1Þf0r ðhtfir þhrftiÞ (B7) In d ¼ 3, Cj ¼ ¼ ð Þ . d 3 gF R Equation (B15) implies a corresponding relation with between the bulk-to-boundary propagators Ka, Kh of 2R2 the metric and gauge field, which is important for our C1 ¼ 2 2 ;fMN ¼ @MaN @NaM: (B8) calculation: gF ffiffiffiffiffiffiffi p rr tt CKa ¼ ggyyg g @rKh: (B20) The canonical momentum for a is then given by ffiffiffiffiffiffiffi i 1 S p rr ii i ¼ ¼ gg g fri Qht (B9) APPENDIX C: SPINOR BULK-TO-BULK C @ a 1 r i PROPAGATOR ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi t p rr rr p rr rr ¼ gg g f0r þð gg g Þð1ÞEr: (B10) In this appendix we derive the spinor bulk-to-bulk propagator. For simplicity of exposition we focus on the The equation for ar is essentially the Gauss law constraint case when the dimension d of the boundary theory is odd; in the bulk and leads to the conservation of this canonical the correspondence between bulk and boundary spinors is momentum, different when d is even, and though a parallel treatment N @ ¼ 0: (B11) can be done we shall not perform it here. We denote by the dimension of the bulk spinor representation; in the case dþ1 Finally, the dynamical equations for ai are that d is odd we have N ¼ 2 2 . Our treatment will pffiffiffiffiffiffiffi essentially apply to any asymptotically AdS spacetime @ þ g@ f ¼ 0: (B12) r with planar slicing and a horizon in the interior; the crite- We turn now to the gravitational fluctuations. At this rion of asymptotically AdS is important only in the precise point it is helpful to specialize to the zero momentum limit; choice of UV boundary conditions and can be easily modi- i.e. all fluctuations depend only on t and r. Now all spatial fied if necessary. directions are the same, and so we pick one direction y (calling it y) and focus only on h , with ¼ðt; rÞ, and 1. Spinor equations where the indices are raised by the background metric. y We begin with the bulk spinor action: We will then find a set of coupled equations for ay and ht Z y pffiffiffiffiffiffiffi (and a and hr, which will be set to zero in the end). The dþ1 M r S ¼ i d x gc ð DM mÞc (C1) relevant equations then become 045016-24 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) where c c y t. From here we can derive the usual Dirac In the above equations we have suppressed the bulk equation, spinor indices which we will do throughout the paper. ~ M The spectral function ðr1;r2; !; kÞ is defined by ð DM mÞc ¼ 0; (C2) ðr; r0; !; k~Þ¼ iðD ðr; r0; !; k~Þ D ðr; r0; !; k~ÞÞ: where the derivative DM is understood to include both the R A spin connection and couplings to background gauge fields (C13) 1 The Euclidean two-point function is related to D by the D ¼ @ þ ! ab iqA : (C3) R M M 4 abM M standard analytic continuation M; N 0 ~ 0 ~ The abstract spacetime indices are ... and the DEðr; r ; i!m; kÞ¼DRðr; r ; ! ¼ i!m; kÞ (C14) abstract tangent space indices are a; b; .... The index with an underline denotes that in tangent space. Thus a and satisfies the spectral decomposition denotes gamma matrices in the tangent frame and M those Z ð 0 ~Þ 0 ~ d! r; r ; !; k in curved coordinates. Note that DEðr; r ; i!m; kÞ¼ : (C15) 2 i!m ! M a M ¼ ea : (C4) We define the corresponding boundary retarded and The nonzero spin connections for (2.2) and (2.3) are advanced Green’s functions as follows: given by GR ðt; x~Þ¼i ðtÞhfO ðt; x~Þ; Oy ð0Þgi (C16) 0 pffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffi 1 gtt rr t 1 gii rr i !t r ¼ g e ;!i r ¼ g e ; (C5) 2 gtt 2 gii A O Oy G ðt; x~Þ¼ i ð tÞhf ðt; x~Þ; ð0Þgi (C17) with where O is the boundary operator dual to the bulk field c 1 1 t 2 i 2 i and , are boundary spinor indices. The boundary e ¼ gttdt; e ¼ giidx : (C6) spectral function B is defined by From the above one finds that GRð!; k~Þ GAð!; k~Þ¼i ð!; k~Þ; (C18) 1 r B ! M ab ¼ @ log ð ggrrÞ UðrÞ r: (C7) 4 abM 4 r and is Hermitian: y In momentum space the Dirac equation (C2) can then be B ¼ B: (C19) written explicitly as From (C16) the linear response relation is t i r c ~ ½ ið!þqAtÞ þiki þ ð@r þUÞ m ð!;k;rÞ¼0 t hOðkÞi ¼ GRðkÞ ðkÞ (C20) (C8) where denotes a source, t is the boundary gamma whose conjugate can be written as matrix, and we have suppressed the spinor indices. t i Q r Our convention for bulk Gamma matrices is that c ½ ið! þ qAtÞ þ iki ð@r þ UÞ m ¼0: (C9) ð aÞy ¼ t a t; ð tÞ2 ¼ 1 (C21) c c Applying (C8) and (C9)to and in (4.31) respec- and for boundary ones tively and using (C7), one can readily derive (4.31). ð tÞ2 ¼ 1; ð tÞy ¼ t: (C22) 2. Green’s functions As mentioned earlier we will focus on odd d, for which 5 We define the retarded and advanced bulk-to-bulk case, there is also a in the bulk which anticommutes a propagator as with all the ’s and satisfies 0 0 ð 5Þy ¼ 5; ð 5Þ2 ¼ 1: (C23) DRðt; x~; r; r Þ¼i ðtÞhfc ðt; x;~ rÞ; c ð0;rÞgi (C10) 0 c c 0 DAðt; x~; r; r Þ¼ i ð tÞhf ðt; x;~ rÞ; ð0;rÞgi (C11) 3. Bulk solutions whose Fourier transform along boundary directions satisfy We begin by recalling how to obtain the boundary the equation retarded Green’s function and some properties of the solu- i tions to the Dirac equation (C2) (see also [28]). ð MD mÞD ðr; r0; !; k~Þ¼ pffiffiffiffiffiffiffi ðr r0Þ: Near the horizon r , it is convenient to choose the M R;A g 0 infalling and outgoing solutions as the basis of wave (C12) functions 045016-25 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) c in;out ~ in;out i! ðrÞ follows that the boundary theory spinor retarded Green’s a ðr; !; kÞ! a e ;r! r0 (C24) R pffiffiffiffiffiffiffiffiffiffiffiffi function GR can be written as where ðrÞ dr g gtt, and are constant basis t 1 rr a ðGR Þ ¼ðBA Þ (C36) spinors, which satisfy the constraint with t the boundary theory gamma matrix. This is the t r in;out ð1 Þ a ¼ 0: (C25) covariant generalization of expressions given previously The index a labels different independent solutions. From for the boundary fermion Green’s function [28], and will be N useful in what follows. One can find the advanced bound- the above equation clearly we have a 2f1... g. 2 ary theory correlator by using outgoing solutions and their Equation (C25) also implies that for any a, b corresponding outgoing expansion coefficient matrices B, c in r c out A in (C36). a b ¼ 0: (C26) We now compute some Wronskians that we will need We will normalize later. Note first that by using the Dirac equation (C2)we can show that for any two radial solutions c ðrÞ, c ðrÞ iny in ¼ ; outy out ¼ : (C27) 1 2 a b ab a b ab evaluated at the same frequency and momentum, the c c Near the boundary r !1it is convenient to consider Wronskian W½ 1; 2 defined as purely normalizable c and purely non-normalizable Y pffiffiffiffiffiffiffiffiffiffiffiffiffiffi W½c ; c ggrr c ð!; kÞ r c ð!; kÞ solutions defined respectively by 1 2 1 2 (C37) is a radial invariant, i.e. @ W ¼ 0. Using (C28) and (C29) c ðr !1Þ! r mR d=2 (C28) r at r ¼1one then finds that þ þmR d=2 W½c ; Y ¼ r þ ¼ þ ¼ i t Y ðr !1Þ! r (C29) ¼ W½Y ; c (C38) where are constant spinors which satisfy r ð1 Þ ¼ 0: (C30) where we have used (C34). It can also be readily checked that Again index labels different solutions and runs from 1 c c Y Y N r W½ ; ¼W½ ; ¼0 (C39) to 2 (as the two different eigenspaces of span the full spinor space). Since the normalizable and non- and normalizable solutions correspond to boundary operator W½c in; c out ¼0; and source respectively, can be interpreted as the bound- (C40) ary theory spinor index. We choose the normalization c in c in c out c out W½ a ; b ¼ ab ¼ W½ a ; b : y ¼ (C31) Also note that c c in t Y c in t and have the following completeness relation: W½ ; a ¼ið AÞ a;W½ ; a ¼ ið BÞ a: X 1 y ¼ ð1 rÞ: (C32) (C41) 2 We can also expand the outgoing solutions as It is also convenient to choose c out ¼ Y ~ þ c ~ þ 5 a A a B a; (C42) ¼ (C33) where 5 was introduced earlier around (C23). The bound- with ary gamma matrices can then be defined as out t ~ W½c ; c a ¼ið AÞ a; (C43) ¼ ið Þy þ: (C34) out t W½Y ; c a ¼ ið B~Þ a: Now expand the infalling solutions in terms of c and Using the above Wronskians we can also write the Y c in Y c c ¼ ic inðAy tÞ ic outðA~y tÞ : (C44) a ¼ A a þ B a (C35) a a a a N N where A and B are both 2 2 matrices that connect the infalling and boundary solutions. Identifying A with the 4. Constructing the propagator source (C20), with t defined as in (C34), one can then We are now ready to construct the bulk-to-bulk retarded 19 hOi check that B can be identified precisely with . It then propagator DR which satisfies Eq. (C12) together with the 0 boundary conditions that as either argument r or r ! r0 19As discussed e.g. in [28], hOi should be identified with the the propagator should behave like an infalling wave in boundary value of the canonical momentum conjugate to c . (C24), and similarly as r or r0 !1the propagator should 045016-26 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) be normalizable as in (C28). Note that the Dirac operator in (C37); the Wronskian of Y with itself vanishes as in (C39), (C12) above acts only on the left index of the propagator and we find then for the left-hand side (which is a matrix in spinor space) and on the argument r; Y c t Y t 2 Y Y if we can demonstrate that the propagator indeed satisfies iW½ ; ¼ ð Þ ¼ ; (C51) this equation then it will also satisfy the corresponding where in the first equality we have used (C38). This is then equation with the differential operator acting from the right consistent with (C50). Similarly contracting (C50) to the and as a function of r0, by the equality of left and right left with c we find inverses. Thus we will only explicitly show that the opera- c Y t c c tor satisfies the equation in r. For the advanced propagator iW½ ; ð Þ ¼ ; (C52) D , the only difference is that the propagator should A c Y behave like an outgoing wave at the horizon. which is again satisfied. Note that since and With the benefit of hindsight, we now simply write down altogether form a complete basis, we have now verified the answer for the bulk-to-bulk retarded and advanced the full matrix equation (C50), and thus the propagator propagator proposed in (C45) is indeed correct. Now given (C45), taking the difference between DR and 0 c R;A c 0 DR;Aðr; r ; !; kÞ¼ ðrÞG ð!; kÞ ðr Þ D , from (C13) and (C18) we thus find that 8 A < Y ðrÞ t c ðr0Þ r 045016-27 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) ! ! in 1 i 0 in rr i!tþikix 1 þmR c ¼ð gg Þ 4e ; ðr !1Þ! r : (D14) 1 0 1 ! (D5) 0 in rr 1 i!tþik xi c ¼ð gg Þ 4e i ; Let us now briefly summarize the low temperature and 2 in c R 2 frequency behavior of and G [14] which are needed for understanding the scaling behavior of the effective in and can in turn be expanded near the boundary as vertex and conductivities. The regime we are interested ! ! 0 1 in is in r!1 mR mR a Aar þ Bar a ¼ 1; 2: (D6) ! 1 0 T ! 0; with w ¼ ¼ fixed: (D15) T We choose the constant spinors in (C28) and (C29)to The discussion proceeds by dividing the radial direction satisfy (C33) 0 1 0 1 into inner and outer regions, which is rather similar to that 1 0 of the vector field in Sec. II B. For definiteness below we B C B C will consider ¼ 1 in (D4) and drop the subscript 1. B 0 C B 0 C B C B C 1 ¼ B C; 2 ¼ B C; @ 0 A @ 1 A 1. Boundary retarded function 0 0 0 1 0 1 (D7) To leading order in T in the limit of (D15), the Dirac 0 0 B C B C equation (D4) in the inner region reduces to that in the B 0 C B 1 C near-horizon metric (2.18) with w as the frequency con- þ B C þ B C 1 ¼ B C; 2 ¼ B C jugate to . In particular, the spinor operator develops an IR @ 0 A @ 0 A scaling dimension given by 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 2 k R and the corresponding boundary Gamma matrices (C34) k mkR2 edq i ;mk m þ 2 : r are given by (D16) t ¼ i 2; x ¼ 1; y ¼ 3: (D8) Near the boundary of the inner region (i.e. ! 0), the The matrices A and B introduced in (C35) are then given by solutions to (D4) behave as k and we can choose the ! ! basis of solutions specified by their behavior near ! 0 0 A B 0 (which also fixes their normalization) ¼ 2 ¼ 1 A ;B (D9) A1 0 0 B2 2 TR k 2 k I ! v ¼ v ; ! 0: (D17) and from (C36) the boundary retarded function is diagonal r r with components given by where v are some constant spinors (independent of R B and !). The retarded solution for the inner region can be G ð!; kÞ¼ ; ¼ 1; 2: (D10) A written as [14] ðretÞ þ G The set of normalizable and non-normalizable I ð ; wÞ¼ I þ kðwÞ I : (D18) solutions introduced in (C28) and (C29) can be written G more explicitly as where kðwÞ is the retarded function for the spinor in the ! ! AdS2 region [14] and will be reviewed at the end of this 20 1 1 1 0 section. c rr 4 c rr 4 1 ¼ð gg Þ ; 2 ¼ð gg Þ (D11) In the outer region we can expand the solutions to (D4) 0 2 in terms of analytic series in ! and T. In particular, the and zeroth order equation is obtained by setting ! ¼ 0 and ! ! T ¼ 0 (i.e. the background metric becomes that of the rr 1 0 rr 1 1 extremal black hole). Examining the behavior the resulting Y ¼ð gg Þ 4 ; Y ¼ ð gg Þ 4 (D12) 1 2 k 2 0 equation near r ¼ r , one finds that ðr r Þ , which matches with those of the inner region in the cross- where 1;2 and 1;2 are two-component bulk spinors over region (2.25). It is convenient to use the basis which is defined by specified by the boundary condition ! 1 mR 20 ðr !1Þ! r (D13) Note that due to normalization difference GkðwÞ defined here 0 differs from (D28) of [14] by a factor T2 k . 045016-28 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) ð0Þ r r k W½ þ; ¼const: (D28) ! v 2 r ! r : (D19) R2 Normalizing so that the constant on the right-hand side Once the zeroth order solutions are specified, higher order of the above equation is !-independent, then after insert- solutions ðnÞðrÞ can then be determined uniquely from ing the ! expansion (D20)of , Eq. (D28) must be saturated by the zeroth order term and all the coefficients ð0Þ using perturbation theory, and the two linearly inde- of higher order terms on the left-hand side must be zero, pendent solutions can be written as21 e.g. at first order in !, X1 n ðnÞ ð0ÞT 2 ð1Þ ð1ÞT 2 ð0Þ ðrÞ¼ ! ðrÞ (D20) þ þ þ ¼ 0: (D29) n Furthermore, equating the value of W½ þ; at r ¼ r where for economy of notation, we have left implicit the and at r ¼1we conclude that expansion in T. Comparing (D17) and (D19), in the over- T 2 lapping region we have the matching W ¼ ivþ v (D30) which is !-independent. $ T k : (D21) I Expanding (D25)in! we find that in the outer region can be expressed in terms of the set of normalizable can be written as and non-normalizable solutions introduced in (D13) and ð0Þ ð1Þ (D14) (recall that all quantities here refer to ¼ 1) ¼ þ ! þ (D31) where ¼ b þ a (D22) 1 ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ where from (D20), a , b can be expanded in perturbative ¼ ðaþ a þ Þ; (D32) series in ! and T, with the zeroth order expressions W ð0Þ ð0Þ denoted by a , b which are functions of k only. and Using (D18), (D21), (D10), and (D22), now the full 1 ð1Þ ð1Þ ð0Þ ð0Þ ð1Þ ð1Þ ð0Þ ð0Þ ð1Þ retarded boundary Green’s function can then be written ¼ ðaþ þ aþ a þ a þ Þ: as [14] W (D33) G 2 k R bþ þ kT b G ð!; kÞ¼ (D23) The expression for in the inner region can then be a þ G T2 k a þ k obtained from matching as which implies that the corresponding spectral function 1 scales with temperature as ð ; w; TÞ¼ ða T k ð ; wÞ a T k þð ; wÞÞ W þ I I R 2 B 2ImG T k : (D24) (D34) with the lowest order term given by 2. Normalizable solution ð0Þ aþ ð ; w; TÞ¼ T k þ : (D35) Let us now turn to the low energy behavior of the bulk W I normalizable solution . Using (D22), can be written in the outer region as 3. Near a Fermi surface 1 ðr; !Þ¼ ðaþð!Þ ðr; !Þ a ð!Þ þðr; !ÞÞ W At a Fermi surface k ¼ kF we have [14] (D25) ð0Þ aþ ðkFÞ¼0 (D36) where and (D35) does not apply. Near kF we have the expansion W aþb a bþ: (D26) aþðk; !; TÞ¼c1ðk kFÞ c2! þ c3T þ (D37) The Wronskian for Eq. (D4)is ð0Þ ð1Þ where c1 ¼ @kaþ ðkFÞ, c2 ¼ aþ ðkFÞ. Thus near kF, T 2 in the inner region the leading behavior for ð ; w; TÞ W½ 1; 2 ¼ 1 2 (D27) becomes where 1;2 are two solutions. Applying it to we find 1 that ð ; w; TÞ¼ ½a ðk; !; TÞT kF ð ; wÞ W þ I ðnÞ ð0Þ kF þ 21 k n a ðkFÞT ð ; wÞ (D38) Note that as r ! r , ðrÞ ðr r Þ . I 045016-29 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) where the coefficient of the first term a ðk; !; TÞ should be T þ g ¼ 2 id u c 2u log þ 2uc ð iuÞþi u þ i now understood as given by (D37). 1 1 Finally let us look at the behavior of the retarded Green’s þ (D47) function (D23) near a Fermi surface (D36), which can be written as ! c where u 2 T qed; is the digamma function; c1, d1 are positive constants23; and denotes terms which are h1 GR ¼ : (D39) real and analytic in ! and T. In the limit w ¼ !=T !1 k k ð!; TÞ ð!; kÞ F Eq. (D47) becomes kFð!; TÞ in (D39) is defined as the zero of (D37), i.e. gðwÞ¼id1w c1w log w þ : (D48) aþðkFð!; TÞÞ ¼ 0 and can be considered as a generalized Fermi momentum 1 c3 APPENDIX E: COUPLINGS TO GRAVITON kFð!; TÞ kF þ ! T þ (D40) AND VECTOR FIELD vF c1 c1 1 In this section we determine the couplings of a spinor to where vF is positive for k > . ð!; kÞ is given by c2 F 2 graviton and gauge field fluctuations; these are necessary to construct the bulk vertex. We consider a free spinor field 2 ! ¼ h T kF G (D41) with the action 2 kF T Z pffiffiffiffiffiffiffi ¼ dþ1 ðc MD c cc Þ and h1, h2 are positive constants whose values are known S d x gi M m numerically. The spectral function can be written as Z pffiffiffiffiffiffiffi ¼ ddþ1 gL (E1) R 2h1Im B ¼ 2ImG ¼ 2 2 : ðk kFð!; TÞ Re Þ þðIm Þ where c ¼ c y t. We now consider a perturbed metric of (D42) the form 2 2 2 2 2 For notational convenience we write ds ¼ g~ttdt þ hðdy þ bdtÞ þ grrdr þ hdxi (E2) ! k with 2 k F ð!; T; kFÞ¼T F g ; (D43) T y 2 b ht ; g~tt ¼ gtt þ hb : (E3) g where the explicit expression for can be obtained from The new spin connections are given by that of Gk given in Appendix D of [14] ¼ r 1 i! !t y f2e (E4) ! k ð þ k þiqedÞ g ; ¼ h ð4 Þ2 k cðkÞ 2 2 T (D44) T 2 ð1 i! þiqe Þ 2 k 2 T d ~ t y !t r ¼ f0e þ f2e (E5) with cðkÞ given by y t ð 2 Þ ð1 þ iqe Þ !y r ¼ f1e þ f2e (E6) cðkÞ¼ k k d k k (D45) ð2 kÞ ð1 k iqedÞ k þ k i !i r ¼ f1e (E7) and mR þ i kRR2 iqe .22 g approaches a constant k 2 r d as w ¼ !=T ! 0 and as w !1 with sffiffiffiffiffiffiffiffiffiffiffiffi gðwÞ!h e i k cðkÞð2wÞ2 k : (D46) 1 h0 pffiffiffiffiffiffiffi 1 h 1 g~0 pffiffiffiffiffiffiffi 2 f grr;f b0; f~ tt grr 1 2 h 2 2 g g~ 0 2 g~ For the marginal Fermi liquid case, ¼ 1 , the above rr tt tt kF 2 expressions should be modified. Instead one finds that (E8) and 22 Note that the sign of the second term in k depends on which component of the spinor we are looking at. Here the sign is for 23They are related by the first component. Also note that the definitions of cðkÞ and h2 differ by a phase factor from those used in [14]. In particular, the d 1 1 definition of h in (D41) ensures it is positive as discussed in 1 c 2 ¼ þ 2Im þ iqed ¼ 2 qe : Appendix D4 of [14]. c1 2 2 1 þ e d 045016-30 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) qffiffiffiffiffiffiffiffiffiffiffi 1 1 XZ d 1 Z t 2 y d k e ¼ g~ttdt; e ¼ h2ðdy þ bdtÞ; Sijð Þ¼T dr gðr Þ trðPjðr ; i ;k~Þ l ð2 Þd 1 1 1 1 l 1 1 (E9) i! r 2 i i m e ¼ grr e ¼ h2dx : ~ i ~ DEðr1;r1;i!m;kÞP ðr1;i l;kÞÞ: (F1) Also note that Here Pi contains the information of the graviton or gauge 1 1 1 t 2 t y 2 t y field propagators and vertex and is deliberately left vague. ¼ g~tt ; ¼ g~tt b þ h 2 : (E10) It is shown in Eq. (G6) in Appendix G that the Matsubara We thus find that the corrections to the Dirac action are sum can be rewritten in terms of an integral over the bulk given by (with a ay): spectral density Z X d! ! 1 y 1 r t y 1 y ~ ~ L c 2 t 2 c T D ðr ;r ; i! ; kÞ¼ tanh ðr ;r ; !; kÞ: 3 ¼ i gtt ht @y þ f2 ih qay ; E 1 2 m 2 2 1 2 4 i!m (E11) (F2) at cubic order. Now as before we express the bulk spectral density in y In (E11) we have restored the indices on b ht , ay terms of the boundary spectral density B and bulk 0 because they make the covariant nature of the expression normalizable wave functions c aðrÞ: ðr; r ;!;kÞ¼ manifest. c ðr; kÞ ð!; kÞc ðr0;kÞ. Away from the Fermi surface 2 c 2 c 2 B At quartic order there are both b and ba terms. the discussion of (F1) parallels that of the main text. In For completeness we list them, although they are not particular, the potential singular T dependence coming required for our calculation. The couplings of the bulk from the IR part of the vertex is compensated by T depen- spinor which are quadratic in the bosonic bulk modes dence of the spectral function, and as a result is nonsingu- (altogether, quartic in fluctuations) are lar. Near a Fermi surface, the eigenvalues of the boundary pffiffiffiffiffiffiffi hb2 spectral density matrix take the form (6.5) and (6.6). As we L ¼ gic ð MD mÞ take T ! 0, since all the other factors in (F1) are analytic 4 2g M ð0Þ tt in momentum k, the k-integral can be schematically b hb 1 t r ~ c written as pffiffiffiffiffiffi Dt iqa þ ðf0Þð2Þ (E12) gtt 2gtt 4 Z Im dk 2 2 (F3) where ðk kFð!; TÞ Re Þ þðIm Þ 2 1 pffiffiffiffiffiffiffi hb with all the other factors evaluated at k ¼ kFð!; TÞ. The ðf~ Þ ¼ grr@ : (E13) 0 ð2Þ 2 r g above integral can then be straightforwardly integrated tt and yields a contribution of order OðT0Þ. Also similar to the discussion in the main text, the potential singular contribution from the effective vertex is suppressed at APPENDIX F: OTHER CONTRIBUTIONS k ¼ kFð!; TÞ, resulting in a nonsingular contribution. In the main text we concentrated on the contributions from a Fermi surface in Fig. 3. Here we consider various 2. Oscillatory region contribution other contributions to the conductivities which we We return to the expression (4.49) for the conductivity as neglected in the main text. These include the contributions an integral over k. In the previous sections we have studied from seagull diagrams depicted in Fig. 5 which arise from the temperature dependence of the region of k near a Fermi quartic couplings involving the graviton (schematically, surface at k . Here we ask whether the ‘‘oscillatory terms like h2 cc and hAcc in the Lagrangian), and F region’’ (values of k such that particle production occurs contributions from the oscillatory region, i.e. the region in the AdS region of the geometry) make significant in momentum space where the IR dimension for the fer- 2 contributions to the conductivity. We will find that their mionic operator is imaginary. We justify our neglect of contribution is finite at T ¼ 0, and hence subleading com- these contributions by showing that they are nonsingular in pared to the T 2 behavior of a Fermi surface. We will not temperature and thus are subleading compared to those worry about numerical factors here. considered in the main text. Our discussion will be For illustration, let us look at the DC conductivity (6.1) schematic. and (6.2), which we copy here for convenience Z 1 Z 1. Seagull diagrams C d 2 d! @fð!Þ 2 2 DC ¼ dkk Bð!; kÞ ð!; k; TÞ: We write the schematic form of a seagull diagram S with 2 0 2 @! external Euclidean frequency l: (F4) 045016-31 FAULKNER et al. PHYSICAL REVIEW D 88, 045016 (2013) Z In the low temperature limit, the fermion spectral density E d ð Þ in the oscillatory region may be written D ði!nÞ¼ : (G4) 2 i!n þ i i e jcj! þ 1 We then find that oscð!; kÞ¼Im 0 ; (F5) ei jcj!i þ 1 Z Z X 1 d 1 1 ! where c!i is the IR Green’s function (with the IR dimen- T DEði!mÞ¼ d! tanh ð Þ i! 2 i 2 1 2 2 sion imaginary) at T ¼ 0. ei ; 0 are phases. This expression m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 is valid in the oscillatory regime k 045016-32 CHARGE TRANSPORT BY HOLOGRAPHIC FERMI SURFACES PHYSICAL REVIEW D 88, 045016 (2013) We thus find that h2 I ¼ 4 Im 1 ¼ 2 h ð! ;K Þ¼2 h ð! ;K Þ (G12) B 1 ð! ! Þþ ð! Þ ð! Þ 1 B 1 1 1 B 2 2 vF 1 2 1 2 with K1 ¼ kFð!2;TÞþ ð!2Þ and K2 ¼ kFð!1;TÞþ ð!1Þ. [1] C. M. 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