NNSF-KITP-15-070, arXiv:1506.05111
Bekenstein-Hawking Entropy and Strange Metals
Subir Sachdev
Department of Physics, Harvard University,Cambridge, Massachusetts, 02138, USAPerimeter Institute for Theoretical Physics,Waterloo, Ontario N2L 2Y5, Canada andKavli Institute for Theoretical Physics,University of California, Santa Barbara CA 93106-4030, USA (Dated: June 19, 2015)
Abstract
We examine models of fermions with infinite-range interactions which realize non-Fermi liquids with acontinuously variable U(1) charge density Q , and a non-zero entropy density S at vanishing temperature.Real time correlators of operators carrying U(1) charge q at a low temperature T are characterized bya Q -dependent frequency ω S = ( q T / (cid:126) )( ∂ S /∂ Q ) which determines a spectral asymmetry. We show thatthe correlators match precisely with those of the AdS horizons of extremal charged black holes. Onthe black hole side, the matching employs S as the Bekenstein-Hawking entropy density, and the lawsof black hole thermodynamics which relate ( ∂ S /∂ Q ) / (2 π ) to the electric field strength in AdS . Thefermion model entropy is computed using the microscopic degrees of freedom of a UV complete theorywithout supersymmetry. a r X i v : . [ h e p - t h ] A ug . INTRODUCTION Holography provides us with powerful tools for investigating models of quantum matter withoutquasiparticle excitations. The best understood among these are strongly-coupled conformal fieldtheories (CFTs) in spatial dimensions d ≥
2. Our understanding of such models is built uponthe foundation provided by the solvable example of maximally supersymmetric Yang-Mills theory,which is known to be holographically dual to string theory on anti-de Sitter space [1].However, many holographic studies [2–4] have focused on experimentally important examplesof strongly-coupled quantum matter which are not CFTs. Of particular interest are compressiblestates without quasiparticles, or ‘strange metals’, in dimensions d ≥
2. Broadly defined, theseare quantum states without quasiparticles in which a conserved U(1) charge density Q can becontinuously varied at zero temperature by a conjugate chemical potential, and the U(1) andtranslational symmetries are not spontaneously broken. Solvable examples of strange metals withholographic duals would clearly be of great interest.Here we consider the strange metal state introduced by Sachdev and Ye [5] (SY) in a modelof fermions with infinite-range interactions. The fermion density Q is conserved and continuouslyvariable, and there is a non-zero entropy density, S , at vanishing temperature [6, 7]. The fermionGreen’s function is momentum independent and so has no Fermi surface (but there is a remnantof the Luttinger theorem, as discussed in Appendix A). The Green’s function is divergent at lowfrequency ( ω ) and temperature ( T ) with a known scaling function [6–8] (the explicit form is inEq. (3) below), determined by the fermion scaling dimension, ∆, its U(1) charge q = 1, and aspectral asymmetry frequency we shall denote by ω S . This frequency determines the asymmetrybetween the particle and hole excitations of the non-Fermi liquid. The values of ∆ and q are fixedand universal (as in traditional critical phenomena), while that of ω S varies with the compressibledensity Q in an apparently non-universal manner. However, the same ω S , scaled by the value of q , appears in the correlators of all operators.One general way to fix the precise value of ω S , without a priori knowledge of the full ω depen-dence of the correlator, is the following. The product of the retarded ( G R ) and advanced ( G A )Green’s functions obeys G R ( ω ) G A ( ω ) = Φ e ( ω − ω S ) , (1)where Φ e ( ω ) is some even function of ω . So the content of Eq. (1) is that G R G A becomes an evenfunction of frequency after the frequency shift, ω S . With this definition, it was found [6–8] thatthere is a surprising general relationship between ω S and the zero temperature entropy S density ω S = q T (cid:126) ∂ S ∂ Q . (2)Such a relationship was first found in the ‘multichannel Kondo’ problem of a local spin degreeof freedom at the boundary of a CFT2 ( i.e. a CFT in 1+1 spacetime dimensions) [6]. It was2ater extended [7, 8] to the fermion model of SY, in which we define S and Q per site, and thereare no explicit CFT2 degrees freedom; instead each fermion site is influenced by a self-consistentenvironment, and this environment plays a role similar to that of the CFT2 in the Kondo problem.Both S and ω S are non-universal functions of the compressible density Q , but they are related asin Eq. (2). It is quite remarkable to have a dynamical frequency determined by a thermodynamicproperty (which is also defined classically) divided by (cid:126) ; other notable instances of connectionsbetween observables characterizing low frequency dissipation and static thermodynamics or fun-damental constants are in Refs. 9 and 10. For the SY state, this value of ω S relies on emergentsymmetries at low energies, but also requires careful regularization of the single-site canonicalfermions present at high energies. In other words, the entropy S density in Eq. (2) counts all thedegrees of freedom in a UV finite fermion model. Indeed, in this context, Parcollet et al. [6] note:“It is tempting to speculate that a deeper interpretation of these facts is still to be found”.As we shall demonstrate in this paper, the above properties of the SY state match preciselywith the quantum theory holographically dual to extremal charged black holes with AdS horizons[11–14]. As a specific example, we will work with the Einstein-Maxwell theory of (planar orspherical) charged black holes embedded in asymptotically AdS d +2 space, with d ≥ ω , T , q , ∆, and ω S as those of the SY state, given in Eq. (3) below, and this agreement can beunderstood by the common conformal and gauge invariances of the two theories [22–24]. However,there is a deeper correspondence between the two theories in that Eq. (2) for the value of ω S also applies in the gravitational theories. The holographic computation of correlators yields thevalue of ω S ( e.g. by using Eq. (1)), while the right-hand-side of Eq. (2) is obtained from aclassical gravitational computation of the Bekenstein-Hawking (or Wald) entropy. The equality inEq. (2) follows from the classical general relativity of AdS horizons of charged black holes (seeSection III B), and this potentially provides the long sought interpretation for the value of ω S .For a general black hole solution, the values of ω S and S depend on Q in a manner differentfrom the SY state, but they all continue to obey Eq. (2). This difference is not surprising, giventhat the Q -dependence of S for the SY state uses its canonical site-fermion structure in the UV, acharacteristic which is not expected to be captured by a gravity dual. But the validity of Eq. (2)in the SY state, and in a wide class of gravity theories, is strong evidence that there is a gravitydual which captures all the universal low energy properties of the SY state.The common two-point correlator of a fermionic operator with U(1) charge q , and scaling3imension ∆, in both the SY and AdS theories is [6–8, 13, 25, 26] G R ( ω ) = G A ∗ ( ω ) = − iCe − iθ (2 πT ) − Γ (cid:18) ∆ − i (cid:126) ( ω − ω S )2 πk B T (cid:19) Γ (cid:18) − ∆ − i (cid:126) ( ω − ω S )2 πk B T (cid:19) , (3)where ∆ = 1 / q = 1 fundamental fermion of the SY state, the amplitude C is a real andpositive, and the angle − π ∆ < θ < π ∆ is given by e πq E = sin( π ∆ + θ )sin( π ∆ − θ ) . (4)Here we have found is convenient to introduce a dimensionless, T -independent, parameter E relatedto ω S by E = 12 πq (cid:126) ω S k B T (5)We have therefore introduced three parameters, ω S , E , and θ , all of which characterize the spectralasymmetry, and they can be determined from each other in Eqs. (4) and (5). The T → E also defines a ‘twist’ in the imaginary time fermioniccorrelator G ( τ ) ∼ (cid:40) − τ − , τ > e − πq E | τ | − , τ < . (6)It is easy to verify that Eq. (3) obeys Eq. (1). We show a plot of Eq. (3) in Fig. 1 which illustratesthe ‘shift’ property of G R ( ω ) G A ( ω ).For the SY state, the previous work [6–8] establishes the additional relation in Eq. (2), whichnow relates the spectral asymmetry parameters ω S , E , and θ to ∂ S /∂ Q . The T = 0 propertiesof the SY state reviewed above are summarized in the left panel of Fig. 2. For our subsequentdiscussion, it is useful to combine Eqs. (2) and (5) in the form ∂ S ∂ Q = 2 π E . (7)In the holographic computation of Eq. (3), the temperature, T , is the Hawking temperature ofthe black hole horizon [27], and the dimensionless spectral asymmetry parameter E appearing inEq. (6) (and Eq. (3)) is determined by the strength of the electric field (see Eq. (57)) supportingthe near-horizon AdS geometry [13, 26, 28] (see Fig. 2). A key observation in the holographicframework is that E , now related to the electric field, obeys an important identity which followsfrom the laws of black hole thermodynamics [29] (see Fig. 2; we set (cid:126) = k B = 1 in the remainingdiscussion): ∂ S BH ∂ Q = 2 π E , (8)4 Im G R ( ! ) Re G R ( ! ) G R ( ! ) G A ( ! ) FIG. 1. Plots of the Green’s functions in Eq. (3) for ∆ = 1 / q = 1, T = 1, A = 1, E = 1 / (cid:126) = k B = 1. Note that while neither Im G R ( ω ) or Re G R ( ω ) have any definite properties under ω ↔ − ω ,the product G R ( ω ) G A ( ω ) becomes an even function of ω after a shift by ω S = 2 πq E T = π/ where S BH is the Bekenstein-Hawking entropy densiy of the AdS horizon. Indeed, Eq. (8) is ageneral consequence of the classical Maxwell and Einstein equations, and the conformal invarianceof the AdS horizon, as we shall show in Section III B. Moreover, a Legendre transform of theidentity in Eq. (8) was established by Sen [15, 16] for a wide class of theories of gravity in theWald formalism [17–21], in which S BH is generalized to the Wald entropy.The main result of this paper is the identical forms of the relationship Eq. (7) for the statisticalentropy of the SY state, and Eq. (8) for the Bekenstein-Hawking entropy of AdS horizons. Thisresult is strong evidence that there is a gravity dual of the SY state with a AdS horizon. Con-versely, assuming the existence of a gravity dual, Eqs. (7) and (8) show that such a correspondenceis consistent only if the black hole entropy has the Bekenstein-Hawking value, and endow the blackhole entropy with a statistical interpretation [30].It is important to keep in mind that (as we mentioned earlier) the models considered here havea different ‘equation of state’ relating E to Q : this is specified for the SY state in Eq. (A5), for theplanar black hole in Eq. (58), and for the spherical black hole in Eq. (B8).The holographic link between the SY state and the AdS horizons of charged black branes hasbeen conjectured earlier [22–24], based upon the presence of a non-vanishing zero temperatureentropy density and the conformal structure of correlators. The results above sharpen this link byestablishing a precise quantitative connection for the Bekenstein-Hawking entropy [31, 32] of theblack hole with the UV complete computation on the microscopic degrees of freedom of the SY5 = 1 N X i D c † i c i E . H = 1(2 N ) / N X i,j,k,` =1 J ij ; k` c † i c † j c k c ` Known ‘equation of state’determines E as a function of Q Boundaryarea A b ;chargedensity Q ⇣ ⇣ = Einstein-Maxwell theory+ cosmological constant L = ↵ D ↵ + m ~x ‘Equation of state’ relating E and Q depends upon the geometryof spacetime far from the AdS Horizon area A h ;AdS ⇥ R d ds = ( d⇣ dt ) /⇣ + d~x Gauge field: A = ( E /⇣ ) dt J , , , J , , , J , , , Microscopic zero temperatureentropy density, S , obeys @ S @ Q = 2 ⇡ E D c i ( ⌧ ) c † i (0) E ⇠ ⇢ ⌧ / , ⌧ > e ⇡ E | ⌧ | / , ⌧ < . ⌦ ( ⌧ ) (0) ↵ ⇠ ⇢ ⌧ / , ⌧ > e ⇡ E | ⌧ | / , ⌧ < . Black hole thermodynamics(classical general relativity) yields @ S BH @ Q = 2 ⇡ E FIG. 2. Summary of the properties of the SY state (Section II) and planar charged black holes (Section III)at T = 0. The spatial co-ordinate (cid:126)x has d dimensions. All results apply also to spherical black holesconsidered in Appendix B. The AdS × R d metric has unimportant prefactors noted in Eq. (55) whichare not displayed above. The fermion mass m has to be adjusted to obtain the displayed power-law. Thespectral asymmetry parameter E appears in the fermion correlators and in the AdS electric field. As thecharge Q is increased, the horizon moves closer to the boundary, and its area, A h , increases. In blackhole thermodynamics, the Bekenstein-Hawking entropy density, S BH is related to area of the horizon via S BH = A h / (4 G N A b ), where G N is Newton’s constant. state.It also worthwhile to note here that in the usual matrix large M limit of the AdS/CFT corre-spondence [1], S and Q are both of order M [13]. So ω S and E both remain of order unity in thislimit. 6e will present an infinite range model and its solution in Section II. An important result hereis the emergent conformal and gauge invariance in Eq. (26), which strongly constrains the lowenergy theory. We then turn to the Einstein-Maxwell theory of charged horizons in Section IIIand show that it also obeys Eqs. (1–8). We conclude with a discussion of broader implications isin Section IV. II. INFINITE RANGE MODEL
SY considered a model of SU( M ) spins with Gaussian random exchange interactions betweenany pair of N sites, followed by the double limit N → ∞ and then M → ∞ . Their Hamiltonian is H = 1( N M ) / N (cid:88) i,j =1 M (cid:88) α,β =1 J ij c † iα c iβ c † jβ c jα , (9)where the c iα are canonical fermions obeying c iα c jβ + c jβ c iα = 0 , c iα c † jβ + c † jβ c iα = δ ij δ αβ , (10)and there is a fermion number constraint1 M (cid:88) α c † iα c iα = Q , (11)on every site i , with 0 < Q <
1. The exchange interactions J ij are independent Gaussian randomnumbers with zero mean and equal variance.Kitaev [24] has recently pointed out that the SY state can also be realized in a simpler model ofMajorana fermions in which only a single large N limit needs to be taken, and which also suppressesspin-glass order [7, 33–35]. We will present our results here using a complex fermion generalizationof Kitaev’s proposal, but we emphasize that essentially all results below apply equally to theoriginal model of SY in Eq. (9). We consider the Hamiltonian of spinless fermions c i H = 1(2 N ) / N (cid:88) i,j,k,(cid:96) =1 J ij ; k(cid:96) c † i c † j c k c (cid:96) − µ (cid:88) i c † i c i , (12)with c i c j + c j c i = 0 , c i c † j + c † j c i = δ ij , (13)and the J ij ; k(cid:96) are complex, independent Gaussian random couplings with zero mean obeying J ji ; k(cid:96) = − J ij ; k(cid:96) , J ij ; (cid:96)k = − J ij ; k(cid:96) , J k(cid:96) ; ij = J ∗ ij ; k(cid:96) | J ij ; k(cid:96) | = J . (14)7ecause there is only a fermion interaction term in H , and no fermion hopping, Eq. (12) can beviewed as ‘matrix model’ on Fock space, with a dimension which is exponentially large N . Theconserved U(1) density, Q is now related to the average fermion number by Q = 1 N (cid:88) i (cid:68) c † i c i (cid:69) , (15)which replaces the on-site constraint in Eq. (11). The value of 0 < Q < µ . The solution described below applies for any µ , and so realizes a compressiblestate.Note that we could equally have defined Q without the 1 /N prefactor in Eq. (15); then wewould have to define S as the total entropy, and both Q and S would be proportional to N in thelarge N limit. The latter scaling would then be similar to the M scaling of these quantities inthe usual matrix large M limit of the AdS/CFT correspondence [1]. But we choose to work herewith an intensive definition of Q and S , and keep the 1 /N in Eq. (15).Introducing replicas c ia , with a = 1 . . . n , we can average over disorder and obtain the replicatedimaginary time ( τ ) action S = (cid:88) ia (cid:90) /T dτ c † ia (cid:18) ∂∂τ − µ (cid:19) c ia − J N (cid:88) ab (cid:90) /T dτ dτ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i c † ia ( τ ) c ib ( τ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ; (16)(here we are neglecting normal-ordering corrections which vanish as N → ∞ ). Following SY,we decouple the interaction by two successive Hubbard-Stratonovich transformations. First, weintroduce the real field Q ab ( τ, τ (cid:48) ) obeying Q ab ( τ, τ (cid:48) ) = Q ba ( τ (cid:48) , τ ) . (17)In terms of this field S = (cid:88) ia (cid:90) /T dτ c † ia (cid:18) ∂∂τ − µ (cid:19) c ia + (cid:88) ab (cid:90) /T dτ dτ (cid:48) (cid:40) N J [ Q ab ( τ, τ (cid:48) )] − N Q ab ( τ, τ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i c † ia ( τ ) c ib ( τ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:41) . (18)A second decoupling with the complex field P ab ( τ, τ (cid:48) ) obeying P ab ( τ, τ (cid:48) ) = P ∗ ba ( τ (cid:48) , τ ) (19)yields S = (cid:88) ia (cid:90) /T dτ c † ia (cid:18) ∂∂τ − µ (cid:19) c ia + (cid:88) ab (cid:90) /T dτ dτ (cid:48) (cid:40) N J [ Q ab ( τ, τ (cid:48) )] + N Q ab ( τ, τ (cid:48) ) | P ab ( τ, τ (cid:48) ) | − Q ab ( τ, τ (cid:48) ) P ba ( τ (cid:48) , τ ) (cid:88) i c † ia ( τ ) c ib ( τ (cid:48) ) (cid:41) (20)8ow we can integrate out the fermions and obtain an action which can be solved in the saddle-pointapproximation in the limit of large N . The saddle-point equations are P ab ( τ, τ (cid:48) ) = (cid:68) c † a ( τ ) c b ( τ (cid:48) ) (cid:69) Q ab ( τ, τ (cid:48) ) = J | P ab ( τ, τ (cid:48) ) | (21)Notice we have dropped the site index on the fermions, because all sites are equivalent and thesaddle-point equations are defined as a single-site problem.We do not expect spin-glass solutions in this model, and so we restrict our attention to replicadiagonal solutions in which P ab ( τ, τ (cid:48) ) = δ ab G ( τ (cid:48) − τ ) , (22)where G ( τ ) is the usual fermion Green’s function. In the operator formalism for the underlyingHamiltonian, this Green’s function is defined in Euclidean time by G ( τ , τ ) = − N (cid:88) i (cid:68) T τ (cid:16) c i ( τ ) c † i ( τ ) (cid:17)(cid:69) (23)where T τ denotes time-ordering, and G ( τ − τ ) = G ( τ , τ ). Now the large N saddle-point equationsbecome [5] G ( iω n ) = 1 iω n + µ − Σ( iω n ) , Σ( τ ) = − J G ( τ ) G ( − τ ) , (24)where ω n is a Matsubara frequency. Although they are innocuously simple in appearance, theseequations contain a great deal of emergent scaling structure.In the low energy scaling limit, ω, T (cid:28) J , the iω n + µ − Σ( iω n = 0) is irrelevant [5]. Then, it isuseful to write these equations in imaginary (Euclidean) time, separating the two time argumentsof the Green’s functions (the self energy has the value of Σ( iω n = 0) subtracted out): (cid:90) dτ G ( τ , τ )Σ( τ , τ ) = − δ ( τ − τ )Σ( τ , τ ) = − J [ G ( τ , τ )] G ( τ , τ ) (25)A crucial property of these equations is that they are invariant under the time re-parameterization τ → σ , under which τ = f ( σ ) G ( τ , τ ) = [ f (cid:48) ( σ ) f (cid:48) ( σ )] − / g ( σ ) g ( σ ) G ( σ , σ )Σ( τ , τ ) = [ f (cid:48) ( σ ) f (cid:48) ( σ )] − / g ( σ ) g ( σ ) Σ( σ , σ ) (26)where f ( σ ) and g ( σ ) are arbitrary functions, corresponding to emergent conformal and U(1) gaugeinvariances. The conformal symmetry of the low energy Green’s functions has been noted earlier98, 22–24], and in the form in Eq. (26) by Kitaev [24] (without the g ( σ ) factors). The gaugetransformation g ( σ ) is a real number in Euclidean time, but it becomes a conventional U(1) phasefactor in Minkowski time. For the original model of SY [5], the gauge invariance was explicitlypresent in the underlying Hamiltonian. In contrast, our Hamiltonian here in Eq. (12) is not gauge-invariant, and only has a global U(1) symmetry; nevertheless, a U(1) gauge invariance emerges inthe low energy theory.Note that the iω n term in Eq. (24) breaks both the conformal and gauge invariances. Althoughthe iω n can mostly be neglected in studying the scaling limit, it is important in selecting the properlow energy solution of Eq. (25) from the highly degenerate possibilities allowed by Eq. (26). A. Low energy Green’s function
We now show that the fermion Green’s function in Eq. (3) follows directly from the conformaland gauge invariances in Eq. (26), when combined with constraints from analyticity and unitarity.This Green’s function was obtained earlier [5–8] by explicit solution of the integral equation inEq. (25) (and also, as discussed in Section III, by the solution [13, 26] of a Dirac equation on athermal AdS background with a non-zero electric field). Given our reliance on conformal andgauge invariances, the computations below are straightforwardly generalized to other operatorswith different values of q .At the Matsubara frequencies, the Green’s function is defined by G ( iω n ) = (cid:90) /T dτ e iω n τ G ( τ ) , (27)and this is continued to all complex frequencies z via the spectral representation G ( z ) = (cid:90) ∞−∞ d Ω π ρ (Ω) z − Ω . (28)The spectral density ρ (Ω) > T . The retarded Green’s function is G R ( ω ) = G ( ω + iη ) with η a positive infinitesimal, while the advanced Green’s function is G A ( ω ) = G ( ω − iη ).At T = 0, given the scale invariance implicit in Eq. (26), we expect G ( z ) to be a power-law of z . More precisely, Eq. (26) implies G ( z ) = C e − i ( π/ θ ) √ z , Im( z ) > , | z | (cid:28) J, T = 0 . (29)Positivity of ρ (Ω) now implies C > − π/ < θ < π/
4. An inverse Fourier transform yields G ( τ ) = − C sin( π/ θ ) √ πτ , τ (cid:29) /J, T = 0 C cos( π/ θ ) √− πτ , − τ (cid:29) /J, T = 0 . (30)10e obtain the non-zero temperature solution by choosing the conformal map in Eq. (26) as τ = 1 πT tan( πT σ ) (31)where σ is the periodic imaginary time co-ordinate with period 1 /T . Applying this map to Eq. (30)we obtain G ( σ ) = − Cg ( σ ) sin( π/ θ ) (cid:18) T sin( πT σ ) (cid:19) / , < σ < TCg ( σ ) cos( π/ θ ) (cid:18) T sin( − πT σ ) (cid:19) / , < − σ < T . (32)The function g ( σ ) is so far undetermined apart from a normalization choice g (0) = 1. We can nowdetermine g ( σ ) by imposing the KMS condition G ( σ + 1 /T ) = − G ( σ ) (33)which implies g ( σ ) = tan( π/ θ ) g ( σ + 1 /T ) . (34)The solution is clearly g ( σ ) = e − π E T σ (35)where the new parameter E and the angle θ are related as in Eq. (4) for ∆ = 1 / q = 1. Thefinal expression determining G ( σ ) is G ( σ ) = − C e − π E T σ √ e − π E (cid:18) T sin( πT σ ) (cid:19) / , < σ < T , (36)and this can be extended to all real σ using the KMS condition. The result in Eq. (3) now followsfrom a Fourier transform.For other fermionic operators with general charge q and scaling dimension ∆, the above argu-ments show that the σ dependence of Eq. (36) will be replaced by G ( σ ) ∼ − e − πq E T σ (cid:18) T sin( πT σ ) (cid:19) , < σ < T , (37)and its Fourier transform will have the frequency shift ω S = 2 πqT E . (38)The T → q and scaling dimension∆. The result in Eq. (3) continues to apply, while Eq. (4) is modified to e πq E = − sin( π ∆ + θ )sin( π ∆ − θ ) . (39)11he constraint on the allowed values of θ is now π ∆ < θ < π (1 − ∆).The constants C and θ (or E ) appearing in Eq. (3) can also be determined exactly for themicroscopic model in Eq. (12), as reviewed in Appendix A; however, their values depend upon thespecific UV completion used here, and do not apply to the holographic model of Section III. Inparticular the ‘equation of state’ for Q as a function of E is in Eq. (A5). B. Entropy
To complete our results for the SY state, we need to establish the connection in Eq. (2) between ω S and the zero temperature entropy density S . This is connection is the focus of our work, andit also relies on the conformal and gauge invariances in Eq. (26). However, in addition, we needinformation on the UV complete nature of the fermion model, and in particular, the fact that theshort-time behavior of the fermion Green’s function is determined by canonical fermions obeyingthe anti-commutation relations in Eq. (13).The computation of the entropy follows Refs. 6 and 7, and relies on the thermodynamic Maxwellrelation (cid:18) ∂ S ∂ Q (cid:19) T = − (cid:18) ∂µ∂T (cid:19) Q . (40)In the T → et al. [6] (Section VI.A.2) show that the right-hand-side of Eq. (40)can be evaluated using the imaginary time Green’s function, and we review their computation here.Their argument requires not only the scaling behavior of the Green’s function at times τ (cid:29) /J given in Eq. (37), but also the short time behavior which is beyond the conformal regime. First,we observe from Eq. (24) the large frequency behavior G ( iω n ) = 1 iω n − µ ( iω n ) + . . . (41)which implies, from Eq. (28), µ = − (cid:90) ∞−∞ d Ω π Ω ρ (Ω) , (42)which makes it evident that µ depends only upon the particle-hole asymmetric part of the spectraldensity. Next, we can relate the Ω integrals to the derivative of the imaginary time correlator µ = − ∂ τ G ( τ = 0 + ) − ∂ τ G ( τ = (1 /T ) − ) . (43)Making the analogy to Eq. (36), we pull out an explicitly particle-hole asymmetric part of G ( τ )by defining G ( τ ) ≡ e − π E T τ g ( τ ) , < σ < T . (44)Note that E was introduced as a parameter in Eq. (36), and then appears in Eq. (3) via Eq. (38).(It might appear that we can absorb the chemical potential, µ , in the Hamiltonian by a temporal12auge transformation, and that such a transformation combined with Eq. (44) implies µ = − π E T and so yields ∂µ/∂T ; however this argument is flawed because µ also includes a Q -dependentpiece at T = 0, which must also be accounted for in the gauge transformation. The non-zero Q ground state is not invariant under gauge transformations.) We proceed in our computation of µ by inserting Eq. (44) into Eq. (43) to obtain µ = 2 π E T (cid:2) G ( τ = 0 + ) + G ( τ = (1 /T ) − ) (cid:3) − ∂ τ g ( τ = 0 + ) − e − π E ∂ τ g ( τ = (1 /T ) − ) (45)For the term in the first square brackets, we have G ( τ = 0 + ) + G ( τ = (1 /T ) − ) = G ( τ = 0 + ) − G ( τ = 0 − ) = − , (46)which follows from the KMS condition and the fermion anti-commutation relation in Eq. (13);also, this is related to the high frequency behavior G ( | z | → ∞ ) = 1 /z . Writing the second termin Eq. (45) in terms of a spectral density ρ g (Ω) for g ( τ ), we obtain µ = − π E T − (cid:90) ∞−∞ d Ω π Ω (cid:2) ρ g (Ω) − e − π E ρ g ( − Ω) (cid:3) e − Ω /T ; (47)(we note that there is a sign error on the right-hand-side of Eq. (65) in Ref. 6, and − ρ g ( − Ω)should be ρ g ( − Ω)). At this point, Ref. 6 argues that at low T and fixed Q , ρ g must be particlehole symmetric with ρ g (Ω) = ρ g ( − Ω), and that the T dependent part of the integral above scalesas T / . We therefore have (cid:18) ∂µ∂T (cid:19) Q = − π E , T → , (48)and then the Maxwell relation in Eq. (40) leads to Eq. (8).Using the relationship between Q and E specified in Appendix A in Eq. (A5), and the limitingvalue S = 0 in the empty state Q = 0, we can integrate Eq. (8) to obtain the full zero temperatureentropy [7]. III. CHARGED BLACK HOLES
This section (apart from Section III B) mainly recalls the results of Faulkner et al. [13, 26] onplanar, charged black holes in AdS d +2 , and makes the correspondence with the properties of theSY state. We will also largely follow their notation, apart from the change d → d + 1 requiredby our definition of d as the spatial dimension (instead of the spacetime dimension). The case ofspherical black holes in global AdS is more complicated and is considered in Appendix B; it has amore complex equation of state, but also obeys all results claimed in Section I. The discussion inthe latter part of Section III B shows how the needed features of Faulkner et al. can be obtainedin a more general class of black hole solutions.13e consider the Einstein-Maxwell theory of a metric g and a U(1) gauge flux F = dA withaction S = 12 κ (cid:90) d d +2 x √− g (cid:20) R + d ( d + 1) R − R g F F (cid:21) , (49)where κ = 8 πG N , R is the Ricci scalar, R is the radius of AdS d +2 , and g F is a dimensionlessgauge coupling constant. The equations of motion of this action have the solution [11, 12] ds = r R (cid:0) − f dt + d(cid:126)x (cid:1) + R r dr f (50)with f = 1 + Θ r d − (cid:18) r d +10 + Θ r d − (cid:19) r d +1 A = µ (cid:18) − r d − r d − (cid:19) dt (51)This solution is expressed in terms of three parameters Θ, r , and µ ; these parameters are deter-mined by the charge density, Q , and temperature, T , of the boundary theory via the relations µ = g F Θ c d R r d − , Q = 2( d − c d Θ κ R d g F T = ( d + 1) r πR (cid:18) − ( d − ( d + 1) r d (cid:19) , c d = (cid:114) d − d . (52)The Bekenstein-Hawking entropy density [31, 32] of this solution is S BH = 2 πκ (cid:16) r R (cid:17) d . (53)We turn next to the holographic implications of this solution at low energy [13, 26, 36], which iscontrolled by the near-horizon geometry. At T = 0, the horizon is at r = [Θ ( d − / ( d + 1)] / (2 d ) ,and so we introduce the co-ordinate ζ by r − (cid:2) Θ ( d − / ( d + 1) (cid:3) / (2 d ) = 1 ζ ; (54)we approach the horizon as ζ → ∞ (see Fig. 2). In terms of ζ , the near horizon geometry at T = 0is ds = R ( − dt + dζ ) ζ + [Θ ( d − / ( d + 1)] /d R d(cid:126)x . (55)The geometry has factorized to AdS × R d , where the AdS radius is given by R = R (cid:112) d ( d + 1) . (56)14n the same low energy limit the gauge field is (see Fig. 2) A = E ζ dt. (57)which determines the strength of the AdS electric field in terms of the dimensionless parameter E . Notice that the value of E in Eq. (57) is invariant under any rescaling of the co-ordinateswhich preserves the ( − dt + dζ ) /ζ structure of the AdS metric. From the present near-horizoncomputation we find the value E = g F sgn( Q ) (cid:112) d ( d + 1) . (58)Eq. (58) is the ‘equation of state’ connecting Q to E , and the analogous expression for the fermionmodel is in Eq. (A5), and for the spherical black hole is in Eq. (B8); the non-analytic Q dependencein Eq. (58) becomes analytic for the spherical black hole in Appendix B. We recall that g F is adimensionless coupling, and so E is also dimensionless, and depends only upon g F and d ; inparticular, E is independent of κ , and so remains of order unity in the matrix large M limit ofholography [1], as noted in Section I.We also take the T = 0 limit of Eq. (53) from Eq. (52), and find S BH = 2 πg F |Q| (cid:112) d ( d + 1) , T → . (59)Comparing Eqs. (58) and (59), we find that Eq. (8) is indeed obeyed. Note that for the presentcase of a planar black hole, we can combine Eqs. (58) and (59) into the simple relationship [13] S BH = 2 π QE . (60)Eq. (60) does not hold for a spherical black hole; but the more fundamental relation for ∂ S BH /∂ Q in Eq. (8) does hold, and is verified in Appendix B, which also derives the different ‘equation ofstate’ relating E and Q for a spherical black hole. A. Fermion correlations
To confirm the link to the fermion model, we need to show that the E obtained above in Eq. (57)is the same as the E (or the related ω S via Eq. (5)) appearing as the spectral asymmetry parameterin the response functions in Eqs. (3) and (6) (see Fig. 2). For this, we need the Green’s functionof matter fields moving on a thermal AdS metric. The finite temperature generalization of theAdS factor in Eq. (55) is [13, 26] ds = R ζ (cid:20) − (cid:0) − ζ /ζ (cid:1) dt + dζ (1 − ζ /ζ ) (cid:21) , (61)15nd that of the gauge field is A = E (cid:18) ζ − ζ (cid:19) dt (62)where T = 12 πζ . (63)The action of a fermionic spinor, ψ , of charge q moving in the backgrounds of Eqs. (61) and (62)is S = i (cid:90) d x √− g (cid:0) ψ Γ α D α ψ − mψψ (cid:1) (64)where m is a bulk fermion mass, Γ α are the Dirac Gamma matrices, and D α is a covariant derivativewith charge q . The correlator of ψ in this thermal AdS [25] plus electric field background hasbeen computed in some detail by Faulkner et al. [13, 26], and their result was already displayedin Eq. (3) in our notation. This computation shows that E = ω S / (2 πqT ) (Eq. (5)) is indeed thesame parameter appearing in Eqs. (57) and (62). In this AdS computation, the scaling dimension∆ is related to the bulk spinor mass by∆ = 12 − (cid:113) m R − q E . (65) B. Black hole thermodynamics
We close this section by noting a significant property of the above solution of classical generalrelativity at all T and Q . From the laws of black holes thermodynamics [29], we deduce that thehorizon area and the chemical potential must obey a thermodynamic Maxwell relation (cid:18) ∂ S BH ∂ Q (cid:19) T = − (cid:18) ∂µ∂T (cid:19) Q , (66)which is the analog of that in the fermion model computation in Eq. (40). And indeed we do findfrom Eqs. (52) and (53) that Eq. (66) is obeyed with (cid:18) ∂µ∂T (cid:19) Q = − π ( d − g F Θ r d c d ( d + 1) r d + c d ( d − d − . (67)In determining the value of ( ∂µ/∂T ) Q as T →
0, rather than explicitly evaluating Eq. (67), itis instructive to use a more general argument which does not use the explicit form of the solutionin Eqs. (51) and (52). From the original action in Eq. (49) and the metric in Eq. (50), Gauss’s lawfor the scalar potential in the bulk is 2 R κ g F ddr (cid:18) r d R d dA t dr (cid:19) = 0 , (68)16nd the constant of integration is the boundary charge density2 R κ g F (cid:18) r d R d dA t dr (cid:19) = Q . (69)We can write the solution of Eq. (69) as A t ( r ) = µ ( T ) − (cid:18) R d − κ g F d − (cid:19) Q r d − , (70)where the r -dependent term in Eq. (70) is independent of T at fixed Q , and the chemical potential µ equals A t ( r → ∞ ) when we choose A t = 0 on the horizon. Now we transform to the near-horizon AdS geometry by making a T -independent change of variables from r to ζ as in Eq. (54), r = r ∗ + 1 /ζ , where r = r ∗ is the position of the horizon at T = 0, but we won’t need the actualvalue of r ∗ . Then Eq. (70) implies that, as ζ → ∞ , the near-horizon scalar potential must of theform in Eq. (62), where now we define ζ = ζ as the position of the horizon at non-zero T , where E is a parameter independent of T , and (cid:18) ∂µ∂T (cid:19) Q = E ∂∂T (cid:18) − ζ (cid:19) Q . (71)The T -dependence of ζ in Eq. (63) follows from the conformal mapping between the T = 0 AdS metric in Eq. (55) and T > (cid:18) ∂µ∂T (cid:19) Q = − π E , T → , (72)which is the same as the fermion model result in Eq. (48). It can be verified that Eq. (72) holds alsoin the spherical geometry of Appendix B. Combining Eq. (72) with Eq. (66), we obtain Eq. (8),which is a special case of results obtained from the Wald formalism [15–21].We note that the above derivation of Eq. (72) relied only on Gauss’s Law and the conformalinvariance of the AdS near-horizon geometry: this implies that such results hold for a wide classof black hole solutions [15–21]. IV. DISCUSSION
In our discussion of the SY state of the infinite-range fermion model in Eq. (12), we notedthat the fermion Green’s function was almost completely determined by the emergent conformaland gauge invariances in Eq. (26). These conformal and gauge invariances also fairly uniquelydetermine the holographic theory of matter moving in curved space in the presence of an electricfield. So, with the benefit of hindsight, we can understand the equivalence of the fermion Green’sfunctions obtained in Sections II and III. 17owever, we have gone beyond the identification of Green’s functions, and also shown that thezero temperature entropy of the SY state can be mapped onto that of the AdS theory (see Fig. 2).Specifically, we chose an appropriate combination of observables in Eqs. (1,2) to allow us to gener-ally define a common frequency ω S , and we showed that this frequency was related to precisely thesame derivative of the entropy in both the SY state and in charged black holes (where the entropywas the Bekenstein-Hawking entropy). In both cases, establishing this relationship required ananalysis of the details of the model, and it did not follow from general symmetry arguments alone.In particular, for the SY state, the entropy computation required careful treatment of the mannerin which the emergent gauge and conformal invariances, present at low energies, were broken bythe on-site canonical fermions, present at high energies.This common relationship between ω S and the entropy indicates an equivalence between thelow-energy degrees of freedom of the two theories in Sections II and III, and strongly supports theexistence of gravity dual of the SY state with a AdS horizon. The present results also imply the c i fermion, with q = 1, of the theory in Eq. (12) is holographically dual to the ψ fermion, with q = 1, [13, 36] of Eq. (64). As the microscopic c i fermion carries all of the Q charge of the theoryin Eq. (12), we expect that ψ also carries a non-negligible fraction of the charge (in the large N limit) behind the AdS horizon. Both models likely also have higher dimension operators, butthese have not been analyzed so far (see however Ref. 24).Note that the above discussion refers to the near-horizon AdS geometry. The larger Reissner-Nordstr¨om-AdS solution is to be regarded here as a convenient (and non-universal) embeddingspace which provides a UV regulation of the gravitational theory. With such an embedding, we areable compute well-defined values for S and Q . Presumably other gravitational UV embeddings, willhave different ‘equations of state’ between E and Q , but the will nevertheless obey the fundamentalrelation in Eq. (8) provided they contain a AdS horizon. We explicitly tested the independenceon the UV embedding in Appendix B by comparing the cases of planar and spherical black holes.The above identification between the c i and ψ fermions differs from that made previously bythe author in Refs. 22 and 23. There, ψ was argued to be dual to a higher dimension compositefermion operator of the original model of SY [5]. This previous identification was based upon therequirement that local bulk operators must be dual to gauge-invariant operators on the boundary,and the original model [5] had a microscopic gauge invariance which did not allow the choice of c i asdual to ψ . However, in the present model in Eq. (12), there is no microscopic gauge invariance, andso we are free to use c i as the dual of the bulk ψ field. It turns out that the low energy boundarytheory for c i does have a gauge invariance (as in Eq. (26)), but this is an emergent gauge invariancewhich is broken by UV terms needed to regularize the theory. The present situation is analogousto the theory of the Ising-nematic quantum critical point in metals, where the regularized modelfor the electrons is not gauge-invariant, but the low energy theory defined on two Fermi surfacepatches does have an emergent gauge structure [37, 38]. And the present situation is different from18hat in the ‘slave particle’ theories of condensed matter, where the gauge structure emerges fromfractionalizing particles into partons, which influenced the reasoning of Refs. 22 and 23. Insteadthe same particle can be gauge-invariant in the underlying theory, and acquire an emergent gaugecharge in the low energy theory. There is some similarity between this interpretation and ideas inRef. 39.Finally, we note recent work [24, 40, 41] on ‘a bound on chaos’ which also related characteristictimes of the real-time dynamics of strongly-coupled quantum systems to thermodynamics, (cid:126) , andblack hole horizons. ACKNOWLEDGMENTS
I thank T. Banks, A. Dabholkar, Wenbo Fu, S. Hartnoll, A. Kitaev, Hong Liu, J. McGreevy,R. Myers, A. Sen, A. Strominger, and W. Witczak-Krempa for valuable discussions, and especiallyA. Georges and O. Parcollet for inspiring discussions on these topics over many years. This researchwas supported by the NSF under Grant DMR-1360789, and also partially by the Templeton Foun-dation. The research at KITP Santa Barbara was supported by the Simons Foundation and NSFGrant PHY11-25915. Research at Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry of Research andInnovation.
Appendix A: Non-universal constants of the fermion model
We compute the constants C and θ (or E ) appearing in Eq. (3) for the microscopic model inEq. (12). The results of this appendix do not apply to the holographic model of Section III.We can compute the self-energy from Eq. (30) and the second equation in Eq. (24)Σ( τ ) = − C J cos(2 θ ) sin( π/ θ )2( πτ ) / , τ (cid:29) J, T = 0 C J cos(2 θ ) cos( π/ θ )2( − πτ ) / , − τ (cid:29) J, T = 0 . (A1)A Fourier transform now leads toΣ( z ) = − J C cos(2 θ ) π e i ( π/ θ ) √ z , Im( z ) > , | z | (cid:28) J, T = 0 . (A2)We now see that Eqs. (29) and (A2) are consistent with the first equation in Eq. (24), providedwe choose the value of C to be C = (cid:18) πJ cos(2 θ ) (cid:19) / . (A3)19inally, the value of θ can be related to the density Q by a computation which parallels theLuttinger-Ward analysis [42] for a Fermi liquid. The present model has no spatial structure, andso no possibility of a Fermi surface. However, if we apply the steps of the Luttinger-Ward proof ofthe volume enclosed by the Fermi surface, we find an expression relating density Q to the spectralasymmetry angle θ . In other words, θ plays a role similar to the Fermi wavevector in a Fermiliquid. And the relationship between Q and θ is [7] Q = 12 − θπ − sin(2 θ )4 . (A4)Note that the constraint − π/ < θ < π/ < Q <
1, as expected. In terms of E , thisrelationship is Q = 14 (3 − tanh(2 π E )) − π tan − (cid:0) e π E (cid:1) . (A5)The right-hand-side is a monotonically decreasing function of E which ranges between 1 and 0, as E increases from −∞ to ∞ . Appendix B: Spherical black holes
We consider the case of spherical black holes in global AdS, following the analysis of Ref. 12.For simplicity, we will limit ourselves to the T = 0 case.Now we choose a solution of the Einstein-Maxwell equations of motion of Eq. (49) with metric ds = − V ( r ) dt + r d Ω d + dr V ( r ) (B1)where d Ω d is the metric of the d -sphere, and V ( r ) = 1 + r R + Θ r d − − Mr d − , (B2)has a zero at r = r so that M = r d − (cid:18) r R + Θ r d − (cid:19) . (B3)The zero temperature case has [12]Θ = r d − [( d − R + ( d + 1) r ]( d − R . (B4)In the near-horizon region, we introduce, as in Section III, the co-ordinate ζ via r − r = R ζ , (B5)20here Eq. (56) is now replaced by R = R (cid:112) d ( d + 1) + ( d − R /r , (B6)and the near-horizon metric becomes AdS × S d , with ds = R (cid:20) − dt + dζ ζ (cid:21) + r d Ω d . (B7)Turning to the gauge field sector, the charge density, Q , and AdS electric field parameter E inEq. (57) are Q = r d − (cid:112) d [( d − R + ( d + 1) r ] κ g F E = g F r (cid:112) d [( d − R + ( d + 1) r ]2 [( d − R + d ( d + 1) r ] . (B8)The ‘equation of state’ obeyed by E and Q is obtained by eliminating r between the equations inEq. (B8); this leads to a very lengthy expression which we shall not write out explicitly.Using the Bekenstein-Hawking entropy density S BH = 2 πκ r d , (B9)and ∂ S BH ∂ Q = ∂ S BH /∂r ∂ Q /∂r , (B10)and evaluating the derivatives via Eq. (B8), we can now verify that Eq. (8) is indeed obeyed. Notethat S BH (cid:54) = 2 π QE here, unlike Eq. (60) for the planar case. [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J.Theor. Phys. , 1113 (1999), arXiv:hep-th/9711200 [hep-th].[2] S. A. Hartnoll, “Horizons, holography and condensed matter,” ArXiv e-prints (2011),arXiv:1106.4324 [hep-th].[3] S. Sachdev, “What can gauge-gravity duality teach us about condensed matter physics?” AnnualReview of Condensed Matter Physics , 9 (2012), arXiv:1108.1197 [cond-mat.str-el].[4] N. Iqbal, H. Liu, and M. Mezei, “Lectures on Holographic Non-Fermi Liquids and Quantum PhaseTransitions,” in String Theory and its Applications - TASI 2010, From meV to the Planck Scale ,edited by M. Dine, T. Banks, and S. Sachdev (2012) pp. 707–815, arXiv:1110.3814 [hep-th].[5] S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet,”Phys. Rev. Lett. , 3339 (1993), cond-mat/9212030.
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