Charge Transport by Holographic Fermi Surfaces

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arXiv:1306.6396v2 [hep-th] 2 Sep 2013 ahsts019 USA. 02139, sachusetts ∗ conclusions and Discussion VII. I.Otieo optto of computation of Outline III. nlaefo:Dprmn fPyis I,Cmrde Mas- Cambridge, MIT, Physics, of Department from: leave On I vlaino Conductivities of Evaluation VI. field spinor a from Conductivity IV. I lc oegoer and geometry hole Black II. .Eetv vertices Effective V. .Introduction I. conductivity conductivity .D conductivity DC A. preparations Some A. formula general A A. description Cartoon A. hole black charged a of Geometry A. calculation the of Set-up A. .Nmrclcoefficients Numerical C. C. conductivity Tree-level C. .Otclconductivity Optical B. behavior temperature Low B. integration Angular B. integrals radial Performing B. propagator bulk to boundary Vector B. .D conductivity DC 2. conductivity Optical 1. surface Fermi from Contribution 2. scaling Temperature 1. 2. 1. M 11 ν ν ieie npriua,i h aeweeteseta dens life spectral transport the where the case and the in possible particular, as h ca In black efficient requires charged lifetime. This as the on is hole. propagator vector dissipation black a charged to AdS correction an loop in fermions probe eitvt slna ntemperature. in linear is resistivity k k 3 in ecmuetecnrbto otecnutvt rmhologr from conductivity the to contribution the compute We etrfrTertclPyis ascuet nttt o Institute Massachusetts Physics, Theoretical for Center ≥ < hmsFaulkner, Thomas d 2 2 1 1 4 3 = eateto hsc,Uiest fClfri tSnDieg San at California of University Physics, of Department 5 hoyGop hsc eatet EN H11 eea23, Geneva CH-1211 CERN, Department, Physics Group, Theory Contents hretasotb oorpi em surfaces Fermi holographic by transport Charge 1 nttt o dacdSuy rneo,N,08540 NJ, Princeton, Study, Advanced for Institute 1 O O ai Iqbal, Nabil ( N (1) 2 ) 2 I-T/36 ENP-H21-5,US/T 13-10,NSF- UCSD/PTH CERN-PH-TH/2013-150, MIT-CTP/4306, IP at abr,C 93106 CA Barbara, Santa KITP, 2 ogLiu, Hong 4 4 2 1 23 23 22 22 20 20 19 18 18 17 17 17 15 15 15 13 10 10 9 8 8 7 5 pounds. liigms ealcsae bevdi aue from nature, ex- in in successful observed very states liquid been metallic has most and plaining system prop- the temperature of low of obtaining erties for picture tool pro- powerful This surface a Fermi vides the to surface. close Fermi quasiparticles fermionic well-defined the interacting the weakly near and are space, quasiparticles momentum excitations in energy surface low Fermi charac- a is by system the fermionic theory, terized interacting liquid an Fermi of In state Lau- ground liquids. on Fermi based of been theory has dau’s metals of properties temperature 3 .Sm sflformulas useful Some G. .Budr pnrseta functions spectral spinor Boundary D. .Rssiiyi la systems clean in Resistivity A. .Sio ukt-ukpropagator bulk-to-bulk Spinor C. .Mxn ewe rvtnadvco field vector and graviton between Mixing B. o vrfit er u nesadn ftelow- the of understanding our years fifty over For .Culnst rvtnadvco field vector and graviton to Couplings E. .Ohrcontributions Other F. onMcGreevy John t sta famria em iud the liquid, Fermi marginal a of that is ity References 3 .Bl solutions Bulk 3. functions Green 2. equations Spinor 1. .Ueu integrals sums Useful Matsubara 2. do to How 1. contribution region Oscillatory 2. diagrams Seagull 1. surface Fermi a Near 3. solution Normalizable 2. function retarded Boundary 1. propagator the Constructing 4. He iecicdswt h single-particle the with coincides time ehooy abig,M 02139 MA Cambridge, Technology, f l emty efidta h current the that find We geometry. ole cltn eti ato h one- the of part certain a lculating pi em ufcsotie from obtained surfaces Fermi aphic ohayfrinbhvo nrr at com- earth rare in behavior fermion heavy to .INTRODUCTION I. ,L ol,C 92093 CA Jolla, La o, ∗ , 4 n ai Vegh David and Switzerland 5 KITP-13-121 35 34 34 34 33 33 33 32 31 31 30 29 29 28 27 27 26 26 25 2

Since the mid-eighties, however, there has been an ac- rather intricate calculation behind them and also give the cumulation of metallic materials whose thermodynamic numerical prefactors. and transport properties differ significantly from those There is one surprise in the numerical results for the predicted by Fermi liquid theory [1, 2]. A prime example prefactors: for certain parameters of the bulk model (co- of these so-called non-Fermi liquids is the strange metal dimension one in parameter space), the leading contri- phase of the high Tc cuprates, a funnel-shaped region bution to the DC and optical conductivities vanishes, in the phase diagram emanating from optimal doping i.e. the actual conductivities are higher order in tem- at T = 0, the understanding of which is believed to be perature than that presented [17]. This happens because essential for deciphering the mechanism for high Tc su- the effective vertex determining the coupling between the perconductivity. The anomalous behavior of the strange fermionic operator and the external DC gauge field van- metal—perhaps most prominently the simple and robust ishes at leading order. The calculation of the leading linear temperature dependence of the resistivity—has re- non-vanishing order for that parameter subspace is com- sisted a satisfactory theoretical explanation for more than plicated and will not be attempted here. 20 years (see [4] for a recent attempt). While photoemis- While the underlying UV theories in which our mod- sion experiments in the strange metal phase do reveal a els are embedded most likely have no relation with the sharp Fermi surface in momentum space, various anoma- UV description of the electronic system underlying the lous behavior, including the “marginal Fermi liquid” [5] strange metal behavior of cuprates or a heavy fermion form of the spectral function and the linear-T resistiv- system, it is tantalizing that they share striking similar- ity imply that quasiparticle description breaks down for ities in terms of infrared phenomena associated with a the low-energy excitations near the Fermi surface [5–7]. Fermi surface without quasiparticles. This points to a Other non-Fermi liquids include heavy fermion systems certain “universality” underlying the low energy behav- near a quantum phase transition [8, 9], where similar ior of these systems. The emergence of an infrared fixed anomalous behavior to the strange metal has also been point and the associated scaling phenomena, which dic- observed. tate the electron scattering rates and transport, could The strange metal behavior of high Tc cuprates and provide important hints in formulating a low energy the- heavy fermion systems challenges us to formulate a low ory describing interacting fermionic systems with a sharp energy theory of an interacting fermionic system with surface but no quasiparticles. a sharp Fermi surface but without quasiparticles (see also [10, 11]). Recently, techniques from the AdS/CFT correspon- A. Set-up of the calculation dence [12] have been used to find a class of non-Fermi liquids [13–18] (for a review see [19]). The low energy In the rest of this introduction, we describe the set-up behavior of these non-Fermi liquids was shown to be gov- of our calculation. Consider a d-dimensional conformal erned by a nontrivial infrared (IR) fixed point which ex- field theory (CFT) with a global U(1) symmetry that hibits nonanalytic scaling behavior only in the time di- has an AdS gravity dual (for reviews of applied hologra- rection. In particular, the nature of low energy excita- phy see e.g. [20–23]). Examples of such theories include tions around the Fermi surface is found to be governed the = 4 super-Yang-Mills (SYM) theory in d = 4, by the scaling dimension ν of the fermionic operator in ABJMN theory in d = 3 [24–26], and many others with 1 the IR fixed point. For ν > 2 one finds a Fermi sur- less supersymmetry. These theories essentially consist face with long-lived quasiparticles while the scaling of of elementary bosons and fermions interacting with non- the self-energy is in general different from that of the Abelian gauge fields. The rank N of the gauge group Fermi liquid. For ν 1 one instead finds a Fermi sur- ≤ 2 is mapped to the gravitational constant GN of the bulk face without quasiparticles. At ν = 1 one recovers the gravity such that 1 N 2. A typical theory may also 2 GN “marginal Fermi liquid” (MFL) which has been used to contain another coupling∝ constant which is related to the describe the strange metal phase of cuprates. ratio of the curvature radius and the string scale. The In this paper we extend the analysis of [13–16] to ad- classical gravity approximation in the bulk corresponds dress the question of charge transport. We compute the to the large N and strong coupling limit in the bound- contribution to low temperature optical and DC conduc- ary theory. The spirit of the discussion of this paper will tivities from such a non-Fermi liquid. We find that the be similar to that of [14, 15]; we will not restrict to any optical and DC conductivities have a scaling form which specific theory. Since Einstein gravity coupled to matter is again characterized by the scaling dimension ν of the fields captures universal features of a large class of field fermionic operators in the IR. The behavior of optical theories with a gravity dual, we will simply work with conductivity gives an independent confirmation of the this universal sector, essentially scanning many possible absence of quasiparticles near the Fermi surface. In par- CFTs. This analysis does involve an important assump- 1 ticular we find for ν = 2 , which corresponds to MFL, tion about the spectrum of charged operators in these the linear-T resistivity is recovered. A summary of the CFTs, as we elaborate below. qualitative scaling behavior has been presented earlier One can put such a system at a finite density by turn- in [17]. Here we provide a systematic exposition of the ing on a chemical potential µ for the U(1) global sym- 3 metry. On the gravity side, such a finite density system is described by a charged black hole in d + 1-dimensional 2 anti-de Sitter spacetime (AdSd+1)[27, 28]. The con- O(N ) served current Jµ of the boundary global U(1) is mapped to a bulk U(1) gauge field AM . The black hole is charged O(N 0) under this gauge field, resulting in a nonzero classical + + + + + + + + background for the electrostatic potential At(r). As we review further in the next subsection, in the limit of zero temperature, the near-horizon geometry of this black hole d 1 is of the form AdS2 R − , exhibiting an emergent scal- FIG. 1: Conductivity from gravity. The horizontal solid line ing symmetry which× acts on time but not on space. denotes the boundary spacetime, and the vertical axis denotes The non-Fermi liquids discovered in [14, 15] were found the radial direction of the black hole, which is the direction by calculating the fermionic response functions in the fi- extra to the boundary spacetime. The dashed line denotes the nite density state. This is done by solving Dirac equa- black hole horizon. The black hole spacetime asymptotes to ×R2 tion for a probe fermionic field in the black hole ge- that of AdS4 near the boundary and factorizes into AdS2 near the horizon. The current-current correlator in (1.1) can ometry (2.2). The Fermi surface has a size of order 0 be obtained from the propagator of the gauge field Ay with O(N ) and various arguments in [14, 15] indicate that the endpoints on the boundary. Wavy lines correspond to gauge fermionic charge density associated with a Fermi surface field propagators and the dark line denotes the bulk propa- 0 should also be of order O(N ). In contrast, the charge gator for the probe fermionic field. The left diagram is the density carried by the black hole is given by the classical tree-level propagator for Ay, while the right diagram includes geometry, giving rise to a boundary theory density of or- the contribution from a loop of fermionic quanta. The contri- 1 2 bution from the Fermi surface associated with corresponding der ρ0 O(GN− ) O(N ). Thus in the large N limit, we will∼ be studying∼ a small part of a large system, with boundary fermionic operator can be extracted from the dia- the background O(N 2) charge density essentially provid- gram on the right. ing a bath for the much smaller O(N 0) fermionic system we are interested in. This ensures that we will obtain The one-loop contribution to (1.1) from gravity con- a well-defined conductivity despite the translation sym- tains many contributions and we are only interested in metry. See Appendix A for further elaboration on this the part coming from the Fermi surface, which can be point. unambiguously extracted. This is possible because con- The optical conductivity of the system can be obtained ductivity from independent channels is additive, and, as from the Kubo formula we will see, the contribution of the Fermi surface is non- 1 1 analytic in temperature T as T 0. We emphasize that σ(Ω) = J (Ω)J ( Ω) Gyy (1.1) → iΩh y y − iretarded ≡ iΩ R the behavior of interest to us is not the conductivity that one could measure most easily if one had an experimental where Jy is the current density for the global U(1) in y di- instantiation of this system and could hook up a battery rection at zero spatial momentum. The DC conductivity 1 to it. The bit of interest is swamped by the contribution is given by from the (N 2) charge density, which however depends analyticallyO on temperature. In the large-N limit which σDC = lim σ(Ω) . (1.2) Ω 0 is well-described by classical gravity, these contributions → appear at different orders in N and can be clearly distin- The right hand side of (1.1) can be computed on the grav- guished. In cases where there are multiple Fermi surfaces, ity side from the propagator of the gauge field A with y we will see that the ‘primary’ Fermi surface (this term endpoints on the boundary, as in Fig. 1. Ina1/N 2 expan- was used in [15] to denote the one with the largest k ) sion, the leading contribution—of O(N 2)—comes from F makes the dominant contribution to the conductivity. the background black hole geometry. This reflects the Calculations of one-loop Lorentzian processes in a presence of the charged bath, and not the fermionic sub- black hole geometry are notoriously subtle. Should one system in which we are interested. Since these fermions integrate the interaction vertices over the full black hole have a density of O(N 0), and will give a contribution geometry or only region outside the black hole? How to to σ of order O(N 0). Thus to isolate their contribution treat the horizon? How to treat diagrams (such as that in we must perform a one-loop calculation on the gravity figure 2) in which a loop is cut into half by the horizon? side as indicated in Fig. 1. Higher loop diagrams can One standard strategy is to compute the corresponding be ignored since they are suppressed by higher powers in correlation function in Euclidean signature, where these 1/N 2. issues do not arise and then obtain the Lorentzian expression using analytic continuation (for Ωl > 0) yy yy GR (Ω) = GE (iΩl =Ω+ iǫ) . (1.3) 1 Note that we are considering a time-reversal invariant system which satisfies σ(Ω) = σ∗(−Ω). Thus the definition below is This, however, requires precise knowledge of the Eu- guaranteed to be real. clidean correlation function which is not available, given 4

AdS 4 the main results. A number of appendices contain addi- tional background material and ﬁne details.

II. BLACK HOLE GEOMETRY AND O(N 2) CONDUCTIVITY AdS 2 2 ! ! + + + + + In this section we first give a quick review of the AdS charged black hole geometry and then consider the leading O(N 2) contribution to the conductivity3. Our moti- FIG. 2: The imaginary part of the current-current correla- vation is twofold. Firstly, this represents a ‘background’ tor (1.1) receives its dominant contribution from diagrams from which we need to extract the Fermi surface contri- in which the fermion loop goes into the horizon. This also bution; we will show that this contribution to the conduc- gives an intuitive picture that the dissipation of current is tivity is analytic in T , unlike the Fermi surface contribu- controlled by the decay of the particles running in the loop, tion. Secondly, the bulk-to-boundary propagators which which in the bulk occurs by falling into the black hole. determine this answer are building-blocks of the one-loop calculation required for the Fermi surface contribution. the complexity of the problem. We will adopt a hybrid approach. We first write down an integral expression yy A. Geometry of a charged black hole for the two-point current correlation function GE (iΩl) in Euclidean signature. We then perform analytic continuation (1.3) to Lorentzian signature inside the inte- We consider a finite density state for CFTd by turning gral. This gives an intrinsic Lorentzian expression for on a chemical potential µ for a U(1) global symmetry. the conductivity. The procedure can be considered as On the gravity side, in the absence of other bulk matter the generalization of the procedure discussed in [29] for fields to take up the charge density, such a finite density system is described by a charged black hole in d + 1- tree-level amplitudes to one-loop. After analytic contin- 4 uation, the kind of diagrams indicated in Figure 2 are dimensional anti-de Sitter spacetime (AdSd+1)[27, 28] . included unambiguously. In fact they are the dominant The conserved current Jµ of the boundary global U(1) is contribution to the dissipative part of current correlation mapped to a bulk U(1) gauge field AM . The black hole function, which gives resistivity. is charged under this gauge field, resulting in a nonzero Typical one-loop processess in gravity also contain UV classical background for the electrostatic potential At(r). divergences, which in Fig. 1 happen when the two bulk For definiteness we take the charge of the black hole to vertices come together. We are, however, interested in be positive. The action for a vector field AM coupled to the leading contributions to the conductivity from exci- AdSd+1 gravity can be written tations around the Fermi surface, which are insensitive 1 d(d 1) R2 to short distance physics in the bulk.2 Thus we are com- S = dd+1x √ g + − F F µν 2κ2 − R R2 − g2 µν puting a UV-safe quantity, and short-distance issues will Z F not be relevant. This aspect is similar to other one-loop (2.1) applied holography calculations [30, 31]. with the black hole geometry given by [27, 28] The plan of the paper is follows. In 2, we first briefly 2 2 2 2 § ds gttdt + grrdr + giid~x review the physics of the finite density state and discuss ≡ − 2 the leading O(N ) conductivity which represents a fore- r2 R2 dr2 ground to our quantity of interest. 3 outlines the struc- = ( hdt2 + d~x2)+ (2.2) R2 − r2 h ture of the one-loop calculation. § 4 derives a general § formula for the DC and optical conductivity respectively with in terms of the boundary fermionic spectral function and 2 d 2 an effective vertex which can in turn be obtained from Q M r − h =1+ , A = µ 1 0 . (2.3) an integral over bulk on-shell quantities. 5 discusses in 2d 2 d t d 2 r − − r − r − ! detail the leading low temperature behavior§ of the effective vertices for the DC and optical conductivities. In tt Note that here we define gtt (and g ) with a positive 6 we first derive the scaling behavior of DC and optical sign. r0 is the horizon radius determined by the largest conductivities§ and then presents numerical results for the prefactors. 7 concludes with some further discussion of §

3 This was also considered in [32–34]. 4 For other ﬁnite density states (with various bulk matter contents, 2 For such a contribution the two vertices are always far apart corresponding to various operator contents of the dual QFT), see along boundary directions. e.g. [35–49] and [50] for an overview. 5 positive root of the redshift factor and the temperature

2 d Q 1 h(r0)=0, M = r0 + d 2 (2.4) T = . (2.16) → r0− 2πζ0 and In this paper we will be interested in extracting the leading temperature dependence in the limit T 0 (but gF Q 2(d 2) with T = 0) of various physical quantities. For→ this pur- µ , cd − . (2.5) ≡ 2 d 2 ≡ d 1 6 cdR r0− r pose it will be convenient to introduce dimensionless vari- − ables It is useful to parametrize the charge of the black hole by 2 a length scale r , defined by T R2 1 ∗ ξ T ζ = , ξ T ζ = , τ Tt ≡ r r 0 ≡ 0 2π ≡ ∗ d d 1 − (2.17) Q r − . (2.6) ≡ d 2 ∗ after which (2.14) becomes r − In terms of r , the density, chemical potential and tem- ∗ 2 2 2 2 perature of the boundary theory are 2 R2 ξ 2 dξ r 2 ds = 1 dτ + 2 + ∗ d~x ξ2 − − ξ2 1 ξ  R2 d 1 0 ξ2 1 r − 1 − 0 ∗ ρ = 2 , (2.7)   (2.18) κ R ed d 2 with d(d 1) r r − µ = ∗ ∗ e , (2.8) − 2 d e ξ d 2 R r0 d − Aτ = 1 . (2.19) 2d 2 ξ − ξ0 dr0 r − T = 1 ∗ (2.9) 4πR2 − 2d 2 r0 − Equation (2.18) can also be directly obtained from (2.2) via a formal decoupling limit with where we have introduced gF ξ, τ = finite,T 0 . (2.20) ed . (2.10) → ≡ 2d(d 1) − Note that in this limit the system is still at a nonzero p 1 In the extremal limit T = 0, r0 = r and the geometry temperature as ξ0 = 2π remains finite. An advantage r r∗ ∗ d 1 near the horizon (i.e. for − 1) is AdS2 R − : of (2.18)–(2.19) is that in terms of these dimensionless r∗ ≪ × variables, T completely drops out of the metric. 2 2 2 R2 2 2 r 2 ds = dt + dζ + ∗ d~x (2.11) ζ2 − R2 B. Vector boundary to bulk propagator where R2 is the curvature of AdS2, and 2 We now calculate the conductivity of the finite density 1 R2 ed R2 = R, ζ = , At = . (2.12) state described by (2.2) using the Kubo formula (1.1), to d(d 1) r r ζ − − ∗ leading order in T in the limit In the extremalp limit the chemical potential and energy Ω density are given by T 0, s = fixed . (2.21) → ≡ T d 1 2 2 d d(d 1) r R − (d 1) R To calculate the two-point function of the boundary µ = − ∗ ed, ǫ = − d 2 R2 κ2 d 2 r current J , we need to consider small fluctuations of the − − ∗ y (2.13) gauge field δAy ay which is dual to Jy with a nonzero with charge density still given by (2.7). frequency Ω and≡k = 0. In the background of a charged r0 r∗ At a finite temperature T µ, −∗ 1 and the black hole, such fluctuations of a mix with the vector ≪ r ≪ y near-horizon metric becomes that of a black hole in AdS2 fluctuations hty of the metric, as we discuss in detail Rd 1 times − in Appendix B. This mixing has a simple boundary in- terpretation; acting on a system with net charges with 2 2 2 2 2 R2 ζ 2 dζ r 2 an electric field causes momentum flows in addition to ds = 1 dt + + ∗ d~x (2.14) 2 2 ζ2 2 charge flows. ζ − − ζ0 1 2  R ζ0 − After eliminating hty from the equations for ay, we find   with that (see Appendix B for details) 2 rr yy yy 2 tt 2 ed ζ R2 ∂r √ gg g ∂ray √ gg meff g Ω ay =0 At = 1 , ζ0 (2.15) − − − − ζ − ζ0 ≡ r0 r (2.22) − ∗ 6 where ay acquires an r-dependent mass given by choose a basis of solutions which are specified as (which also fixes their normalization) 2d 2 2 2 r − m R =2 ∗ . (2.23) 1 3 eff 2 2 2 r r r − ± 1 3 η±(ξ; s) − ∗ = ξ 2 ∓ 2 , ξ 0 . (2.28) I → T R2 → The small frequency and small temperature limit of the 2 solution to equation (2.22) can be obtained by the match- Note that since the metric (2.18) has no explicit T - ing technique of [15]; the calculation parallels closely that dependence, as a function of s (2.21), ηI±(s, ξ) also have of [15], and was also performed in [32]. The idea is to no explicit T -dependence. This will be important for our divide the geometry into two regions in each of which discussion in Sec. V. the equation can be solved approximately; at small fre- The retarded solution (i.e. ay is in-falling at the hori- quency, these regions overlap and the approximate solu- zon) for the inner region can be written as [15] tions can be matched. For this purpose, we introduce a (ret) + crossover radius rc, which satisfies a (ξ; s)= η + (s)η− . (2.29) y I Gy I 2 rc r T R2 where y is the retarded function for ay in the AdS2 − ∗ 1, ξc 1 (2.24) G 3 r ≪ ≡ rc r ≪ region, which can be obtained by setting ν = 2 and q =0 ∗ − ∗ in equation (D27) of Appendix D of [15]6 and will refer to the region r > rc as the outer (or UV) region and the region r0

5 Note that here the IR marginality of Jt only applies to zero momentum. That Jt must be marginal in the IR for any compress- 6 We copy it here for convenience

ible system leads to general statements of the two-point function 1 is 1 y Γ(−2ν)Γ + ν − + iqed Γ + ν − iqed of J in the low frequency limit [51]. For example for a system G (s) = (4π)2ν 2 2π 2 . R 1 is 1 whose IR limit is characterized by a dynamical exponent z, then Γ(2ν)Γ − ν − + iqed Γ − ν − iqed y d−2 2 2π 2 J must have IR dimension 2 + z . For d = 3 this further (2.30) implies for any z the two point function of Jy must scale like Ω3 Note that the above equation diﬀers from of Appendix D of [15] at zero momentum. by a factor of T 2ν due to normalization diﬀerence in (2.28). 7

From (2.29), the matching (2.33), and (2.34), we thus Note that this equation assumes the normalization of the find the full bulk-to-boundary propagator is then gauge field specified in (2.32).

(0) 3 (0) η (r)+ y(s)T η (r) + G − (0) 3 (0) outer region a + y (s)T a + G − KA(r;Ω) =  .  + −  η + y (s)η  I G I T (0) 3 (0) inner region a + y (s)T a + G − C. Tree-level conductivity  (2.35)  Note that in the above expression all the T -dependence is made manifest. (0) We now proceed to study the low-frequency and low- The leading order solutions η in the outer region can temperature conductivity at tree level in the charged ± be determined analytically [32]; with black hole, using the boundary to bulk propagator just discussed. d 2 (0) r r − η (r)= ∗ 1 ∗ (2.36) + (d 2)R2 − r The O(N 2) conductivity is given by the boundary 2 − value of the canonical momentum conjugate to ay (in and thus terms of r foliation) evaluated at the solution (2.35)

(0) (0) r b+ d 2 a = ∗ , = r − . (2.37) + 2 (0) 2 yy rr (d 2)R2 a − ∗ 2R √ gg g ∂rKA − + σ(Ω) = lim − (2.40) r 2 2 − →∞ gF κ iΩ Useful relations among a(0), b(0) can be obtained from the constancy in r of the Wronskian± ± which gives W [a ,a ] a √ggyygrr∂ a a √ggyygrr∂ a (2.38) 1 2 ≡ 1 r 2 − 2 r 1 where a1,2 are solutions to (2.22). In particular, equating 3 d (0) 3 (0) (0) (0) 2R − 1 b+ + yT b W [η , η ] at boundary and horizon gives σ(Ω) = (d 2) G − . (2.41) + − g2 κ2 iΩ (0) 3 (0) − F a+ + yT a G − (0) (0) (0) (0) 2ν d 3 2 3 d 3 2 b a+ a b+ = r − R = r − R . − − − d 2 ∗ d 2 ∗ − − (2.39) We thus ﬁnd

3 d (0) (0) (0) (0) (0) 2R − 1 b+ 3 b a+ a b+ σ(Ω) = (d 2) + yT − − − + (2.42) − g2 κ2 iΩ (0) G (0) 2 ··· F a+ (a+ ) ! d 2 d 3 d 5 r − i d 2 − µ − 2 2 = (d 2) ∗ + − Ω + (2πT ) + (2.43) K − R2 Ω K d(d 1) e ··· − d where we have used (2.37), (2.39) as well as (2.13) and resents a ballistic contribution to the conductivity for a − 2Rd 1 clean charged system with translational and boost invari- introduced g2 κ2 . , which also appears in the K ≡ F K vacuum two-point function, specifies the normalization ance, as we review in Appendix A. It is also interesting of the boundary current and scales as O(N 2). to note that from the bulk perspective the delta func- Using (2.13) and (2.7), the first term in (2.43) can also tion in the conductivity is a direct result of the fact that be written as the fluctuations of the bulk field ay are massive, as is clear from the perturbation equation (2.22). This is not ρ2 i a breakdown of gauge-invariance; rather the gauge field σ(Ω) = + (2.44) ǫ + P Ω ··· acquired a mass through its mixing with the graviton. ǫ In the absence of such a mass term the radial equation where P = d 1 is the pressure. When supplied with the − of motion is trivial in the hydrodynamic limit (as was standard iǫ prescription, this term gives rise to a contri- shown in [52]) and there is no such delta function. bution proportional to δ(Ω) in the real part of σ(Ω). This delta function follows entirely from kinematics and rep- In (2.43), the important dynamical part is the second 8 term, which gives dissipative part of the conductivity. theory retarded Green’s function of a fermionic operator 2 2 This part, being proportional to Ω + (2πT ) , is analytic has a Fermi-surface-like pole at ω = 0 and k = kF of the in both T and Ω and the DC conductivity goes to zero kind described in [14, 15]. We will neglect many “com- in the T 0 limit. This has a simple physical inter- plications,” including spinor indices, matrix structures, pretation;→ the dissipation of the current arises from the gauge-graviton mixing, and a host of other important de- neutral component of the system, whose density goes to tails, which turn out to be inessential in understanding zero in the T 0 limit, leaving us with only the bal- the structure of the calculation. listic part (2.44→) for a clean charged system. We also note that for a clean system such as this, there is no non- The important bulk Feynman diagram is depicted in trivial heat conductivity or thermoelectric coefficient at Fig. 3. Note that it is structurally very similar to the k = 0, independent of the charge conduction. This is particle-hole bubble which contributes to the Fermi liq- because momentum conservation is exact, and just as in uid conductivity (see e.g. [53, 54]): an external current the discussion of Appendix A, there can therefore be no source creates a fermion-antifermion pair, which then re- dissipative part of these transport coefficients. combine. The calculation differs from the standard Fermi liquid calculation in two important ways. The first, obvious difference is that the gravity amplitude involves III. OUTLINE OF COMPUTATION OF O (1) integrals over the extra radial dimension of the bulk ge- CONDUCTIVITY ometry. It turns out, however, that these integrals can be packaged into factors in the amplitude (called Λ be- As explained in the introduction, in order to obtain low) that play the role of an effective vertex. The second the contribution of a Fermi surface to the conductivity, main difference is that actual vertex correction diagrams we need to extend the tree-level gravity calculation of 2 in the bulk are suppressed by further powers of N − and the previous section to the one-loop level with the cor- are therefore negligible, at least in the large N limit in responding bulk spinor field running in the loop. This which we work. one-loop calculation is rather complicated and is spelled out in detail in the next section. In this section we outline the main ingredients of the computation suppressing We now proceed to outline the computation. While it the technical details. is more convenient to perform the tree-level calculation of the last section in the Lorentzian signature, for the one- loop calculation it is far simpler to work in Euclidean signature, where one avoids thorny conceptual issues re- garding the choice of vacuum and whether interaction D (i ! +i" ) E l m vertices should be integrated through the horizon or not. ! ! A(i l) A(-i l) Our strategy is to first write down an integral expression 7 yy for the Euclidean two-point function function GE (iΩl) r ! r ! 1 DE(i "m)! 2 and then analytically continue to Lorentzian signature inside the integral (for Ωl > 0)

FIG. 3: The bulk Feynman diagram by which the spinor yy yy G (Ω) = G (iΩl =Ω+ iǫ) (3.1) contributes to the conductivity. R E

which will then give us the conductivity via the Kubo A. Cartoon description formula (1.1) and (1.2).

In this subsection we will describe a toy version of the We now turn to the evaluation of the diagram in Fig- one-loop conductivity. We will assume that the boundary ure 3, which works out to have the structure

Gyy(iΩ ) T d~kdr dr D (r , r ; iΩ + iω ,~k)K (r ; iΩ )D (r , r ; iω ,~k)K (r ; iΩ ) . (3.2) E l ∼ 2 2 E 1 2 l n A 1 l E 2 1 n A 2 − l iω Xn Z

The ingredients here require further explanation. DE(r1, r2; iωn, k) is the spinor propagator in Euclidean 9 space. KA(r; iΩl) is the boundary-to-bulk propagator We now turn to the evaluation of the expression (3.2). for the gauge field (i.e. Euclidean analytic continuation of (2.35)); it takes a gauge field source localized at the boundary and propagates it inward, computing B. Performing radial integrals its strength at a bulk radius r. The vertices have a great deal of matrix structure that we have suppressed, and the actual derivation of this expression from the fundamental Inserting (3.3) into (3.2) we can now use standard formulas of AdS/CFT requires a little bit of manipula- manipulations from finite-temperature field theory to tion that is discussed in next section, but its structure rewrite the Matsubara sum in (3.2) in terms of an inte- should appear plausible. The radial integrals dr should gral over Lorentzian spectral densities. The key identity be understood as including the relevant metric factors to is (see Appendix G for a discussion), make the expression covariant, and d~k denotes integration over spatial momentum along boundary directions. 1 1 f(ω1) f(ω2) T = − We would now like to perform the Euclidean frequency i(ωm +Ωl) ω1 iωm ω2 ± ω1 iΩl ω2 ωm − − − − sum. This is conveniently done using the spectral repre- X (3.5) sentation of the Euclidean green’s function for the spinor, with

1 dω ρ(r , r ; ω,~k) f(ω)= (3.6) D (r , r ; iω ,~k)= 1 2 , (3.3) eβω 1 E 1 2 m (2π) iω ω ± Z m − where the upper (lower) sign is for fermion (boson). Us- ~ where ρ(r1, r2; ω, k) is the bulk-to-bulk spectral density. ing the above identity (3.2) can now be written as As we discuss in detail in Appendix C, ρ(r1, r2; ω,~k) can be further written in terms of boundary theory spectral yy ~ f(ω1) f(ω2) density ρ (ω, k) as (again schematically, suppressing all GE (iΩl) dω1dω2dkdr1dr2 − B ∼ ω1 ω2 iΩl × indices) Z − − ρ(r , r ; ω,~k)K (r ; iΩ )ρ(r , r ; iΩ ,~k)K (r ; iΩ ) . 1 2 A 1 l 2 1 l A 2 − l ρ(r, r′; ω,~k)= ψ(r; ω,~k)ρB(ω, k)ψ(r′; ω,~k) (3.4) (3.7) where ψ(r) is the normalizable spinor wavefunction8 to We then analytically continue the above expression to the the Dirac equation in the Lorentzian black hole geome- Lorentzian signature by setting iΩl = Ω+ iǫ. We now try. Equation (3.4) can be somewhat surprising to some realize the true power of the spectral decomposition (3.3) readers and we now pause to discuss it. The bulk-to- and (3.4); the bulk-to-bulk propagator factorizes into a bulk spectral density factorizes in the radial direction; product of spinor wavefunctions, allowing us to do each thus in some sense the density of states is largely deter- radial integral independently. In this way all of the radial mined by the analytic structure of the boundary theory integrals can be repackaged into an eﬀective vertex spectral density ρB(ω,~k). We will see that this means that despite the presence of the extra radial direction Λ(ω ,ω , Ω,~k)= dr ψ(r; ω ,~k)K (r; Ω)ψ(r; ω ,~k), in the bulk, the essential form of one-loop calculations 1 2 1 A 2 in this framework will be determined by the boundary Z (3.8) theory excitation spectrum, with all radial integrals sim- where the propagator KA(r;Ω) has now become a ply determining the structure of interaction vertices that Lorentzian object that propagates infalling waves, and appear very similar to those in ﬁeld theory. we are left with the formula

f(ω ) f(ω ) Gyy(Ω) dω dω d~k 1 − 2 ρ (ω ,~k)Λ(ω ,ω , Ω,~k)ρ (ω ,~k)Λ(ω ,ω , Ω,~k) . (3.9) R ∼ 1 2 ω ω Ω iǫ B 1 1 2 B 2 2 1 Z 1 − 2 − −

This formula involves only integrals over the boundary Computing the effective vertex requires a complete so- theory spectral densities; the radial integral over spinor lution to the bulk wave equations; however, we will show and gauge field wavefunctions simply provides an exact that in the low temperature and low frequency limit, the derivation of the effective vertex Λ that determines how conductivity (3.9) is dominated by the contribution near strongly these fluctuations couple to the external field the Fermi surface, where we can simply replace Λ by theory current, as shown in Fig. 4. a constant. Thus if one is interested in extracting low 10

! A. A general formula

We consider a free spinor ﬁeld in (2.2) with an action " " S = dd+1x√ gi(ψ¯ΓM ψ mψψ¯ ) (4.1) ! − − DM − Z FIG. 4: Final formula for conductivity; radial integrals only determine the eﬀective vertex Λ, with exact propagator for boundary theory fermion running in loop.

temperature DC and optical conductivities in the low frequency regime, the evaluation of (3.9) reduces to a familiar one as that in a Fermi liquid (without vertex corrections). FIG. 5: The ‘seagull’ diagram coming from quartic vertices involving terms quadratic in bosonic perturbations. The boundary is represented by a circle. IV. CONDUCTIVITY FROM A SPINOR FIELD

¯ t In this section we describe in detail the calculation where ψ = ψ†Γ and leading to (3.9), paying attention to all subtleties. For 1 ab readers who wants to skip the detailed derivation, the fi- M = ∂M + ωabM Γ iqAM (4.2) nal results for the optical and DC conductivities are given D 4 − by (4.49) and (4.50), with the relevant vertices given by The abstract spacetime indices are M,N and the ab- (for d =3) (4.53)–(4.56). stract tangent space indices are a, b, . The··· index with Before going into details, it is worth mentioning some an underline denotes that in tangent··· space. Thus Γa to important complications which we ignored in the last sec- denote gamma matrices in the tangent frame and ΓM tion: those in curved coordinates. According to this convention, for example, 1. As already discussed in Sec. IIB, the gauge field perturbations on the black hole geometry mix with ΓM =Γae M , Γr = √grrΓr . (4.3) the graviton perturbations; a boundary source for a the bulk gauge field will also lead to perturba- We will make our discussion slightly more general, ap- tions in metric, and as a result the propagator KA plicable to a background metric given by the first line in (3.2) should be supplemented by a graviton com- of (2.2) with gMN depending on r only. Also for nota- ponent. Thus the effective vertex Λ is rather more tional simplicity we will denote involved than the schematic form given in (3.8). dd 1k 2. Another side effect of the mixing with graviton is = − (4.4) (2π)d 1 that, in addition to Fig. 3, the conductivity also Zk Z − receives a contribution from the ‘seagull’ diagram of Fig. 5, coming from quartic vertices involving We are interested in computing the one-loop correction terms quadratic in metric perturbations (given in to the retarded two-point function of the boundary vec- Appendix E). We will show in Appendix F that tor current due to a bulk spinor field. In the presence of such contributions give only subleading corrections a background gauge field profile, the fluctuations of the bulk gauge field mix with those of the metric. Consider in the low temperature limit and will be omitted. j j small perturbations in aj δAj and ht = δgt . In Ap- 3. The careful treatment of spinor fields and associ- pendix E we find that the corrections≡ to the Dirac action ated indices will require some care. are given at cubic order by

1 δS [a ,hj , ψ]= i dd+1x√ gψ¯ hjΓt∂ + g ∂ hj Γrtj iqa Γj ψ (4.5) 3 j t − − − t j 8 jj r t − j j X Z where Γrtj ΓtΓrΓj . Note that in the above equation and below the summations over boundary spatial inde- ≡ 11 cies will be indicated explicitly (with no summation as- where δS3 can be written in momentum space as sociated with repeated indices). There are also quartic corrections (involving terms quadratic in bosonic ﬂuctu- ations) which are given in Appendix E. These terms give only subleading corrections, as we discuss in Appendix F. δS = T 2 dr√ g Now we go to Euclidean signature via 3 − − ω k Xm XΩl Z Z t it ω iω A iA iS S . ψ¯(r; iωm + iΩl,~k) B(r; iΩl,~k) ψ(r; iωm,~k) (4.8). E E t τ E × →− → → →− (4.6) It’s helpful to keep in mind that ψ and ψ† don’t change under the continuation, and we do not change Γt. The Euclidean spinor action can then be written as Here the kernel B(r; iΩl,~k) contains all dependence on the gauge and metric ﬂuctuations, which we have also S = i dd+1x√g ψ¯ ΓM D(0) m ψ +δS + (4.7) Fourier expanded: E M − 3 ··· Z

g B(r; iΩ ,~k)= i ik hj (r; iΩ )Γt + jj ∂ hj (r; iΩ )Γrtj iqa (r; iΩ )Γj . (4.9) l − − j t l 8 r t l − j l j X

Note that since we are only interested in calculating the the free Dirac equation in the black hole geometry9. conductivity at zero spatial momentum, we have taken ρB(ω, k) is the boundary theory spectral density of the j aj and ht to have zero spatial momentum. holographic non-Fermi liquid, and was discussed in de- We now evaluate the one-loop determinant by integrat- tail in [15] (see Appendix D for a review). Note that ing out the fermion field. We seek the quadratic depen- in (4.12) bulk spinor indices are suppressed and α, γ in dence on the gauge and graviton fields; the relevant term ψ label independent normalizable solutions and as dis- in the effective action is given by the Feynman diagram cussed in Appendix C can be interpreted as the boundary in Figure 3 and is spinor indices. Now introduce various momentum space Euclidean T 2 Γ[a ,hj ]= dr g(r )dr g(r ) boundary to bulk propagators for the gauge field and j t − 2 1 1 2 2 ω ,Ω k graviton. Xm l Z Z p p tr DE(r1, r2; iωm + iΩl,~k)B(r2;Ωl,~k) × aj(r; iΩl)= KA(r; iΩl)Aj (iΩl), j D (r , r ; iω ,~k)B(r ; Ω ,~k) (4.10) h (r; iΩl)= Kh(r; iΩl)Aj (iΩl) (4.13) E 2 1 m 1 − l t where the tr denotes the trace in the bulk spinor indices where Aj (iΩl) is the source for the boundary conserved and DE(r1, r2; iωm,~k) denotes the bulk spinor propaga- current in Euclidean signature. These are objects which tor in the Euclidean signature. As always we suppress propagate a gauge field source at the boundary to a gauge bulk spinor indicies. The bulk spinor propagator is dis- field or a metric fluctuation in the interior, and so should A A cussed in some detail in Appendix C. The single most perhaps be called KA and Kh ; we drop the second ‘A’ important property that we use is its spectral decompo- label since we will never insert metric sources in this pa- sition per. KA(r; iΩl) and Kh(r; iΩl) go to zero (for any nonzero ~ dω ρ(r1, r2; ω, k) Ωl) at the horizon and they do not depend on the in- DE(r1, r2; iωm,~k)= , (4.11) 2π iωm ω dex j due to rotational symmetry. Kh and KA are not Z − independent; in Appendix B we show that where ρ(r1, r2; ω,~k) is the bulk spectral function. As is shown in show in (C53) of Appendix C, the bulk spectral d+1 ∂ K = √g g g− 2 K , (4.14) function can be written in terms of that of the boundary r h −C rr tt ii A theory as

ρ(r, r ; ω,~k)= ψ (r) ραγ (ω,~k) ψ (r ) (4.12) ′ α B γ ′ 9 For a precise deﬁnition of the normalizable Lorentzian wave function, see again Appendix C. In this section, to avoid clutter, where the boundary spectral function ρB is Hermitian we will use the boldface font to denote the normalizable solution. and ψ is the normalizable Lorentzian wave function for The non-normalizable solution will not appear. 12

j with new object Q (r; ,iΩl,~k) that has the source explicitly extracted: =2κ2ρ (4.15) C j B(r; iΩl,~k)= Q (r; iΩl,~k)Aj (iΩl) (4.16) where ρ is the background charge density (2.7). j Now using the deﬁnition of the propagators (4.13) we X can write the kernel B(r; iΩl,~k) in (4.9) in terms of a with

g Qj (r; iΩ ,~k)= i ik K (r; iΩ )Γt + jj ∂ K (r; iΩ )Γrtj iqK (r; iΩ )Γj . (4.17) l − − j h l 8 r h l − A l Plugging (4.16) into (4.10), we can now express the entire expression in terms of the boundary gauge ﬁeld source Aj (iΩl). Taking two functional derivatives of this expression with respect to Aj , we ﬁnd that the the boundary Euclidean current correlator can now be written as

ij ~ i ~ ~ j ~ GE (iΩl)= T dr1 g(r1)dr2 g(r2) tr DE(r1, r2; iωm + iΩl, k)Q (r2; iΩl, k)DE(r2, r1; iωm, k)Q (r1; iΩl, k) − k − ωm Z Z X p p (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 ωm using the techniques that was outlined earlier in (3.5)–(3.7). We find that equation (4.18) can be further written as

dω dω f(ω ) f(ω ) Gij (iΩ ) = 1 2 dr g(r )dr g(r ) 1 − 2 E l − 2π 2π 1 1 2 2 ω iΩ ω Zk Z Z 1 − l − 2 tr ρ(r , r ; ω ,~k)Qpi(r ; iΩ ,~kp)ρ(r , r ; ω ,~k)Qj (r ; iΩ ,~k) , (4.19) × 1 2 1 2 l 2 1 2 1 − l Now plugging (4.12) into (4.19) we ﬁnd that

dω dω f(ω ) f(ω ) Gij (iΩ ) = 1 2 1 − 2 ραβ(ω ,~k)Λi (ω ,ω ,iΩ ,~k) ργδ(ω ,~k)Λj (ω ,ω , iΩ ,~k) (4.20) E l − 2π 2π ω iΩ ω B 1 βγ 1 2 l B 2 δα 2 1 − l Zk Z 1 − l − 2 with Λi (ω ,ω ,iΩ ,~k)= ij ij βγ 1 2 l GR (Ω) = GE (iΩl =Ω+ iǫ) (4.22) ~ i ~ ~ dr√g ψβ(r; ω1, k) Q (r; iΩl; k) ψγ(r; ω2, k)(4.21). We will suppress the iǫ in equations below for notational Z simplicity but it is crucial to keep it in mind. For sim- Let us now pause for a moment to examine this expres- plicity of notations we will denote the Lorentzian analytic sion. In the last series of manipulations we replaced continuation of various quantities only by their argument, the interior frequency sum with an integral over real e.g. Lorentzian frequencies; however by doing this we ex- K (r; iΩ ) K (r; Ω) (4.23) ploited the fact that the spectral density factorizes in A l |iΩl=Ω → A r. This allowed us to absorb all radial integrals into Λ, i which should be thought of as an effective vertex for the We also make the analogous replacements also for Kh,Q virtual spinor fluctuations. Now only the boundary the- and Λαβ. Note that KA(r; Ω) and Kh(r; Ω) have now become retarded functions which are in-falling at the hori- ory spectral density ρB appears explicitly in the expression. This form for the expression is perhaps not sur- zon and satisfy prising from a field-theoretical point of view; however it K∗ (r, Ω) = K (r, Ω),K∗(r, Ω) = K (r, Ω) . is interesting that we have an exact expression for the A A h h − − (4.24) vertex, found by evaluating radial integrals over normal- We thus find that the retarded Green function for the izable wave functions. currents can be written as We now obtain the retarded Green function for the ij currents by starting with GE (iΩl) for Ωl > 0 and ana- ij dω1 dω2 f(ω1) f(ω2) ij GR (Ω) = − lytically continuing GE (iΩl) to Lorentzian signature via − 2π 2π ω Ω ω iǫ × Zk Z 1 − − 2 − 13

αβ ~ i ~ γδ ~ j ~ ρB (ω1, k)Λβγ(ω1,ω2, Ω, k) ρB (ω2, k)Λδα(ω2,ω1, Ω, k) where it is important to note that this expression makes (4.25) sense only when sandwiched between the two on-shell spinors in Λ. This manipulation replaced the Kh with where its radial derivative ∂rKh, and one can now use the relation between gauge and graviton propagators (4.14) to i ~ i eliminate ∂rKh in favor of KA, leaving us with Λβγ(ω1,ω2, Ω, k)= dr√g ψβ Q ψγ (4.26) Z j j iY2kj r rtj with Q (r;Ω,~k)= KA(r; Ω) Y1Γ + Γ + iY3Γ ω ω 1 − 2 j t (4.33) Q (r;Ω,~k)= i ikjKh(r;Ω)Γ − − where gjj rtj j + ∂rKh(r;Ω)Γ iqKA(r;Ω)Γ . (4.27) 8 − 1 d+1 1 d − 2 − 2 − 2 Y1 = qgjj , Y2 = gjj √gtt, Y3 = gjj . i ~ − −C 8 C Note that in (4.20) both Λ (ω1,ω2, iΩl, k) analytically (4.34) i ~ 10 ± continue to Λ (ω1,ω2, Ω, k). This is the form of Qj that will be used in the remain- The complex, frequency-dependent conductivity is der of this calculation. In (4.33), the -dependent terms ( was given in (4.15)) can be interpretedC as giving a Gij (Ω) C σij (Ω) R (4.28) “charge renormalization” resulting from mixing between ≡ iΩ the gauge ﬁeld and graviton. which through (4.25) is expressed in terms of intrinsic boundary quantities; Λ can be interpreted as an eﬀective vertex. Note that from (4.24) one can readily check that B. Angular integration

i t i t (Q (r;Ω,~k))† =Γ Q (r; Ω,~k)Γ (4.29) We will now use the spherical symmetry of the under- − lying system to perform the angular integration in (4.25). which implies that For this purpose we choose a reference direction, say, ~ i i with kx = k k and all other spatial components of Λ ∗ (ω ,ω , Ω,~k)=Λ (ω ,ω , Ω,~k) . (4.30) ≡ | | βγ 1 2 γβ 2 1 − ~k vanishing. We will denote this direction symbolically We now make a further manipulation on the expression as θ = 0 below. Then from the transformation properties for the effective vertex (4.26) to rewrite the first term of spinors it is easy to see that there in terms of ∂rKh, which can then be related simply ρ (~k) = U(θ)ρ (k,θ = 0)U †(θ), to KA via (4.14). To proceed note that the wave func- B B ~ tion ψ satisfies the Dirac equation (C2), which implies Λi(k) = Rij (θ)U(θ)Λj (k,θ = 0)U †(θ) (4.35) that11 (see Appendix C1 for details) where Rij (θ) is the orthogonal matrix which rotate a vec- t ψβ(r; ω1)Γ ψγ (r; ω2) tor ~k to θ = 0 and U is the unitary matrix which does i 1 r the same rotation on a spinor (i.e. in α, β space). The = ∂r √g ψβ(r; ω1)Γ ψγ (r; ω2) (4.31). ω1 ω2 √g angular integral in (4.25) is reduced to − We now use this identity in the first term of Qj in (4.27) 1 dd 2θ R (θ)R (θ)= Cδ δ (4.36) and integrate by parts. We can drop both boundary d 1 − ik jl ij kl (2π) − terms: the term at infinity vanishes since the ψ are Z normalizable, and the term at the horizon vanishes be- d 2 where C is a normalization constant and d − θ denotes i cause the graviton wavefunction ht (and thus Kh) van- the measure for angular integration. Note that C = 1 12 4π ishes there. We then find that (4.27) can be rewritten 1 for d = 3 and C = 12π2 for d = 4. The conductivity can as now be written as k j ~ j r ij ij Q (r;Ω, k)= i ∂rKh(r;Ω)Γ (4.32) σ (Ω) = δ σ(Ω) (4.37) − −ω ω 1 2 g − + ii ∂ K (r;Ω)Γrtj iqK (r;Ω)Γj , with 8 r h − A 14

C ∞ d 2 dω1 dω2 f(ω1) f(ω2) αβ i γδ i σ(Ω) = dkk − − ρB (ω1, k)Λβγ(ω1,ω2, Ω, k) ρB (ω2, k)Λδα(ω2,ω1, Ω, k) −iΩ 2π 2π ω1 Ω ω2 iǫ Z0 Z − − − i X (4.38) i where ρB(ω, k) and Λδα(ω2,ω1, Ω, k) in (4.38) and in all expressions below should be understood as the corresponding quantities evaluated at θ = 0 as in (4.35). αβ α αβ Equation (4.38) can now be further simpliﬁed in a basis in which ρB is diagonal, i.e. ρB = ρB δ , leading to

C ∞ d 2 dω1 dω2 f(ω1) f(ω2) α γ σ(Ω) = dkk − − ρ (ω , k) M (ω ,ω , Ω, k) ρ (ω , k) (4.39) −iΩ 2π 2π ω Ω ω iǫ B 1 αγ 1 2 B 2 Z0 Z 1 − − 2 −

where (there is no summation over α, γ below) Let us now look at the Ω 0 limit of (4.43), for which 1 → the ω ω term in (4.33) has to be treated with some care. i i 1− 2 Mαγ(ω1,ω2, Ω, k)= Λαγ(ω1,ω2, Ω, k)Λγα(ω2,ω1, Ω, k) Naively, it appears divergent; however, note that i X (4.40) i 1 r with Λ given by (4.26). From (4.30) we also have lim ψβ(r; ω +Ω,~k)Γ ψγ (r; ω,~k) Ω 0 Ω → r Mαγ∗ (ω1,ω2, Ω, k)= Mαγ(ω1,ω2, Ω, k) . (4.41) = ψ (r; ω,~k)Γ ∂ ψ (r; ω,~k) (4.44) − − β ω γ The DC conductivity can now be obtained by taking the Ω 0 limit in (4.39), which can be written as is ﬁnite because (see equation (C39) of Appendix C) → C ∞ dω ∂f(ω) d 2 r σDC = dkk − ψ (r; ω,~k)Γ ψ (r; ω,~k)=0 . (4.45) − 2 2π ∂ω β γ α,γ Z0 Z α X γ ρB(ω, k) αγ (ω, k) ρB(ω, k)+ (4.42) M S Introducing where i i λβγ (ω, k) lim Λβγ(ω Ω, ω, Ω, k) (4.46) αγ(ω, k) lim Mαγ (ω +Ω, ω, Ω, k) (4.43) Ω 0 Ω 0 ≡ → ± M ≡ → is real (from (4.41)). Note that the term written explic- we thus have itly in (4.42) is obtained by taking the imaginary part of 1 in (4.39). The rest, i.e. the part proportional ω1 Ω ω2 iǫ i i − − − 1 αγ(ω, k)= λαγ (ω, k)λγα(ω, k) (4.47) to the real part of ω Ω ω iǫ , is collectively denotes as M 1 2 i . We will see in Sec.− VI− B−that such contribution van- X Sishes in the low temperature limit, so we will neglect it from now on. and from (4.44) and (4.33)

λj (ω,~k)= dr√ gK (r;Ω=0) ψ (r; ω,~k) Y Γj + iY Γrtj ik Y Γr∂ ψ (r; ω,~k) . (4.48) βγ − A β 1 3 − j 2 ω γ Z

The above expressions (4.39) and (4.42) are very gen- at discrete momenta k = kF , only one of the eigenval- 1 eral, but we can simplify them slightly by using some ex- ues of ρB, say ρB, develops singular behavior. We can plicit properties of the expression for ρB. We seek singu- extract the leading singularities in the T 0 limit by → 1 2 lar low-temperature behavior in the conductivity, which simply taking the term in (4.39) proportional to (ρB ) . will essentially arise from low-frequency singularities in Thus (4.39) simpliﬁes to ρB. At the holographic Fermi surfaces described in [15], 15

C ∞ d 2 dω1 dω2 f(ω1) f(ω2) 1 1 σ(Ω) = dkk − − ρ (ω , k) M (ω ,ω , Ω, k) ρ (ω , k) (4.49) −iΩ 2π 2π ω Ω ω iǫ B 1 11 1 2 B 2 Z0 Z 1 − − 2 − and we will only need to calculate M11. Similarly, for the one-loop DC conductivity,

C ∞ d 2 dω ∂f(ω) 1 1 σ = dkk − ρ (ω, k) (ω, k) ρ (ω, k) . (4.50) DC − 2 2π ∂ω B M11 B Z0 Z

C. M11 in d = 3 if the wave-functions Φ1,2 are known. Y1,2,3 were introduced here in (4.34). For definiteness, let us now focus on d = 3 and choose As this was a somewhat lengthy exposition, let us the following basis of gamma matrices briefly recap: after a great deal of calculation, we find the optical and DC conductivities are given by (4.49) σ3 0 iσ1 0 and (4.50), with (for d = 3) M given by (4.53)–(4.54) Γr = − , Γt = , 11 0 σ3 0 iσ1 and given by (4.55)–(4.56). − M11 σ2 0 0 σ2 Γx = , Γy = . (4.51) −0 σ2 σ2 0 V. EFFECTIVE VERTICES and write In this section we study in detail the analytic proper- rr 1 Φ1 rr 1 0 ψ1 = ( gg )− 4 , ψ2 = ( gg )− 4 ties of the effective vertices (4.53)–(4.56) appearing re- − 0 − Φ2 (4.52) spectively in the expressions for optical and DC conductivities (4.49) and (4.50) in the regime of low frequencies where Φ1,2 are two-component bulk spinors. As discussed in Appendix D, in the basis (4.51), the fermion and temperatures. We will restrict to d = 3. spectral function is diagonal and the subscript 1, 2 For simplicity of notations, from now on we will suppress various superscripts and subscripts in in (4.52) can be interpreted as the boundary spinor in- x x dices. Also note that in this basis the Dirac equation is M11, Λ11, 11, λ11 and Φ1, and denote them simply as M, Λ, ,M λ and Φ. Recall that under complex conjuga- real in momentum space and Φ1,2 can be chosen to be M real. tion It then can be checked that Λx (evaluated at θ = 0) Λ∗(ω ,ω , Ω, k)=Λ(ω ,ω , Ω, k) (5.1) only has diagonal components while Λy only has off- 1 2 2 1 − M ∗(ω ,ω , Ω, k)= M(ω ,ω , Ω, k) (5.2) diagonal components. From (4.40), we then find that 1 2 1 2 − and both λ(ω, k) and (ω, k) are real. Introducing scal- x x ing variables M M11 =Λ11(ω1,ω2, Ω, k)Λ11(ω2,ω1, Ω, k) . (4.53) ω ω Ω where w 1 , w 2 , s (5.3) 1 ≡ T 2 ≡ T ≡ T Λx (ω ,ω , Ω, k)= dr√g K (Ω) 11 1 2 rr A × we will be interested in the regime Z T 3 iY2k 2 1 Φ1 (ω1, k) Y1σ σ + Y3σ Φ1(ω2, k) w1, w2,s = fixed,T 0 . (5.4) − ω1 ω2 → − (4.54) A. Some preparations Similarly, for the DC conductivity,

x x 11 = λ11(ω, k)λ11(ω, k) (4.55) As in the discussion of Sec. IIB it is convenient to sep- M arate the radial integral in (4.54) into two parts, coming with from IR and UV region respectively, i.e.

x λ (ω, k)= dr√grr KA(r;Ω=0) Λ=Λ +Λ (5.5) 11 × IR UV T Z 3 2 Φ1 (r; ω, k) Y1σ + Y3σ1 + ikY2σ ∂ω Φ1(r; ω, k) . with

(4.56) rc ∞ Λ = dr√g , Λ = dr√g Equations (4.53)–(4.56) are a set of a complete and self- UV rr ··· IR rr ··· Zrc Zr0 contained expressions that can be evaluated numerically (5.6) 16 where r0 is the horizon at a finite temperature and rc not have any explicit dependence on T . The above is the crossover radius specified in (2.24) and (2.25). In expression, however, does not apply near a Fermi the inner (IR) region it is convenient to use coordinate ξ (0) surface k = kF where a+ (kF ) is zero. Near a introduced in (2.17), and then Fermi surface we have instead (see discussion in Appendix D around (D38)) ξ0 ΛIR = dξ√gξξ (5.7) 1 νkF ξc ··· Φ(ξ; w,T ) = a (k,ω,T )T − Ψ−(ξ; w) Z W + I (0) νkF + In the limit (5.3), as discussed around (2.25), we can take a (kF )T ΨI (ξ, w) (5.13) − − rc r and ξc 0 in the integrations of (5.6) and (5.7). i Note,→ however,∗ → this limit can only be straightforwardly where taken provided that the integrals of (5.6) are convergent a+(k,ω,T ) = c1(k kF ) c2ω + c3T + as rc r . Below we will see in some parameter range − − ··· → ∗ = c1(k kF (ω,T )) (5.14) this is not so and the limit should be treated with care. − Now let us look at the integrand of the vertex (4.54) with real coefficients c1,c2,c3. Again in (5.13) all in the limit (5.3). For this purpose let us first review the the T -dependence has been made manifest. Here behavior of the vector propagator KA and spinor wave we have also introduced a “generalized” Fermi mo- function Φ in the IR and UV regions, which are discussed mentum kF (ω,T ) defined by respectively in some detail in Sec. IIB and Appendix D 1 c3 (please refer to these sections for definitions of various kF (ω,T )= kF + ω T + (5.15) notations below): vF − c1 ··· c1 with vF = . 1. From equation (2.35), we find in the outer region c2 3. We now collect the asymptotic behavior of various (0) η+ functions appearing in the effective vertices (4.54) K (Ω) = + iO(T 3) A (0) and (4.56): a+ r r 3 = − ∗ + O(T )+ iO(T ) (5.8) (a) For r , r → ∞ mR Φ r− ,KA(Ω) O(1) with the leading term independent of Ω and T and ∼1 1 ∼ 1 real. In the above we have also indicated the lead- √g , Y , Y (5.16) rr r 1 r 2,3 rd ing temperature dependence of the imaginary part ∼ ∼ ∼ (from (2.31)). In the inner region from the second and thus the UV part of the integrals are al- line of equation (2.35) we have ways convergent as r . Note that in our convention mR > 1 →with ∞ the negative mass 4 − 2 KA(Ω) = TKA(s, ξ)+ O(T ) (5.9) corresponding to the alternative quantization. ··· (b) For r r , in the outer region, where → ∗ 1 1 + √grr , Y1, Y3 O(1), Y2 r r . (5.17) − KA(s, ξ)= (0) (ηI + y(s)ηI ) (5.10) ∼ r r ∼ ∼ − ∗ G ∗ a+ − From (5.8), KA(Ω) r r . From (D32) has no explicit T -dependence. and (D33), ∼ − ∗

2. In the outer region, to lowest order in T , the nor- (0) 1 (0) (0) (0) (0) Φ = (a+ Ψ a Ψ+ ), (5.18) malizable spinor wave function Φ can be expanded W − − − in ω as, and (0) (1) Φ = Φ + ωΦ + (5.11) (1) 1 (1) (0) (0) (1) (1) (0) (0) (1) ··· Φ = a+ Ψ + a+ Ψ a Ψ+ a Ψ+ W − − − − − − where Φ(0) and Φ(1) are deﬁned respectively (5.19) in (D32) and (D33) and are T -independent. where

(n) νk n In the inner region, to leading order, Φ can be writ- Ψ (r r )± − , r r . (5.20) ten as (from (D35)) ± ∼ − ∗ → ∗ (c) Near the event horizon ξ ξ0, (0) → a s + νk i Φ(ξ, w)= T − Ψ− + (5.12) KA(s, ξ) = (ξ0 ξ) 4π (1 + ) , (5.21) W I ··· − ···

(0) and where a+ and W0 are some k-dependent constants w w i 4π i 4π (but independent of ω and T ), and Ψ±(ξ, w) do ΨI±(w, ξ) c (ξ0 ξ) + c∗ (ξ0 ξ)− (5.22) I ∼ ± − ± − 17

1 where c some ξ-independent constant spinors 1. νk < 2 (which depend± on w and k). Also note

1 1 Let us ﬁrst look at the inner region contribution. Equa- gξξ , Y1,3 O(1), Y2 (ξ0 ξ) 2 . (5.23) ∼ ξ0 ξ ∼ ∼ − tions (5.9) and (5.12) give the leading order temperature − dependence as One can then check that the integrals for the IR part (5.7) are always convergent near the horizon

ξ0. (0) 2 1 2νk λ (a ) T − + (5.24) ∝ + ···

B. Low temperature behavior where we have also used that √gtt T . The outer region ∝ 0 With the preparations of last subsection, we will now contribution λUV starts with order O(T ) and we can proceed to work out the low temperature behavior of the thus ignore (5.24) at leading order. The full vertex can effective vertices (4.54) and (4.56), which in turn will play be written as an essential role in our discussion of the low temperature behavior of the DC and optical conductivities in Sec. VI. 1 2νk The stories for (4.54) and (4.56) are rather similar. For λ(ω, k)= λ0(k)+ O(T − ) (5.25) illustration we will mainly focus on (4.56) and only point out the differences for (4.54). The qualitative behavior of the vertices will turn out to depend on the value of νk. where λ0(k) is given by the zero-th order term of λUV We will thus treat different cases separately. and can be written as

3 3 ∞ r r (0)T qR 3 R (0) R (0)T 2 (1) λ (k)= dr√g − ∗ Φ σ + C σ Φ ik √h Φ σ Φ (5.26) 0 rr r − r 8r3 1 − C r3 Zr∗

where we have taken rc r in the lower limit of the The divergence is of course due to our artificial separation integral (as commented below→ ∗ (5.7)), and have plugged of the whole integral into the IR and UV regions and in the explicit form of Y1,2,3 From (5.16)–(5.20) it can there should be a corresponding divergence in λIR in the also be readily checked that the integral is convergent on limit ξ 0 to cancel the one from (5.26). What the c → both ends. Note that λ0(k) is independent of both T and divergence signals is that the leading contribution to the ω and real. full effective vertex now comes from the IR region, as Similarly for (4.54), one has the UV region integral is also dominated by the IR end. Thus for a generic momentum k, the effective vertex λ 1 2νk Λ(ω1,ω2, Ω; k)=Λ0(w1, w2,s; k)+ O(T − ) (5.27) has the leading behavior

(0) 2 1 2νk λ(ω, k) (a ) T − + . (5.30) with the leading term Λ0 given by ∼ + ··· A slightly tricky case is ν = 1 , for which λ has a loga- Λ (w , w ,s; k)= λ (k) . (5.28) k 2 0 0 1 2 0 rithmic divergence and could lead to a log T contribution once the divergence is canceled. We have not checked its existence carefully, as it will not affect the leading be- 2. ν ≥ 1 k 2 havior of the DC and optical conductivities (as will be clear in the discussion of next section). Thus in what fol- 1 1 0 When νk 2 , the inner region contribution (5.24) is lows, it should be understood that for νk = 2 , the O(T ) no longer negligible≥ for generic momentum. Closely re- behavior in (5.30) could be log T . lated to this, the leading outer region contribution, which It can also be readily checked that for generic k, the is given by (5.26), now becomes divergent at the lower effective vertex Λ has the same temperature scaling as λ. end (near r ). More explicitly, from equations (5.17)– At a Fermi surface k = k , a(0)(k ) = 0 [15], for which ∗ F + F (5.20) we find that as r r , the integrand of (5.26) the leading order term in (5.30) vanishes. Thus near a → ∗ behaves schematically as Fermi surface, which is main interest of this paper, we need also to examine subleading terms. Plugging (5.13) (0) 2 2νk 0 (a+ ) (r r )− + O((r r ) )+ . (5.29) into the expression (4.56) for the vertex, we find that near − ∗ − ∗ ··· 18 kF the temperature dependence of λ (including both IR as from (5.29) λ0 is finite at kF . Note that expres- 13 1 and UV contributions) can be written as sion (5.32) applies to all νkF including νkF < 2 .

2 1 2νk From (5.32), note that at the Fermi surface k = kF , λ(ω, k)= B(ω, k)(a+(k,ω,T )) T − + λ0,finite + O(T ) (5.31) where B(k,ω) is a smooth function of k and non- 0 3 O(T ) νkF < 2 vanishing near kF . At low temperatures it scales with λ(ω, kF ) (5.34) 3 2νkF 3 0 ∼ (O T − νkF 2 temperature as O(T ). λ0,finite denotes the finite part ≥ of (5.26), which is again ω and T -independent, and a smooth function of k (also at k ). For k k . O(T ), F − F and the vertex develops singular temperature dependence using (5.14) we can further write (5.31) as for ν 3 . However, at the generalized Fermi momen- kF ≥ 2 2 2 1 2νk tum kF (ω,T ), the singular contribution is suppressed. λ(ω, k)= B(ω, k )c (k k (ω,T )) T − F +λ (k )+ F 1 F 0 F This structure will be important below in understand- − (5.32)··· ing the low temperature behavior of the DC and optical where k (ω,T ) was the “generalized Fermi momentum” F conductivities. introduced in (5.15). In (5.32) we have also used Similarly the vertex (4.54) can be written for k k . − F λ0,finite(kF )= λ0(kF ) (5.33) O(T ) as

2 1 2νk Λ(ω ,ω , Ω, k)= B˜(ω ,ω , Ω, k )c (k k (ω ,T ))(k k (ω ,T ))T − F + λ (k )+ (5.35) 1 2 1 2 F 1 − F 1 − F 2 0 F ···

where B˜(ω1,ω2, Ω, k) is a smooth function of k, which in Appendix F2. We will continue to follow the notations 0 ω 1 scales with temperature as O(T ), and we have introduced in (5.3) with below w = T and f(w)= ew +1 . used (5.28). To summarize the main results of this section: A. DC conductivity 1 0 1. For νk < 2 , the eﬀective vertices are O(T ) for all momenta. For both the DC and optical conductiv- Let us ﬁrst consider the leading low temperature de- ities they are given by λ0(k) of (5.26), which is a pendence of the DC conductivity (4.50) which we copy smooth function of k. here for convenience 2. For ν 1 , the vertices develop singular temper- dw ∂f(w) k 2 σDC = I(w,T ) (6.1) ≥ 1 2νk − 2π ∂w ature dependence for generic momenta as T − . Z But near the Fermi surface (more precisely at the ω with w = T and (dropping all the super and subscripts) generalized Fermi momentum kF (ω,T )) the singular contribution is suppressed. C ∞ d 2 2 2 I(w,T ) dkk − ρB(w,k,T ) λ (w,k,T ) . (6.2) ≡ 2 0 3. For all values of νkF , the vertices for the DC and Z optical conductivities are given by (5.32) and (5.35) Note that as a function w, the Fermi function f(w) is respectively. independent of T , thus all the T -dependence of σDC is contained in the momentum integral I(w,T ). As discussed in [15] (and reviewed in Appendix D), for VI. EVALUATION OF CONDUCTIVITIES generic momentum, the spinor spectral function ρB has leading low temperature dependence

With the behavior of the effective vertices in hand we 2νk ρB T , (6.3) can now finally turn to the main goal of the paper: the ∼ leading low temperature behavior of the DC and optical Using (6.3), (5.25) and (5.30), we find that for a generic conductivities. We will first present the leading temper- momentum (up to possible logarithmic corrections) ature scaling and then calculate the numerical prefactors 4νk 1 in the last subsection. In the discussion below we will 2 2 T νk < 2 ρBλ 2 1 (6.4) only consider a real νk. Depending on the values of q and ∼ T ν ( k ≥ 2 m, there could be regions in momentum space where νk is imaginary, referred to as oscillatory regions in [14, 15]. where the leading contribution of the first line (for νk < 1 We consider the contribution from an oscillatory region 2 ) comes from the UV part of the vertex, while for the 19

second line the leading contribution comes from the IR kF /µ) which can be obtained from the retarded function part of the vertex. in AdS2 evaluated at kF (see (D43)–(D47) for explicit Near a Fermi surface as reviewed at the end of Ap- expressions). pendix D Now let us consider the momentum integral (6.2) near 2h ImΣ a Fermi surface. For this purpose it is convenient to ρ = 1 . (6.5) B (k k (ω,T ) Re Σ)2 + (Im Σ)2 introduce a new integration variable − F −

where h1 is a positive constant, kF (ω,T ) is given y = k k (ω,T ) (6.7) by (5.15) and − F 2ν Σ= T kF g(w) . (6.6) in terms of which (6.2) can be written to leading order g(w) is a T -independent scaling function (depending on as

2 2 d 2 ∞ ImΣ 2 2 1 2νk 2 I(w,T ) =2Ch k − dy B(k )c y T − F + λ (k ) + (6.8) FS 1 F (y Re Σ)2 + (Im Σ)2 F 1 0 F ··· Z−∞ −

where we have used (5.32). Now the key is that since We emphasize that in the above derivation it is crucial 2νk 2νk Σ T F , by scaling y T F y, the term propor- that the same k kF (ω,T ) appears in both the spec- tional∼ to A in the last parenthesis→ becomes proportional tral function (6.5)− and the effective vertex (5.32). As a 1+2ν to T kF and can be ignored. Now the integral can be result the leading contribution to the DC conductivity straightforwardly evaluated and we find that is dominated by the UV part of the effective vertex due 1 to suppression at kF (ω,T ), despite that for νk > 2 the

C′ 2νk vertex is generically dominated by the IR part. I(w,T ) = T − F (6.9) FS 2Im g(w) Finally note that in writing down (6.11) we have as- sumed there is a single Fermi surface. In the presence where of multiple Fermi surfaces, the contribution from each of them can be simply added together and the one with the 2 d 2 2 C′ =2πCh1kF− λ0(kF ) . (6.10) largest νkF dominates.

Clearly (6.9) dominates over the contribution from regions of momentum space away from a Fermi surface B. Optical conductivity which from (6.4) can at most be O(T 0).14

Plugging (6.9) into (6.1), we then ﬁnd that for all νkF Let us now look at the optical conductivity, which the DC conductivity has the following leading low tem- from (4.49) can be written as perature behavior C dω1 dω2 f(ω1) f(ω2) σ(Ω) = − I(ω1,ω2, Ω,T ) 2νk −iΩ 2π 2π ω1 Ω ω2 iǫ σ = αT − F (6.11) DC Z − − − (6.13) where where α is a numerical prefactor given by ∞ d 2 I(ω1,ω2, Ω,T ) dkk − (ω1,ω2, Ω, k) (6.14) C′ dw ∂f(w) 1 ≡ I α = . (6.12) Z0 − 2 2π ∂w Im g(w) Z with

We will discuss the numerical evaluation of α in Sec. VI C. (ω1,ω2, Ω, k) ∂f I Note that since both and Im g(w) are positive and = ρB(ω1, k) Λ(ω1,ω2, Ω, k) Λ(ω2,ω1, Ω, k)ρB(ω2, k) . − ∂w even, the integral in the above expression is manifestly (6.15) positive. Recall that the vertex Λ satisﬁes (5.1) which implies that

σ(Ω) = σ∗( Ω) (6.16) − 14 See Appendix F 2 for a discussion of the contribution from oscil- as one would expect since the system has time-reversal latory regions which is again at most of order O(T 0). symmetry. 20

1. Temperature scaling ing contribution comes from the IR part of the vertex. Thus one could at most get All the temperature dependence of (6.13) is in I(ω1,ω2, Ω,T ) which we examine ﬁrst. As in (6.3)–(6.4) I(ω ,ω , Ω,T ) O(T 0) (6.18) one ﬁnds that away from a Fermi surface has the fol- 1 2 ∼ lowing leading T -dependence I from regions of momentum space away from a Fermi sur- 4νk 1 T νk < 2 face. (ω1,ω2, Ω, k) 2 1 (6.17) I ∼ T νk Near a Fermi surface Λ has the low temperature expan- ( ≥ 2 sion (5.35) and ρB is given by (6.5)–(6.6). Introducing 1 where for νk < 2 the leading contribution comes from the y = k kF (ω1,T ), then the integral has the following UV part of the vertex, while for the second line the lead- structure−

ImΣ1 ImΣ2 I(ω1,ω2, Ω,T ) FS dy 2 2 2 2 | ∼ (y Re Σ1) + (ImΣ1) (y + δ Re Σ2) + (ImΣ2) Z − 1 2νk − 1 2νk b y(y + δ)T − F + λ (k ) b y(y + δ)T − F + λ (k ) (6.19) × 1 0 F 2 0 F where The corresponding temperature scaling of (6.19) depends 2ν on the range of δ. For δ O(T kF ), one has 1 ∼ δ (ω1 ω2), (6.20) ≡ vF − 2νk 2νk I(ω ,ω , Ω,T ) T − F , δ O(T F ) (6.21) 1 2 |FS ∼ ∼ Σ1,2 Σ(ω1,2), and b1,2 are some y-independent functions≡ of ω ,ω , Ω which scale with temperature as O(T 0). while for δ O(T ), one ﬁnds 1 2 ∼ For y O(T 0) (i.e. away from the Fermi surface) the in- ∼ tegrand scales as (6.17). Near the Fermi surface, i.e. in 2 2νkF 1 λ0T − νkF < 2 the range y . O(T ), as in the analysis of (6.8), due to I(ω1,ω2, Ω,T ) FS . (6.22) 2 2νkF 2 1 2νk | ∼ (λ0T − νkF 2 that Σ1,2 T F , the dominant contribution in the y- ≥ ∼ 2ν integral comes from the region y O(T kF ). One then ﬁnds from a simple scaling that∼ the term proportional To summarize, the contribution from near the Fermi 2 to λ0(kF ) (i.e. the UV part of the vertex) is dominating. surface is given by

2 Cλ0(kF ) d 2 dω1 dω2 f(ω1) f(ω2) σ(Ω) = dkk − − ρ (ω , k)ρ (ω , k)+ (6.23) − iΩ 2π 2π ω Ω ω iǫ B 1 B 2 ··· Z Z 1 − − 2 − which is of the form of that for a Fermi liquid in the The latter integral can be done straightforwardly (see absence of vortex corrections. Appendix G2 for details) and gives

(FS) 2 d 2 2. Contribution from Fermi surface I (ω1,ω2, Ω,T ) = 2πh1λ0(kF )kF− ρB(ω2,K2) 2 d 2 = 2πh1λ0(kF )kF− ρB(ω1,K1) Now let us look at the contribution from the Fermi (6.25) surface in detail and work out the explicit frequency dependence. As discussed above we only need include the where K k (ω ,T )+Σ∗(ω ) and K k (ω ,T )+ UV part of the eﬀective vertex, which gives 2 ≡ F 1 1 1 ≡ F 2 Σ∗(ω2). We now plug (6.25) into (6.13) and evaluate one (FS) 2 d 2 of the frequency integral as follows. Split the integrand I (ω1,ω2, Ω,T )= λ0(kF )kF− dkρB(ω1, k)ρB(ω2, k) into two terms; in the one with the f(ω2), we use the Z (6.24) second line of (6.25) and do the ω1 integral, which can 21

be written as with c(kF ) given by (D45). In obtaining (6.31) we have made a change of variable w = s (y 1) in (6.30) and 2 − dω1 ρB(ω1,K1) R taken the large s limit. Note that the falloﬀ in (6.31) = G (ω2 +Ω,K1) 2π ω1 Ω ω2 iǫ is much slower than the Lorentzian form familiar from Z − −h − = 1 (6.26) Drude theory. The behavior (6.29) and (6.31) are indica- Ω +Σ (ω ) Σ(ω + Ω) vF ∗ 2 2 tive of a system without a scale and with no quasiparti- − − cles. where in the ﬁrst line we used the spectral decomposition 1 2. νkF > : in this case there are two regimes: of the boundary fermionic retarded function GR and the 2

Ω 2νkF 1 second line used (D39). Similarly for the term with f(ω1) 2a. with u = 2ν = fixed and s = uT − 0, we T kF → we can use the first line of (6.25) and do the ω2 integral, find (6.28) becomes which gives 2νk σ(Ω) = T − F F2(u) (6.33) dω2 ρB(ω2,K2) h1 = Ω . with 2π ω1 Ω ω2 iǫ + Σ(ω1) Σ∗(ω1 Ω) Z − − − vF − − (6.27) C ∂f(w) 1 F (u)= ′ dw (6.34) Combining them together we thus find that 2 2πi ∂w u +2iIm g(w) Z vF C′ dω f(ω) f(ω + Ω) Since ∂f is peaked around w = 0, we can approx- σ(Ω) = Ω − (6.28) ∂w iΩ 2π +Σ∗(ω) Σ(ω + Ω) imate the above expression by setting g(w) to its Z − vF − value at w = 0, leading to a Drude form where C′ was introduced before in (6.10). It is now man- 2ν 2 iC T kF 1 ω ifest from the above equation that in the Ω 0, we σ(Ω) ′ − = p (6.35) recover (6.11). This confirms the claim below (4.43→ ) that ≈ 2π u +2iIm g(0) 1 iΩ vF τ in (4.42) vanishes at leading order at low temperatures. − S We now work out the qualitative Ω-dependence with

of (6.28) which has a rich structure depending on the vF C′ 1 2 2νkF value of ν . As stated earlier we work in the low tem- ωp , 2Im g(0)vF T . (6.36) kF ≡ 2π τ ≡ perature limit T 0 with s = Ω ﬁxed. → T This behavior is consistent with charge transport 1. 1 : νkF < 2 in this case given (6.6), to leading order we from quasiparticles with a transport scattering rate can ignore the term proportional to Ω in the downstairs 2νk given by τ T − F . Furthermore we could inter- of the integrand of (6.28). Then σ(Ω) can be written in ∝ pret C′ as proportional to the quasiparticle density. a scaling form Indeed from (6.10) it is proportional to the area of

2νk the Fermi surface. Note that λ0(kF ) in C′ can be σ(Ω) = T − F F1(Ω/T ) (6.29) interpreted as the effective charge of the quasiparticles. with F1(s) a universal scaling function given by Ω 2b. For s = T = fixed, the two Σ terms in the down- dw f(w + s) f(w) 1 stairs of the integrand of (6.28) are much smaller F1(s)= C′ − 2π is g(w + s) g∗(w) than the Ω term, and we can then expand in power Z − (6.30) series of Σ, with the lowest two terms given by with g given by (D44). In the s 0 limit we recover → iω2 the DC conductivity (6.11). In the limit s , which p 2νk 1 σ(Ω) = 1+ T F − k(s)+ (6.37) corresponds to the regime T Ω µ, using→ ∞ (D46) we Ω ··· 15 ≪ ≪ find that with

2νk σ(Ω) = C′′( iΩ)− F (6.31) vF − k(s)= dw (f(w) f(w + s))(g∗(w) g(w + s)) . s2 − − Z where C′′ is a real constant given by (6.38) In the large s limit using (D46) we ﬁnd 1 C′ dy C′′ = 2νk 2νk 2νk 1 4vF h2Im c(kF ) −4πih2 1 c(kF )(1 + y) F c∗(kF )(1 y) F k(s) a( 2is) F − , a = (6.39) Z− − − (6.32) →− − 2νkF +1 in which case σ(Ω) (i.e. for T Ω µ) can be written as ≪ ≪ 2 15 Note that in our setup µ is a UV cutoﬀ scale, thus we always iω p 2 2νk 2 σ(Ω) = 2aω ( 2iΩ) F − + . (6.40) assume Ω ≪ µ. Ω − p − ··· 22

The leading 1/Ω piece in (6.37) gives rise to a compute σ(Ω) for a fixed m and q, we need the following term proportional to δ(Ω) with a weight consistent quantities: with (6.35). The subleading scaling behavior may Fermi momentum: k (m, q). be interpreted as contribution from the leading ir- • F 1 relevant operator. At T = 0, Re G− (k,ω = 0) changes its sign at the Fermi momentum. We determine the location Note that in both regimes discussed above the temper- of this sign change using the Newton method (up ature scalings are consistent with those identified ear- to 40 iterations). The algorithm needs an initial k lier in (6.21)–(6.22). For real part of σ(Ω) we have value where the search starts. This initial value was δ ω ω Ω as constrained by the delta function re- 1 2 set by empirical linear fits on kF (m, q). When there sulting∝ − from∝ the imaginary part of 1 in (6.23), ω1 Ω ω2 iǫ were multiple Fermi surfaces, we picked the pri- − − − while for the imaginary part of σ(Ω) the dominant term mary Fermi surface (the one with the largest kF ). i (i.e. the term proportioal to Ω ) comes from the region 1 2νk Computing G− involves solving the Dirac ω1 ω2 O(T F ). equation in the bulk. We used Mathemat- − ∼1 3. νkF = 2 : the Marginal Fermi liquid, for which the Ω ica’s NDSolve to solve the differential equation term in the downstairs of (6.28) is of the same order as using AccuracyGoal/PrecisionGoal = 12...22, Σ, and we have and WorkingPrecision = 70. Typical IR and UV 12 20 25 40 cutoffs are 10− ... 10− and 10− ... 10− , re- 1 Ω T σ(Ω) = T − F3 , log (6.41) spectively. The resulting kF values are typically T µ th accurate to the 10 digit. where F2 can be written as Numerator of the Green’s function: h (m, q). • 1 dw f(w + s) f(w) 1 The numerator of the fermionic Green’s function F3 = C′ − s is determined by fitting a parabola on six data 2π is + g(w + s) g∗(w) vF 1 Z − points of G− (k,ω = 0) near the Fermi surface (i.e. (6.42) 5 5 k = k 10− ...k + 10− ), and then taking the with g(w) now given by (D47). Due to time reversal F − F derivative of the parabola at k = kF . The compu- symmetry the real part σ1(Ω) of σ(Ω) is an even function in Ω and thus for Ω/T < 1, one can again approximate tation of redundant data points makes the result- σ(Ω) by a Drude form with the transport scattering time ing h1 value somewhat more accurate, but its main τ 1 . For Ω T , using (D48) we find that function is to monitor the stability of the numerics: ∝ T ≫ whenever the six points are not forming an approx- 1 C 1 1 1+ iπ 1 imately straight line, we know that the kF finding σ(Ω) = ′ + + + algorithm has failed. In this case, we need to go Ω Ω 2 Ω 2πic1 log (log ) 2 vF c1 ··· T T ! back and “manually” obtain the value of the Fermi (6.43) momentum. which is analogous to (6.31), but with logarithmic modi- fications. Recall that there are no logarithmic corrections Self-energies: Σ(ω; m,q,T ). • for the DC conductivity (6.11). Let Σ(ω) denote the self-energy at the Fermi surface with the linear ω term included. Then, vF C. Numerical coefficients e h1 GR(ω, k)= (6.45) (k k ) Σ(ω) In this section, we discuss the numerical computation − F − of the conductivities in 2+1 boundary dimension (d = 3). Since we already know h1 andekF , we determine Σ by computing the fermionic Green’s function at 1. Optical conductivity k = kF . Computing both Σ∗(ω; m,q,T ) and Σ(ω + Ω;e m,q,T ) then gives the denominator of (6.44). e e The optical conductivity is given by (6.28) that we Effective vertex: Λ(ω1,ω2, Ω, k; m, q). copy here for convenience • The numerical code computed the frequency- i C′ dω f(ω) f(ω + Ω) dependent Λαβ(ω1,ω2, Ω, k; m, q) (see (4.26) for an σ(Ω) = − (6.44) Ω explicit formula) instead of the simpler λ0(kF ). iΩ 2π +Σ∗(ω) Σ(ω + Ω) Z − vF − We determine Λ(ω1,ω2, Ω, k) by first numeri- 2 d 2 2 where C′ = 2πCh1kF− λ0(kF ). The formula implicitly cally computing KA(r, Ω), which is the bulk- depends on the bulk fermion mass m and charge q. We to-boundary gauge field propagator with ingoing are using this version since it only contains one ω inte- boundary conditions at the horizon. (Note that at gral and it is easier to evaluate than (4.39). In order to Ω = 0 this may be done analytically.) We then 23

compute the spinor propagator with normalizable coefficient is computed by a fit using α(m, q) = 2νk UV boundary conditions at both ω1 and ω2 and σDC (m,q,T )T F . In the m q space, the resolution also compute the Λ integral using a single NDSolve was 45 45 with computation time− approx. 22 hours. The call. The integration proceeds towards the horizon results× are seen in FIG. 8. The plot shows log α(m, q) us- where it oscillates somewhat before converging. ing a color code. The numerically unstable areas (with error larger than 3%) are colored gray in the figure. For

νkF > 1.1 (in the G2 component) the numerical inaccu- 500 2.0 racies became too large. For q < 0.3, the automated kF 400 1.5 0.008 300 1.0 1 200 Re I, Im I Re I, Im I 0.5 F 0.006 100

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 , Im

1 0.004

Ω Ω F

Re 0.002 FIG. 6: Typical functions whose integral gives the conduc- ~ H-iΩL-2 Ν ∞ tivity: σ(Ω) = R−∞ I(ω)dω. The two ﬁgures correspond to 0.000 Ω ∼ T and Ω ≫ T . The real and imaginary parts are indi- -6 -4 -2 0 2 4 6 8 cated by blue and orange colors, respectively. logHΩTL

0.015 2

By using the above quantities, we compute the con- F Σ ~ 0 ductivity at a ﬁxed Ω and T by performing the integral 0.010 1 - i ΤΩ , Im over ω in (6.44). The Fermi functions suppress the in- 2 F

tegral exponentially outside a certain window set by the Re 0.005 parameters, see FIG. 6. The size of this window can - ~ iΩ 1 be determined and is used to automatically set the in- 0.000 tegration limits. The integrand is computed at 15 ... 30 -6 -4 -2 0 2 4 points. We used Mathematica’s parallel computing ca- logHΩT 2 ΝL pabilities in order to compute three data points at the same time. FIG. 7: These ﬁgures show the scaling functions for the op- Fig. 7 show the scaling functions F1,2 deﬁned in the tical conductivity. Top: Real and imaginary parts of the 1 1 scaling function F1 for m = 0, q = 1, where the IR fermion ex- previous section for νkF < and νkF > . The behavior 2 2 ν ≈ . Bottom: in limiting cases agrees with the analysis above. ponent is kF 0 24. Real and imaginary parts of the scaling function F2 for m = 0, q = 2, where the IR fermion

exponent is νkF ≈ 0.73 and hence we are in the regime with a 2. DC conductivity stable quasiparticle. As indicated in the ﬁgure, both real and imaginary parts resemble the Drude behavior.

The DC conductivity can be computed using (6.11).

ω 2 2 kF C ∂f T h1 λ (ω; T ) σDC (T )= dω (6.46) ﬁnding algorithm typically failed and we had to deter- − 2 ∂ω ImΣ(ω, kF ,T ) Z mine kF manually.

The computation of kF , h1 and Σ was detailed in the Note the deep blue line in the G1 spinor component. previous subsection. The ω derivative inside λ(ω; T ) is At these points, the effective vertex λ0(kF ) changes sign computed by taking the difference of the wavefunctions and therefore the leading contribution to the DC conduc- 6 8 with ∆ω = 10− ... 10− . The numerical AC conductiv- tivity vanishes (so does the leading contribution to the ity in the zero frequency limit matched the output of the optical conductivity since it is also proportional to λ0). DC conductivity code. Since the DC effective vertex is real, this happens along a For a given pair (m, q), the DC conductivity is codimension one line in the m q plane. This ‘bad metal’ − computed at different temperatures between T = line crosses the νkF = 1/2 line at around m 0.18 and 4 8 ∼ 10− ... 10− . Then, the temperature-independent α q 3.4 (not in the figure). ∼

VII. DISCUSSION AND CONCLUSIONS One can package all radial integrals into eﬀective vertices in a way that makes it manifest that the actual conduc- Despite the complexity of the intermediate steps, the tivity is completely determined by the lifetime of the one- result that we ﬁnd for the DC conductivity is very simple. 24

FIG. 8: The plots show the coeﬃcient log α(m, q) for the primary Fermi surface for both spinor components. In the white regions there is no Fermi surface. Black lines indicate half-integer νkF (m, q). In the two gray regions in G2 the numerical computations were unreliable (with error greater than 3%).

scattering rate [65]. Such systems are therefore better Ψ$ Ψ$ metals than one would have guessed from the single- !" "$ particle lifetime. In our calculation, the current dissipation is more efficient. To understand why, note that in our gravity treatment the role played by the gapless boson in the above example is instead filled by the AdS2 region. From a field FIG. 9: The system can be described by a low energy ef- theoretical point of view, our system can be described fective action where fermionic excitations Ψ around a free by a low energy effective action [15, 72, 73] in which fermion Fermi surface hybridize with those in a strongly cou- fermionic excitations Ψ around a free fermion Fermi sur- pled sector (labelled as SLQL in the figure) described on the face hybridize with those of a strongly coupled sector, gravity side by the AdS2 region [15, 72, 73] (see [19] for a which can be considered as the field theory dual of the more extensive review). AdS2 region and was referred to as a semi-local quantum liquid (SLQL) in [74]. See Fig. 9. The SLQL provides a set of fermionic gapless modes to which the excita- particle excitations, as is clear from the formula (6.23). tions around the Fermi surface can decay. In our bulk treatment this process has a nice geometric interpreta- We should stress that from a field-theoretical point of tion in terms of the fermion falling into the black hole, view this conclusion is not a priori obvious, as the single- as in Fig. 2. The crucial point is that because of the particle lifetime measures the time needed for the par- semi-local nature of the SLQL–as exhibited by the self- ticle to decay, whereas the conductivity is sensitive to energy (6.6)–there are gapless fermionic modes for any the way in which it decays. For example, if the Fermi momentum16. Thus the phase space for scattering is not quasiparticles are coupled to a gapless boson (as is the sensitive to the momentum transfer, and the conductiv- case in many field-theoretical constructions of non-Fermi- ity is determined by the one-particle lifetime. Note that liquids; see e.g. [1, 55–71] and references therein), small- this scenario for strange metal transport is rather similar momentum scattering is strongly preferred because of to that discussed before in [75, 76] as reviewed in [77]. the larger phase space available to the gapless boson at smaller momenta. However, this small-momentum scattering does not degrade the current and so contributes differently to the conductivity than it does to the single- 16 This is similar to that postulated for the bosonic fluctuation particle lifetime, meaning that the resistivity grows with spectrum in the MFL description of the cuprates [5]. But an im- temperature with a higher power than the single-particle portant distinction is that here the gapless modes are fermionic. 25

We find that the conductivity at the Marginal Fermi frame where 1 Liquid point νkF = 2 – when the single-particle spectral function takes the MFL form – is consistent with a linear jt ρ =0,T tt = ǫ =0,T ii = P, ji =0, πi T ti = 0 ; resistivity, just as is observed in the strange metals. The ≡ 6 6 ≡ (A1) correlation between the single-particle spectral function ǫ is the energy density and P is the pressure. Boost18 by and the collective behavior and transport properties is a a velocity ui to a frame where strong and robust prediction of our framework. While it is fascinating that this set of results is self-consistent, we ji = uiρ, πi = ui(ǫ + P ) . (A2) 1 do stress that the marginal νkF = 2 point is not special from our gravity treatment, and more work needs to be This gives done to understand if there is a way to single it out in holography.17 ρ ji = πi , (A3) ǫ + P Acknowledgements which is effectively a constitutive relation. In a non- relativistic system, the enthalpy ǫ + P reduces to the We thank M. Barkeshli, E. Fradkin, T. Grover, S. Hart- mass of the particles. Combining this with conservation noll, G. Kotliar, P. Lee, J. Maldacena, W. Metzner, of momentum (Newton’s law) S. Sachdev, T. Senthil, and B. Swingle for valuable i i discussions and encouragement. During its long ges- ∂tπ = ρE (A4) tation, this work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under and Fourier transforming gives cooperative research agreements DE-FG0205ER41360, DE-FG02-92ER40697, DE-FG0205ER41360, and DE- i ρ2 SC0009919, and the OJI program, in part by the Al- ji(Ω) = Ei(Ω) (A5) fred P. Sloan Foundation, and in part by the National Ω (ǫ + P ) Science Foundation under Grant No. NSF PHY05-51164 and PHY11-25915. All the authors acknowledge the hos- and hence pitality of the KITP, Santa Barbara at various times, in πρ2 particular the miniprogram on Quantum Criticality and Re σ(Ω) = δ(Ω) (A6) the AdS/CFT Correspondence in July, 2009, and the pro- ǫ + P gram Holographic Duality and Condensed Matter Physics plus, in general, dissipative contributions. Note that in in Fall 2011. TF was also supported by NSF grant PHY systems where the relation J~ = ρ ~π is an operator 0969448, PHY05-51164, the UCSB physics department ǫ+P and the Stanford Institute for Theoretical Physics. DV equation, momentum conservation implies that there are was also supported by the Simons Institute for Geometry no dissipative contributions, and the conductivity is ex- and Physics and by a COFUND Marie Curie Fellowship, actly given by (A6). This is indeed consistent with the and acknowledges the Galileo Galilei Institute for Theo- leading term we obtained in (2.44). retical Physics in Florence for hospitality during part of The Fermi surface contribution which is main result of this work. the paper does not contain a delta function in Ω, as the Fermi surface current can dissipate via interactions with the O(N 2) bath. Although the total momentum (of the Fermi surface plus bath) is conserved, the time it takes the bath to return momentum given to it by the Fermi Appendix A: Resistivity in clean systems surface degrees of freedom is parametrically large in N, as in probe-brane conductivity calculations [80]. The DC conductivity we obtain is averaged over a long time that In a translation-invariant and boost-invariant system is of (N 0). O at finite charge density (without disorder or any other Our discussion here is somewhat heuristic, but a more mechanism by which the charge-carriers can give away careful hydrodynamic analysis that also takes into ac- their momentum), the DC resistivity is zero. An applied count the leading frequency dependence was performed electric field will accelerate the charges. This statement e.g. in [79], where it was explicitly shown that the pres- is well-known but we feel that some clarification will be ence of impurities broadens this delta function into a useful. It can be understood as follows. Start with a Drude peak. uniform charge density at rest and in equilibrium, in a

18 We perform a small boost ui ≪ c, so we can use the Galilean 17 See [51, 78] for recent work in this direction. transformation, even if the system is relativistic. 26

Appendix B: Mixing between graviton and vector Finally, the dynamical equations for ai are field ∂ πµ + √ g∂ f νµ =0 (B12) − r − ν In this section we construct the tree-level equations of We turn now to the gravitational fluctuations. At this motion for the coupled vector-graviton fluctuations about point it is helpful to specialize to the zero momentum the charged black brane background. The action can be limit, i.e. all fluctuations depend only on t and r. Now written in the usual form, all spatial directions are the same, and so we pick one 2 y 1 R direction (calling it y) and focus only on hα, with α = S = dd+1x √ g 2Λ F F µν (B1) 2κ2 − R− − g2 µν (t, r), and where the indices are raised by the background Z F metric. We will then find a set of coupled equations for y y with background metric given by ay and ht (and ar and hr , which will be set to zero in the end). The relevant equations then become 2 2 2 2 ds = gttdt + grrdr + giidxi (B2) − y rr yy y π = √ gg g ay′ ht (B13) and a nonzero background profile A0(r). It is convenient − − −Q to work with the radial background electric field Er = ∂rA0, which satisfies the equation of motion y yy tt 2 ∂rπ √ gg g ω ay =0 (B14) ∂ (E √ ggttgrr)=0 . (B3) − − r r − We will denote the r-independent quantity 2 rr tt y 2 2κ C1 ay = √ ggyyg g ∂rht (B15) tt rr 2 gF Q − − Er√ gg g = κ ρ (B4) Q≡− − 2R2 Taking a derivative of the first equation with respect to where we have used (2.7). r one can derive an equation for ay alone Now consider small fluctuations rr yy 2 2 grrgtt yy tt 2 ∂r(√ gg g ay′ )+ 2κ C1 √ gg g ω ay =0 A A δ + a , g g + h . (B5) − Q √ ggyy − − M 0 M0 M MN MN MN → → − (B16) We seek to determine the equations of motion for these Note now that using fluctuations; we first use our gauge freedom to set 1 g = fr2, g = , g = r2 (B17) hrM = ar =0 . (B6) tt rr r2f ii At quadratic level in fluctuations the Maxwell action we find that (B16) becomes can now be written as 2 d 5 d 1 d 1 ω r − d+1 1 MN ∂ r − fa′ + r− − a =0 (B18) SEM = C1 d x √ gfMN f r y CQ − f y − 4 − 1 Z E √ ggttgrr + √ ggtrgtr E In the last expression we introduced the constant −2 r − − (2) r tt rr Er(g g √ g)(1)f0r 2 2 2κ C1 =2κ ρ . (B19) − i −i C ≡ Q htfir + hrfti (B7) −Q 4√3 r∗ 2 In d = 3, d=3 = . with C| gF R Equation (B15) implies a corresponding relation be- 2 2R tween the bulk-to-boundary propagators Ka,Kh of the C1 = 2 2 , fMN = ∂M aN ∂N aM (B8) gF κ − metric and gauge field, which is important for our calculation: The canonical momentum for aµ is then given by rr tt Ka = √ ggyyg g ∂rKh . (B20) i 1 δS rr ii i C − − π = = √ gg g fri ht (B9) C1 δ∂rai − − −Q Appendix C: Spinor bulk-to-bulk propagator πt = √ ggrrgrrf + √ ggrrgrr E (B10) − 0r − (1) r 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 in the bulk and leads to the conservation of this case when the dimension d of the boundary theory is odd; canonical momentum, the correspondence between bulk and boundary spinors is different when d is even, and though a parallel treat- µ ∂µπ =0 (B11) ment can be done we shall not perform it here. We denote 27 by the dimension of the bulk spinor representation; in 2. Green functions N d+1 the case that d is odd we have =2 2 . Our treatment N will essentially apply to any asymptotically AdS space- We define the retarded and advanced bulk-to-bulk time with planar slicing and a horizon in the interior; propagator as the criterion of asymptotically AdS is important only in D (t, ~x; r, r′) = iθ(t) ψ(t, ~x, r), ψ¯(0, r′) (C10) the precise choice of UV boundary conditions and can be R { } easily modified if necessary. D (t, ~x; r, r′) = iθ( t) ψ(t, ~x, r), ψ¯(0, r′) (C11) A − − { } whose Fourier transform along boundary directions sat- 1. Spinor equations isfy the equation

M i (Γ m)D (r, r′; ω,~k)= δ(r r′) . (C12) We begin with the bulk spinor action: DM − R,A −√ g − − In the above equations we have suppressed the bulk S = i dd+1x√ gψ¯ ΓM m ψ (C1) − − DM − spinor indices which we will do throughout the paper. Z ~ The spectral function ρ(r1, r2; ω, k) is defined by t where ψ¯ ψ†Γ . From here we can derive the usual ≡ ~ ~ ~ Dirac equation, ρ(r, r′; ω, k)= i(DR(r, r′; ω, k) DA(r, r′; ω, k)) . − − (C13) M (Γ M m)ψ =0, (C2) The Euclidean two-point function is related to DR by the D − standard analytic continuation where the derivative is understood to include both DM ~ ~ the spin connection and couplings to background gauge DE(r, r′; iωm, k)= DR(r, r′; ω = iωm, k) (C14) fields and satisfies the spectral decomposition

1 ab M = ∂M + ωabM Γ iqAM . (C3) dω ρ(r, r′; ω,~k) D 4 − DE(r, r′; iωm,~k)= . (C15) 2π iωm ω The abstract spacetime indices are M,N and the ab- Z − We define the corresponding boundary retarded and stract tangent space indices are a, b, . The··· index with advanced Green functions as follows an underline denotes that in tangent··· space. Thus Γa to denote gamma matrices in the tangent frame and ΓM R Gαβ(t, ~x) = iθ(t) α(t, ~x), β† (0) (C16) those in curved coordinates. Note that {O O } A D E † M a M Gαβ(t, ~x) = iθ( t) α(t, ~x), β(0) (C17) Γ =Γ ea (C4) − − {O O } D E where is the boundary operator dual to the bulk field The nonzero spin connections for (2.2)–(2.3) are given by ψ and Oα, β are boundary spinor indices. The boundary spectral function ρB is defined by 1 gtt′ rr t 1 gii′ rr i ωtr = √g e , ωir = √g e , (C5) R A −2 gtt 2 gii G (ω,~k) G (ω,~k)= iρ (ω,~k), (C18) − B with and is Hermitian

1 1 t 2 i 2 i ρB† = ρB . (C19) e = gttdt, e = gii dx . (C6) From (C16) the linear response relation is From the above one ﬁnds that t r (k) = GR(k)γ χ(k) (C20) 1 M ab Γ rr r hO i ω Γ Γ = ∂ log( gg ) U(r)Γ . (C7) t 4 abM 4 r − ≡ where χ denotes a source, γ is the boundary gamma matrix, and we have suppressed the spinor indices. In momentum space the Dirac equation (C2) can then be Our convention for bulk Gamma matrices is that written explicitly as a t a t t 2 (Γ )† =Γ Γ Γ , (Γ ) = 1 (C21) t i r ~ − i(ω + qAt)Γ + ikiΓ +Γ (∂r + U) m ψ(ω, k; r)=0 and for boundary ones − − (C8) t 2 t t whose conjugate can be written as (γ ) = 1, (γ )† = γ . (C22) − − As mentioned earlier we will focus on odd d, for which ψ¯ i(ω + qA )Γt + ik Γi (←−∂ + U)Γr m =0 . − t i − r − case, there is also a Γ5 in the bulk which anticommutes h i (C9) with all the Γa’s and satisﬁes Applying (C8) and (C9) to ψ and ψ¯ in (4.31) respec- γ β 5 5 5 2 tively and using (C7), one can readily derive (4.31). (Γ )† =Γ , (Γ ) =1 . (C23) 28

3. Bulk solutions Now expand the in-falling solutions in terms of of ψα and the Yα We begin by recalling how to obtain the boundary re- ψin = Y A + ψ B (C35) tarded Green function and some properties of the solu- a α αa α αa tions to the Dirac equation (C2) (see also [29]). where A and B are both N N matrices that connect 2 × 2 Near the horizon r0, it is convenient to choose the in- the infalling and boundary solutions. Identifying A with falling and out-going solutions as the basis of wave func- the source χ (C20), with γt defined as in (C34), one can tions then check19 that B can be identified precisely with . hOi in,out in,out iωσ(r) It then follows that the boundary theory spinor retarded ψ (r; ω,~k) ξ e± , r r (C24) a → a → 0 Green’s function GR can be written as tt t 1 where σ(r) dr grrg , and ξa are constant basis (GRγ )αβ = (BA− )αβ (C36) spinors, which≡ − satisfy the constraint R p with γt the boundary theory gamma matrix. This is the t r in,out covariant generalization of expressions given previously 1 Γ Γ ξa =0 . (C25) ∓ for the boundary fermion Green’s function [29], and will The index a labels different independent solutions. From be useful in what follows. One can find the advanced boundary theory correlator by using outgoing solutions the above equation clearly we have a 1.. N2 . Equa- tion (C25) also implies that for any a,b∈ and their corresponding outgoing expansion coefficient matrices B, A in (C36). ¯in r out ψa Γ ψb =0 . (C26) We now compute some Wronskians that we will need later. Note first that by using the Dirac equation (C2) we We will normalize can show that for any two radial solutions ψ1(r), ψ2(r) in in out out evaluated at the same frequency and momentum, the ξ †ξ = δ , ξ †ξ = δ . (C27) a b ab a b ab Wronskian W [ψ1, ψ2] defined as

rr r Near the boundary r it is convenient to consider W [ψ1, ψ2] √ gg ψ1(ω, k)Γ ψ2(ω, k) (C37) purely normalizable ψ →and ∞ purely non-normalizable Y ≡ − solutions defined respectively by is a radial invariant, i.e. ∂rW = 0. Using (C28) and (C29) at r = one then finds that mR d/2 ∞ ψα(r ) ζα−r− − (C28) r + + t → ∞ → W [ψ , Y ]= ζ¯−Γ ζ = ζ¯−ζ = iγ = W [Y , ψ ] + +mR d/2 α β α β α β αβ α β Y (r ) ζ r − (C29) − α → ∞ → α (C38) where we have used (C34). It can also be readily checked where ζ are constant spinors which satisfy α± that r (1 Γ ) ζ± =0 . (C30) ∓ α W [ψα, ψβ]= W [Yα, Yβ]=0 (C39) Again index α labels different solutions and runs from and r 1 to N (as the two different eigenspaces of Γ span 2 in out in in out out the full spinor space). Since the normalizable and non- W [ψ , ψ ]=0, W [ψa , ψb ]= δab = W [ψa , ψb ]. − normalizable solutions correspond to boundary opera- (C40) tor and source respectively, α can be interpreted as the Also note that boundary theory spinor index. We choose the normaliza- W [ψ , ψin]= i(γtA) , W [Y , ψin]= i(γtB) tion α a αa α a αa − (C41) We can also expand the outgoing solutions as ζα†±ζβ± = δαβ (C31) ψout = Y A˜ + ψ B˜ . (C42) and have the following completeness relation, a α αa α αa with 1 r ζ±ζ†± = (1 Γ ) . (C32) α α 2 ± W [ψ , ψout]= i(γtA˜) , W [Y , ψout]= i(γtB˜) α α a αa α a αa X − (C43) It is also convenient to choose Using the above Wronskians we can also write in t out t + 5 ψα = iψ (A†γ )aα iψ (A˜†γ )aα . (C44) ζα =Γ ζα− (C33) a − a where Γ5 was introduced earlier around (C23). The boundary gamma matrices can then be defined as 19 As discussed e.g. in [29], hOi should be identified with the bound- µ µ + γ = i(ζ−)†Γ ζ . (C34) ary value of the canonical momentum conjugate to ψ. αβ − α β 29

4. Constructing the propagator that the propagator indeed satisfies this equation then it will also satisfy the corresponding equation with the We are now ready to construct the bulk-to-bulk re- differential operator acting from the right and as a func- tarded propagator DR which satisfies the equation (C12) tion of r′, by the equality of left and right inverses. Thus together with the boundary conditions that as either ar- we will only explicitly show that the operator satisfies gument r or r′ r the propagator should behave like the equation in r. For the advanced propagator DA, the → 0 an in-falling wave in (C24), and similarly as r or r′ only difference is that the propagator should behave like the propagator should be normalizable as in (C28).→ Note ∞ a outgoing wave at the horizon. that the Dirac operator in (C12) above acts only on the With the benefit of hindsight, we now simply write left index of the propagator (which is a matrix in spinor down the answer for the bulk-to-bulk retarded and ad- space) and on the argument r; if we can demonstrate vanced propagator

t R,A Yα(r)γαβ ψβ(r′) r < r′ DR,A(r, r′; ω, k)= ψα(r)Gαβ (ω, k)ψβ(r′) t (C45) − (ψα(r)γαβ Yβ(r′) r > r′

We now set out to prove that the above propagators where in the first equality we have used (C38). This is have all of the properties required of them, very few of then consistent with (C50). Similarly contracting (C50) which are manifest in this form. We will discuss DR to the left with ψσ we find explicitly, with exactly parallel story for D . For r > r A ′ t αβ we have iW [ψσ, Yα](γ ) ψβ = ψσ, (C52) R t DR(r, r′; ω, k)= ψα(r) Gαβ(ω, k)ψβ(r′) γαβYβ(r′) which is again satisfied. Note that since ψσ and Yα − (C46) altogether form a complete basis, we have now verified which satisfies (C12) inr, as well as the boundary con- the full matrix equation (C50), and thus the propagator dition that the solution be normalizable as r , as proposed in (C45) is indeed correct. → ∞ the dependence on r is simply that of the normalizable Now given (C45), taking the difference between DR solution ψα. For r < r′ we have and DA, from (C13) and (C18) we thus find that R t B DR(r, r′; ω, k)= ψα(r)Gαβ (ω, k) Yα(r)γαβ ψβ(r′) ρ(r, r′; ω, k)= ψα(r)ραβ (ω, k)ψβ(r′) (C53) − (C47) Now using (C36) and (C35) we can write the above equa- where ρ and ρB are respectively the bulk and boundary tion as spectral density. This is the expression used in (4.12). in 1 t DR(r, r′; ω, k)= ψa (r)(A− γ )aβψβ(r′) (C48) − Appendix D: Boundary spinor spectral functions which satisfies both the defining equation (C12) and the infalling boundary condition for r < r′, as the dependence on r is now simply that of the in-falling solution. In this appendix we specialize the discussion of the previous appendix to d = 3 in an explicit basis and review We now verify that the discontinuity across r = r′ is consistent with the delta function in (C12), which when the boundary retarded Green function derived in [15]. We choose the following basis of bulk Gamma matrices integrated across r = r′ becomes rr r σ3 0 iσ1 0 √ gg Γ (DR(r + ǫ, r) DR(r, r + ǫ)) = i . (C49) Γr = , Γt = , − − − −0 σ3 0 iσ1 Inserting (C45) into this equation we thus need to show 2 − 2 x σ 0 y 0 σ Γ = − 2 , Γ = 2 (D1) rr r t t 0 σ σ 0 i√ gg Γ ψα(r)γαβ Yβ(r) Yα(r)γαβ ψβ(r) = 1, − − − (C50) with where the right hand side is an identity matrix in the 2 bulk spinor space. To show it we first contract both sides 5 0 iσ Γ = 2 . (D2) from the left with Yσ(r). The right-hand side becomes iσ 0 − just Yσ. The left-hand side then becomes a sum of two Wronskians (C37); the Wronskian of Y with itself van- Writing ishes as in (C39), and we find then for left-hand side 1 i rr iωt+ikix Ψ1 t t 2 ψ = ( gg )− 4 e− (D3) iW [Yσ, ψα]γ Yβ = (γ ) Yβ = Yσ, (C51) − Ψ2 − αβ − σβ 30 and choosing the momentum to be along the x-direction where Φ1,2 and φ1,2 are two-component bulk spinors de- with kx = k, the corresponding Dirac equation (C2) can fined by be written as 1 mR Φα(r ) r− (D13) √grr∂ + mσ3 Ψ = i gttσ2u + ( 1)α giikσ1 Ψ → ∞ → 0 r α − α (D4) 0 +mR p p φα(r ) r . (D14) with u = ω + qAt and α =1, 2. Note that (D1) is chosen → ∞ → 1 so that Ψ1,2 decouple from each other and equation (D4) is real for real ω, k. Let us now briefly summarize the low temperature and in frequency behavior of ψ and GR [15] which are needed The in-falling solutions ψ1,2 can be written in terms of α those of (D4) for understanding the scaling behavior of the effective vertex and conductivities. The regime we are interested

1 i in in is in rr iωt+ikix Ψ1 ψ = ( gg )− 4 e− , 1 − 0 ω T 0, with w = = fixed . (D15) 1 i in rr iωt+ikix 0 → T ψ = ( gg )− 4 e− (D5) 2 − Ψin 2 The discussion proceeds by dividing the radial direction and Ψin can in turn be expanded near the boundary as into inner and outer regions, which is rather similar to α that of the vector field in Sec. IIB. For definiteness below we will consider α = 1 in (D4) and drop the subscript 1. in r mR 0 mR 1 Ψ →∞ A r + B r− a =1, 2 . a ≈ a 1 a 0 (D6) 1. Boundary retarded function We choose the constant spinors in (C28)–(C29) to satisfy (C33) To leading order in T in the limit of (D15), the Dirac 1 0 0 0 equation (D4) in the inner region reduces to that in the 0 0 + 0 + 1 near-horizon metric (2.18) with w as the frequency con- ζ− = , ζ− = , ζ = , ζ = 1 0 2 1 1 0 2 −0  jugate to τ. In particular, the spinor operator develops an IR scaling dimension given by 0 0 1  0        (D7) k2R2 and the corresponding boundary Gamma matrices (C34) 2 2 2 2 2 2 νk mkR2 edq iǫ, mk m + 2 . (D16) are given by ≡ − − ≡ r q ∗ γt = iσ2, γx = σ1, γy = σ3 . (D8) Near the boundary of the inner region (i.e. ξ 0), the νk → − − − solutions to (D4) behave as ξ± and we can choose the basis of solutions specified by their behavior near ξ 0 The matrices A and B introduced in (C35) are then given → by (which also fixes their normalization)

νk T R2 ∓ 0 A2 B1 0 2 νk A = , B = (D9) ΨI± v = v ξ∓ , ξ 0 . (D17) A1 0 0 B2 → ∓ r r ∓ → − − ∗ where v are some constant spinors (independent of ξ and from (C36) the boundary retarded function is diag- ± onal with components given by and ω). The retarded solution for the inner region can be written as [15] R Bα Gαα(ω, k)= , α =1, 2 . (D10) (ret) + A Ψ (ξ; w)=Ψ + k(w)Ψ− . (D18) α I I G I The set of normalizable and non-normalizable solu- where (w) is the retarded function for the spinor in the Gk tions introduced in (C28)–(C29) can be written more ex- AdS2 region [15] and will be reviewed at the end of this plicitly as section.20 In the outer region we can expand the solutions to (D4) rr 1 Φ1 rr 1 0 in terms of analytic series in ω and T . In particular, ψ1 = ( gg )− 4 , ψ2 = ( gg )− 4 − 0 − Φ2 the zero-th order equation is obtained by setting ω = 0 (D11) and T = 0 (i.e. the background metric becomes that of and

rr 1 0 rr 1 φ1 Y1 = ( gg )− 4 , Y2 = ( gg )− 4 − φ2 − − 0 20 Note that due to normalization difference Gk(w) defined here (D12) differs from (D28) of [15]) by a factor T 2νk . 31

the extremal black hole). Examining the behavior the where resulting equation near r = r , one finds that Ψ (r νk ∗ ∼ − W a b a b . (D26) r )± , which matches with those of the inner region in + + ∗ ≡ − − − the crossover region (2.25). It is convenient to use the The Wronskian for equation (D4) is basis which are specified by the boundary condition T 2 W [η1, η2]= η1 σ η2 (D27) νk (0) r r ± ∗ Ψ v −2 r r . (D19) where η are two solutions. Applying it to Ψ we find ± → ∓ R → ∗ 1,2 2 that ± Once the zero-th order solutions are specified, higher or- (n) W [Ψ+, Ψ ] = const (D28) der solutions Ψ (r) can then be determined uniquely − (0) ± Normalizing Ψ so that the constant on the right hand from Ψ using perturbation theory, and the two ± linearly-independent± solutions Ψ can be written as21 side of the above equation is ω-independent, then after ± inserting the ω expansion (D20) of Ψ , equation (D28) must be saturated by the zero-th order± term and all the ∞ coefficients of higher order terms on the left hand side Ψ (r)= ωnΨ(n)(r) (D20) ± ± must be zero, e.g. at first order in ω, n X (0)T 2 (1) (1)T 2 (0) Ψ+ σ Ψ +Ψ+ σ Ψ =0 . (D29) where for economy of notation, we have left implicit the − − expansion in T . Comparing (D17) and (D19), in the Furthermore, equating the value of W [Ψ+, Ψ ] at r = r overlapping region we have the matching and at r = we conclude that − ∗ ∞ νk T 2 Ψ T ± Ψ± . (D21) W = iv+σ v (D30) ± ↔ I − − Ψ can be expressed in terms of the set of normalizable which is ω-independent. and± non-normalizable solutions introduced in (D13)– Expanding (D25) in ω we find that in the outer region (D14) (recall that all quantities here refer to α = 1) Φ can be written as Φ = Φ(0) + ωΦ(1) + (D31) Ψ = b Φ + a φ (D22) ··· ± ± ± where where from (D20), a ,b can expanded in perturbative series in ω and T , with± the± zero-th order expressions de- (0) 1 (0) (0) (0) (0) Φ = (a+ Ψ a Ψ+ ), (D32) noted by a(0),b(0) which are functions of k only. W − − − Using (D18± ),± (D21), (D10) and (D22), now the full and retarded boundary Green function can then be written (1) 1 (1) (0) (0) (1) (1) (0) (0) (1) as [15] Φ = a+ Ψ + a+ Ψ a Ψ+ a Ψ+ . W − − − − − − (D33) 2νk R b+ + kT b − The expression for Φ in the inner region can then be G (ω, k)= G 2ν (D23) a+ + kT k a obtained from matching as G −

which implies that the corresponding spectral function 1 νk νk + Φ(ξ; w,T )= (a+T − Ψ−(ξ; w) a T Ψ (ξ, w)) scales with temperature as W I − − I (D34) ρ 2Im GR T 2νk . (D24) with the lowest order term given by B ≡ ∼ a(0) + νk Φ(ξ; w,T )= T − ΨI− + . (D35) 2. Normalizable solution W ···

Let us now turn to the low energy behavior of the bulk 3. Near a Fermi surface normalizable solution Φ. Using (D22), Φ can be written in the outer region as At a Fermi surface k = kF we have [15]

1 (0) Φ(r; ω)= (a+(ω)Ψ (r; ω) a (ω)Ψ+(r; ω)) (D25) a+ (kF ) = 0 (D36) W − − − and (D35) does not apply. Near kF we have the expansion

(n) ± − 21 Note that as r → r∗, Ψ (r) ∼ (r − r∗) νk n. a (k,ω,T )= c (k k ) c ω + c T + (D37) ± + 1 − F − 2 3 ··· 32

(0) (1) 1 where c1 = ∂ka+ (kF ), c2 = a+ (kF ). Thus near kF , For the Marginal Fermi Liquid case, νkF = 2 , the above in the inner region the leading− behavior for Φ(ξ; w,T ) expressions should be modified. Instead one finds that becomes T 1 νk g =2πid1u πc1 2u log +2uψ( iu)+ iπu + i + Φ(ξ; w,T ) = a (k,ω,T )T − F Ψ−(ξ; w) − µ − ··· W + I (0) ν + (D47) kF ω a (kF )T ΨI (ξ, w) (D38) where u qe , ψ is the digamma function, c , d − − ≡ 2πT − d 1 1 i are positive constants23, and denotes terms which are where the coefficient of the first term a+(k,ω,T ) should real and analytic in ω and T .··· In the limit w = ω/T be now understood as given by (D37). equation (D47) becomes → ∞ Finally let us look at the behavior of the retarded

Green function (D23) near a Fermi surface (D36), which g(w)= id1w c1w log w + . (D48) can be written as − ···

R h1 G = . (D39) Appendix E: Couplings to graviton and vector field k k (ω,T ) Σ(ω, k) − F − kF (ω,T ) in (D39) is defined as the zero of (D37), i.e. In this section we determine the couplings of a spinor to a+(kF (ω,T )) = 0 and can be considered as a generalized graviton and gauge field fluctuations; these are necessary Fermi momentum to construct the bulk vertex. We consider a free spinor 1 c3 field with the action kF (ω,T ) kF + ω T + (D40) ≡ vF − c1 ··· d+1 M d+1 S = d x√ gi(ψ¯Γ M ψ mψψ¯ )= d √ g c1 1 where vF is positive for νk > . Σ(ω, k) is given − − D − − L c2 F 2 Z Z by ≡ (E1) t where ψ¯ = ψ†Γ . We now consider a perturbed metric of 2ν ω Σ= h T kF (D41) the form 2 GkF T ds2 = g˜ dt2 + h(dy + bdt)2 + g dr2 + hdx2 (E2) and h1,h2 are positive constants whose values are known − tt rr i numerically. The spectral function can be written as with 2h ImΣ ρ = 2Im GR = 1 . y 2 B (k k (ω,T ) Re Σ)2 + (Im Σ)2 b ht , g˜tt = gtt + hb (E3) − F − ≡ (D42) The new spin connections are given by For notational convenience we write r ωty = f2e (E4) 2νk ω kF Σ(ω,T ; kF )= T F g ; (D43) T µ ω = f˜ et + f ey (E5) tr − 0 2 y t where the explicit expression for g can be obtained from ωyr = f1e + f2e (E6) that of k given in Appendix D of [15] i G ωir = f1e (E7) Γ( 1 + ν iω + iqe ) ω k 2νk 2 k 2πT d g , = h2(4π) c(k) − with T µ Γ 1 ν iω + iqe 2 − k − 2πT d (D44) 1 h 1 h 1 g˜ ′ rr ˜ tt′ rr with c(k) given by f1 √g , f2 b′, f0 √g ≡ 2 h ≡ 2sgrrg˜tt ≡ 2 g˜tt Γ( 2ν )Γ(1+ ν iqe ) η ν (E8) c(k)= − k k − d k − k (D45) Γ(2ν )Γ(1 ν iqe ) η + ν and k − k − d k k 1 1 1 1 kRR2 22 t 2 y 2 r 2 i 2 i and η mR +i iqe . g approaches a constant e =g ˜ttdt, e = h (dy + bdt), e = grr e = h dx k ≡ 2 r∗ − d as w = ω/T 0 and as w (E9) → → ∞ Also note that iπνk 2νk g(w) h2 e− c(k) (2w) . (D46) 1 1 1 → t 2 t y 2 t y Γ =˜g− Γ , Γ = g˜− bΓ + h− 2 Γ (E10) tt − tt

22 Note that the sign of the second term in ηk depends on which component of the spinor we are looking at. Here the sign is for 23 They are related by the first component. Also note that the definitions of c(k) and d 1 1 π h2 differ by a phase factor from those used in [15]. In particular, 1 = π + 2Im ψ + iqe = . d −2πqe the definition of h2 in (D41) ensures it is positive as discussed in c1 2 2 1+ e d Appendix D4 of [15]. 33

We thus ﬁnd the corrections to the Dirac action are cause they make the covariant nature of the expression given by (with a ay): at cubic order manifest. ≡ At quartic order there are both b2ψ2 and baψ2 terms. 1 1 2 y t 1 rty y For completeness we list them, although they are not δ = iψ¯ g− h Γ ∂ + f Γ ih− 2 qa Γ ψ . L3 − − tt t y 4 2 − y required for our calculation. The couplings of the bulk (E11) spinor which are quadratic in the bosonic bulk modes In (E11) we have restored the indices on b hy,a be- (altogether, quartic in ﬂuctuations) are ≡ t y

hb2 b hb 1 √ ¯ M t r ˜ 4 = giψ Γ M m (0) Γ Dt iqa + Γ f0 ψ (E12) L − 2gtt D − − √gtt 2gtt − 4 (2) where integral over the bulk spectral density

1 hb2 ~ dω βω ~ ˜ rr T DE(r1, r2; iωm, k)= tanh ρ(r1, r2; ω, k) f0 = √g ∂r . (E13) 2π 2 (2) 2 gtt iωm Z X (F2) Now as before we express the bulk spectral density in terms of the boundary spectral density ρB and Appendix F: Other contributions bulk normalizable wave functions ψa(r): ρ(r, r′,ω,k) = αβ ψα(r, k) ρB (ω, k) ψβ(r′, k). Away from the Fermi sur- In the main text we concentrated on the contributions face the discussion of (F1) parallels to that of the main from a Fermi surface in Fig. 3. Here we consider various text. In particular, the potential singular T dependence other contributions to the conductivities which we ne- coming from the IR part of the vertex is compensated glected in the main text. These include the contributions by T -dependence of the spectral function, and as a result from seagull diagrams depicted in Figure 5 which arise is non-singular. Near a Fermi surface, the eigenvalues from quartic couplings involving the graviton (schemat- of the boundary spectral density matrix take the form ically, terms like h2ψψ¯ and hAψψ¯ in the Lagrangian), (6.5)–(6.6). As we take T 0, since all the other factors → and contributions from the oscillatory region, i.e. the re- in (F1) are analytic in momentum k, the k-integral can gion in momentum space where the IR dimension for the be schematically written as fermionic operator is imaginary. We justify our neglect ImΣ of these contributions by showing that they are nonsin- dk (F3) (k k (ω,T ) Re Σ)2 + (Im Σ)2 × · · · gular in temperature and thus are subleading compared Z − F − to those considered in the main text. Our discussion will with all the other factors evaluated at k = kF (ω,T ). The be schematic. above integral can then be straightforwardly integrated and yields a contribution of order O(T 0). Also similar to the discussion in the main text, the potential singular contribution from the eﬀective vertex is suppressed at 1. Seagull diagrams k = kF (ω,T ), resulting a non-singular contribution.

We write the schematic form of a seagull diagram S with external Euclidean frequency Ωl: 2. Oscillatory region contribution

dd 1k We return to the expression (4.49) for the conductiv- Sij (Ω )= T − dr g(r ) l (2π)d 1 1 1 × ity as an integral over k. In the previous sections we iω − Xm Z Z p have studied the temperature-dependence of the region j i tr P (r1; iΩl,~k)DE(r1, r1; iωm,~k)P (r1; iΩl,~k) of k near a Fermi surface at kF . Here we ask whether − the “oscillatory region” (values of k such that particle (F1) production occurs in the AdS2 region of the geometry) make significant contributions to the conductivity. We Here P i contains the information of the graviton or will find that their contribution is finite at T = 0, and 2ν gauge field propagators and vertex and is deliberately hence subleading compared to the T − behavior of a left vague. It is shown in equation (G6) in Appendix G Fermi surface. We will not worry about numerical fac- that the Matsubara sum can be rewritten in terms of an tors here. 34

For illustration, let us look at the DC conductiv- where the upper (lower) sign is for fermion (boson). ity (6.1)–(6.2), which we copy here for convenience One can apply this kind of technique for the frequency sums involving spinor bulk-to-bulk propagator. As an ex- C ∞ d 2 dω ∂f(ω) 2 2 ample consider the spectral decomposition of a Euclidean σDC = dkk − ρB(ω, k) λ (ω,k,T ) . − 2 0 2π ∂ω correlation function Z Z (F4) In low temperature limit, the fermion spectral density in dΩ ρ(Ω) DE(iω )= . (G4) the oscillatory region may be written n 2π iω +Ω Z n iθ iλ e c ω +1 We then find that ρ (ω, k)=Im ′| | . (F5) osc eiθ c ωiλ +1 | | 1 dΩ ∞ 1 βω T D (iω ) = dω tanh ρ(Ω) where cωiλ is the IR Green’s function (with the IR di- E m ′ 2πi 2π 2 2 iθ,θ iωm Z Z−∞ mension imaginary) at T = 0. e are phases. This X 1 1 expression is valid in the oscillatory regime k < kosc = (G5) 2 2 2 2 × ω + iǫ Ω − ω iǫ Ω q ed mkR2 (see eqn (68) of [15]). The important − − − point now− is that as a function of ω, the object (F5) is p The bracketed factor reduces to a delta function, and we bounded. In fact, it can be bounded uniformly in k (i.e. find we can find a constant A˚ such that A>ρ˚ osc(ω, k) for all k < k ). Numerical evidence for this statement is fig- 1 dΩ βΩ osc T D (; iω )= tanh ρ(Ω) . (G6) ure 7 of [14]. In the oscillatory region (5.25) still applies E m 2 2π 2 iωm Z except that νk is now imaginary. Thus we see that in the X oscillatory region the effective vertex λ is also nonsingu- Similarly consider lar in the limit T 0. → We thus conclude that the contribution from the oscil- E E S(Ωl) T D1 (ωm +Ωl)D2 (ωm) latory region to σDC is nonsingular in the low tempera- ≡ ωm ture limit. X dω dω ρ (ω ) ρ (ω ) = T 1 2 1 1 2 2 , 2π 2π i(ω +Ω ) ω iω ω ω m l 1 m 2 Xm Z − − Appendix G: Some useful formulas 2πm where ωm = β with m a half integer (an integer) for 2πl Here we compile some standard and useful identities fermions (bosons), while Ωl = β with l an integer. Then that are used in the main text. using (G2) we find that

dω1 dω2 f(ω1) f(ω2) S(Ωl)= − ρ1(ω1) ρ2(ω2) . 1. How to do Matsubara sums ± 2π 2π ω1 iΩl ω2 Z − − (G7) A standard trick is perform a Matsubara sum over discrete imaginary Euclidean frequencies is to rewrite the sum over frequencies as a contour integral (we consider 2. Useful integrals fermionic frequencies here) We now give details for some integrals which we en- 1 1 βω T dω tanh (G1) countered in the main text. First consider the integral → 2πi 2 2 iω C in (6.24) Xm Z where we take the contour C to encircle all the poles. A I dk ρ (ω , k)ρ (ω , k) (G8) convenient deformation of the contour is to make it into B ≡ B 1 B 2 two lines, one running left to right just above the real Z axis and the other running right to left just below. In where the fermionic case this encircles all the poles. Exactly h1 parallel manipulations can be used to obtain the identity ρB(ω, k) = 2Im (G9) 2πl k kF (ω,T ) Σ (Ωl = β with l an integer) − − 1 1 f(ω ) f(ω ) The above integral has the form T = 1 − 2 i(ωm +Ωl) ω1 iωm ω2 ± ω1 iΩl ω2 ωm − − − − 1 1 1 1 X (G2) I(a,b)= dk − k a − k a∗ k b − k b∗ with Z − − − −(G10) 1 which can be carried out straightforwardly by opening f(ω)= (G3) eβω 1 the parenthesis and evaluating each term using contour ± 35 integration. Note that since both a and b lie in the upper = 2πh1ρB(ω1,K1)=2πh1ρB(ω2,K2) (G12) half plane, only two among the four terms contribute and we ﬁnd 1 I(a,b)=4πIm (G11) b a ∗ − We thus ﬁnd that

2 with K1 = kF (ω2,T )+Σ∗(ω2) and K2 = kF (ω1,T )+ h1 IB = 4πIm 1 Σ∗(ω1). (ω1 ω2)+Σ∗(ω1) Σ(ω2) vF − − !

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