A Comment on Holographic Luttinger Theorem

Robustness of the Luttinger theorem for fermionic liquids is examined in holography. The statement of the Luttinger theorem, the equality between the fermion charge density and the volume enclosed by the Fermi surface, can be mapped to a Gauss's law in the gravity dual, a la Sachdev. We show that various deformations in the gravity dual, such as inclusion of magnetic fields, a parity-violating theta-term, dilatonic deformations, and higher-derivative corrections, do not violate the holographic derivation of the Luttinger theorem, as long as the theory is in a confining phase. Therefore a robustness of the theorem is found for strongly correlated fermions coupled with strongly coupled sectors which admit gravity duals. On the other hand, in the deconfined phase, we also show that the deficit appearing in the Luttinger theorem is again universal. It measures a total deficit which measures the charge of the deconfined ("fractionalized") fermions, independent of the deformation parameters.

Abstract: Robustness of the Luttinger theorem for fermionic liquids is examined in holography. The statement of the Luttinger theorem, the equality between the fermion charge density and the volume enclosed by the Fermi surface, can be mapped to a Gauss's law in the gravity dual, a la Sachdev. We show that various deformations in the gravity dual, such as inclusion of magnetic fields, a parity-violating θ-term, dilatonic deformations, and higher-derivative corrections, do not violate the holographic derivation of the Luttinger theorem, as long as the theory is in a confining phase. Therefore a robustness of the theorem is found for strongly correlated fermions coupled with strongly coupled sectors which admit gravity duals. On the other hand, in the deconfined phase, we also show that the deficit appearing in the Luttinger theorem is again universal. It measures a total deficit which measures the charge of the deconfined ("fractionalized") fermions, independent of the deformation parameters.

Luttinger theorem and holography
The Luttinger theorem [1,2] is one of the key fundamental relations in condensed matter physics and it states that the volume enclosed by the Fermi-surface is equal to the charge density. This theorem, which is originally derived by Luttinger and Ward for Landau's Fermi liquids, is non-trivial in the sense that this theorem is about the volume enclosed by the Fermi-surface. Remember that in the Landau's Fermi-liquid picture, we have a quasi-particle description for the spectrum near the Fermi-surface, but generically the quasi-particle description is not valid for the spectrum far away from the Fermi-surface, therefore the spectrum deep inside the Fermi-surface does not always allow the quasi-particle description generically. The non-trivial point of the Luttinger theorem is that it relates the spectrum not only near the Fermi-surface (where quasi-particle picture holds) but also deep inside the Fermi-surface (where quasi-particle picture does not hold), to the charge density.
It is widely known that the theorem holds for Fermi liquids having a Fermi surface, and there is a general non-perturbative proof of the Luttinger theorem for Fermi liquids [3] (the original proof by Luttinger and Ward was with perturbation of Fermi liquids). The proof of [3] is based on a U (1) gauge symmetry, Fermi-liquids description near the Fermi-surface and a mild assumption for dynamical degrees of freedom, namely, all the momentum and charge carrying degrees of freedom are quasiparticles near the Fermi-surfaces. See also [4,5] for further developments concerning the proof of the Luttinger theorem.
On the other hand, in nature there are quite interestingly materials, such as high T c superconductors or heavy fermions, where its normal phase shows non-Fermi liquid behavior, and in addition, the standard quasi-particle description breaks down. In such situations whether the theorem holds or not is to be better understood.
Recent progress in applications of string theory, the holographic principle [6,7,8], to condensed matter systems brought an insight about the Luttinger theorem in strongly correlated fermion systems. In [9,10,11], it was pointed out that a holographic system with a charged bulk fermion, which is called "electron star" in the literature, exhibits the Luttinger theorem of the boundary fermion theory. This is based on the observation that the bulk fermions obey a bulk Luttinger theorem at each radius for the electron star. Then, Hartnoll pointed out [12] that the flux emanating from the black hole horizon will equal the deviation from the Luttinger relation. Furthermore, Sachdev clarified [13] that in a simple holographic set-up for fermions with fermion-number chemical potential, the Luttinger relation follows simply from the Gauss's law in the bulk and that it holds in confined phase (thermal gas phase) but breaks down in deconfined phase (black hole phase). However it is also true that their argument uses a specific holographic setup like neglecting higher derivative corrections. Therefore it is natural to ask how universal the nonperturbative Luttinger theorem is for fermions.
The holographic principle has been widely applied to various gravity setups, and robust correspondence has been thoroughly studied. Among many variations of the holographic models, some of the most popular and meaningful ones are: (i) higher-derivative corrections in the bulk gravity + Maxwell theory, (ii) inclusion of θ term and magnetic field, (iii) inclusion of a dilaton to have dilatonic gravity models. 1 Each corresponds, in terms of condensed matter theory language, to: (i) Sub-leading terms concerning the strong coupling expansion, (ii) Parity-violating terms inducing quantum Hall effects under magnetic fields, and (iii) Drastically different infra-red behavior, for example having a Lifshitz-like scaling near quantum critical points, and more realistic systems with vanishing entropy at zero temperature.
We would like to study whether the holographic derivation of the Luttinger theorem a la Sachdev can survive against the deformations, to find a universality of the holographic Luttinger theorem. In this paper, we examine these popular deformations and show the Luttinger theorem to hold for all of these deformations, in the case of confining phases. 2

Holographic derivation of the Luttinger theorem
We follow the beautiful argument of Sachdev's holographic derivation of the Luttinger relation [13], to show the robustness of the holographic Luttinger theorem. We generalize Sachidev's derivation [13], in particular concerning the following points: (i) In the gravity side we allow for a generic action for the bulk gauge fields including higher-derivative corrections. (ii) We allow an axion coupling (responsible for a θterm) in the action and we include a magnetic field in the background. (iii) The gravity action is coupled to a dilaton with a generic form of its couplings to the gravity and the bulk gauge fields, resulting in generic background geometry which is different from the cut-off AdS 4 used in [13]. Specifically, we consider the following action Here, the spacetime dimension is 4 which is dual to a condensed matter system in 2+1 dimensions. The 2 + 1 dimensions are spanned by x, y and t, while the z direction is the emergent space coming out of the holographic principle. The subspacetime z = 0 is the boundary of the bulk geometry. The geometry typically is an asymptotic AdS 4 geometry, but in this paper we do not rely on any specific metric.
and includes higher derivative terms as a form of F 4 and higher multiples. In holography, higher-derivative corrections may be related directly to a physical consequence; for example, the famous calculation on the shear viscosity of the quark-gluon plasma [18,19] can be corrected by higher-derivative terms to have lower values, but a physical constraint on the form of the higher-derivative terms coming from a bulk causality may forbid the value of the viscosity (divided by the entropy density) to go lower [20,21]. In generic holographic setting, even if we take large N limit, these higher derivative corrections are non-negligible.
The system couples to the bulk scalar field φ which is a dilaton in string theory. Explicit solutions (such as the ones with horizons studied in, for examples, [22] - [34] ) are not necessary in the following. In the second line of the action (2.1), we have the parity-violating θ-term (the axion coupling), which is relevant for a quantum Hall effect under the magnetic field, see for examples, [35,36,37,29,38].
The essence of the Sachdev's derivation is to consider a bulk fermion one-loop path-integration in the total free energy as a semi-classical approximation. This affects the scalar potential of the bulk gauge field, to minimize the free energy (which is nothing but the on-shell effective action in the gravity side). A radial integral of the bulk Gauss's law turns out to be nothing but the Luttinger relation. Once one puts φ = a = 0 and L = (1/4)F 2 , and takes a cut-off AdS space in the derivation below, everything reduces to the Sachdev's original derivation.
The bulk free energy per a unit volume in our case is We have integrated out the bulk fermion ψ, to have the last term. 4 We ignore the back reaction of the fermions to the bulk geometry and the dilaton and the axion; we only consider a back reaction to the gauge field (a possible justification is presented in the next section).
As for the gauge field configuration, we assume the homogeneity and the isotropy in the (x, y) directions. Then non-vanishing components of the static gauge field strengths are only F tz and F xy . Using the Jacobi identity in the (x, y, z) space, we obtain ∂ z F xy = 0 which means that B ≡ F xy is constant. So, in the A z = 0 gauge, we are left with the gauge configuration The charge density of the boundary theory is given by Here we have defined the chemical potential as µ = Φ(z = 0) where we denote the temporal component of the gauge field as A 0 ≡ iΦ. Note that the Φ dependence in the fermion loop in (2.3) does not contribute to the definition of the charge (2.5), because the bulk fermion wave functions vanish at the boundary z = 0 where the chemical potential is defined, due to the normalizability. To explicitly perform the fermion one-loop integral to evaluate (2.3), one just needs a formal expression characterizing the discrete energy eigen modes of the bulk fermions, which looks D n χ l,n (z) = E l,n χ l,n (z). (2.6) The operator D n is nothing but a spatial part of the covariant Dirac operator in the curved geometry. χ l,n and E l,n are the eigenfunctions and eigenvalues of the operator D n , respectively. 5 Here the integer l labels the Kaluza-Klein modes of the bulk fermion in the curved space, and n labels the Landau levels of the fermion wave function in the x-y space as the magnetic field is present. Below, we shall use only the two facts: First, the operator D n includes a trivial term q Φ coming from the minimal coupling in the Dirac operator, and second, the normalization is given as where no summation for l and n is imposed. 6 See Appendix A for the explicit evaluation of the states with the operator D.
Using this energy eigenvalue, generically the trace log term in the free energy can be evaluated as The step function is necessary to count only the residues appearing in the shift of the poles in the path integral in the off-shell k 0 space. See Appendix B for the detailed calculations. The factor qB/2π is the unit volume of the discretized momentum space (k x , k y ) due to the magnetic field. Now, to find a saddle point of the free energy with respect to the bulk field Φ, we consider the bulk on-shell equation To calculate this with the fermion loop term, we use (2.7) and (2.6) to rewrite the fermion free energy (2.8) as With the fact that Φ dependence of the operator D is just linear in qΦ as it is a Dirac operator (see Appendix A for the details), the minimization of the free energy 5 For deconfined geometry l can be a continuous parameter, but here we formally write the generic label as l. In addition to that, in the deconfined geometry, the energy eigenvalues are generically complex, thus the amplitude of the wave function damps in time exponentially, as in the case of quasi-normal modes. Here formally we regard our calculation performed in a time scale shorter than the decay time scale. 6 We consider the cases where bulk metric takes the form as Then we make an integration over the z space, which leads us to the Luttinger relation, as follows. For that, we need one more information for the geometry at the IR. Let us first consider a generic confining geometry, for which the geometry consistently ends at z = z IR . The Gauss's law at the IR end z = z IR shows that the electric flux D z vanishes there. Then, from the z integration of (2.11), we obtain This is the Luttinger relation, since the right hand side is the volume enclosed by the Fermi surface. Note that since we turn on the magnetic field, the Landau levels appear and the x-y momentum space is discretized, and resultantly the unit volume of the x-y momentum space qB/2π appears. For deconfined geometries, the IR boundary condition differs, and in particular the electric flux does not vanish there. There appears a deficit in the Luttinger relation. We discuss the situation in the next section.
One should have noticed that the derivation here is almost identical with what Sachdev gave in [13]. However, we find it intriguing that the derivation by Sachdev is so robust that the theorem is valid against various deformations of the system. In particular, the inclusion of the higher derivative corrections corresponds to the direction toward a weak coupling where it is plausible that the Luttinger theorem is valid. Furthermore, the inclusion of the background magnetic field is interesting, as it not only introduces a nice regularization of the momentum space but also is involved with quantum Hall effects. The dilatonic corrections are related with existence of different scaling at the IR, and even in those systems our generalized derivation shows that the Luttinger theorem holds.
Although in this paper we worked in four spacetime dimensions in the gravity theory (which corresponds to three spacetime dimensions for the liquid system), we can generalize the derivation to higher dimensions, in a straightforward manner. A possible obstacle would be the F ∧ F term in the bulk, which should be generalized to a Chern-Simons term in higher dimensions, then one cannot impose the isotropy which we have employed in the derivation above. Another concern may be on the fermion integral, since generically in higher dimensions the theory becomes non-renormalizable. However, the leading quantum loop which we considered in this paper is just an effect of the chemical potential of the one-loop diagram of fermions, which can be defined without any problem. Therefore we claim that higher-dimensional generalization of the derivation can be done accordingly.

Discussions
The essential statement which we would like to make in this paper is just the robustness of the holographic Luttinger theorem, which was already shown in the previous section. We end this short paper with two important observations: First, the validity of our calculation in view of quantum corrections in string theory and AdS/CFT, and second, the emergent dependence on deformation parameters only in the deconfinement phase.

Quantum corrections in string theory
In the derivation of the holographic Luttinger theorem, Sachdev and we computed the fermion one-loop diagram and considered its back-reaction to the gauge potential. However, in general in string theory, other loops involving gravity and other fields may contribute, so the effect on the gauge potential may not be only from the fermion one-loop. Here we shall point out that a proper large N scaling in AdS/CFT can avoid this mixing problem 8 .
We assume that the bulk fermion is from a space-filling D-brane. This fermion is often called "mesino" since it may be a fermion counterpart of mesons, when the D-brane is identified with the flavor D-branes in the AdS/CFT correspondence [40]. The gauge potential A µ also comes from a space-filling D-brane. This means that the effective gauge coupling q is of order O(1/ √ N ), because in the AdS/CFT correspondence, the string coupling constant g s scales as ∼ 1/N , and the coupling on the Dbrane is the open string coupling √ g s while the coupling in the bulk geometry (gravity and the dilaton φ and the axion a) is g s . Denoting the graviton/dilaton/axion fluctuation as δg, and the gauge fluctuation as δA µ , and the fermion fluctuation as ψ, then the generic dependence in N in general AdS/CFT is written as Looking back our derivation of the holographic Luttinger theorem, we have performed the ψ integral in this action. From the generic action written above, we observe that this one-loop integral involves the term 1 √ N (δA µ )ψψ, so it shifts the action by a term of order O(1/ √ N ). This is a leading order effect, compared to the quantum 8 See also [13] for the argument to include quantum fluctuations beyond the fermion one-loop.
corrections involving the graviton, the dilaton and the axion, 9 since in the action above those corrections come with a coupling of O(1/N ). Therefore, we conclude that, assuming that the origin of the fermions and the gauge fields is D-branes in the bulk, the fermion one-loop integral is the leading order in 1/N expansion in AdS/CFT, so we can consistently ignore the other quantum corrections.

Deconfined phase
As first pointed out by Hartnoll [12] and also studied in detail by Iqbal and Liu recently [15], the presence of the black hole horizon significantly alters the result; in the final line of the derivation of the Luttinger theorem (2.12), we have used the fact that the electric displacement D z at the IR endpoint vanishes due to the confining geometry. However in the presence of the black hole the IR boundary condition is different and there exists in general an electric flux emanating from the black hole horizon, for charged black holes.
So, there exists a Luttinger deficit for the deconfined phase [12,15]. Here D z is the electric displacement in the bulk at the IR of the geometry, which is the electric flux penetrating the horizon of the black hole. This comes from the first term of (2.11).
In the presence of the deformations which we consider, for the deconfined phase, there appears a dependence on the magnetic field and the axion field at the horizon. The axion field corresponds to the parity-violating θ term, so in general, the Luttinger theorem in the deconfinement phase is violated with a deficit dependent on the θ term and the magnetic field in addition to the deficit [15] of the electric charge carried by the "fractionalized" fermions [41,42,17] (which are deconfined quarks in the standard holographic QCD terminology).
The electric displacement includes a contribution from the parity-violating term, the second term of the (3.3), This term directly responds to the magnetic field present in the system, and the effective value of the coefficient, θ eff , in the Luttinger theorem is given by It is intriguing that the parameter θ eff is not given by the UV geometry but the IR geometry. The IR geometry is not directly related to the parameters of the fermion liquid system defined at UV, and it is rather determined by the gravitational dynamics corresponding to strongly coupled sectors, which these fractionalized fermions couple. This dependence on IR geometry reflects the fact that the Luttinger relation is a phenomena at low energy. Note that even with no electric field F 0z , once the parity-violating term and the magnetic field F xy is turned on, the Luttinger deficit appears. The deficit appearing in the Luttinger relation (3.2) appears to depend explicitly on the deformation parameters we introduced in the gravity dual. The parameters are in the nonlinear electrodynamics L[F ] and the dilaton-axion couplings to the gauge fields in the gravity side. However, in (3.2), the deficit depend only on the electric displacement D z at the black hole horizon. The effect of the parity-odd term (θ eff B) is also included in the electric displacement. So, we conclude that the effect of the deconfinement phase to the deviation from Luttinger theorem can be summarized into electric displacement D z , which is determined by the total charge of the "fractionalized" fermions, even in the presence of the deformations considered in this paper. Note that we have not assumed the existence of the quasi-particle picture in our derivation.

A. Bulk fermion eigen-states
In this appendix we shall explicitly calculate (2.6) with the background gauge field. We work in the Lorentzian signature, while the Euclidean signature (which we employed in the derivation) can be easily obtained by an analytic continuation. First, we derive the Dirac operator. The fermion action in the bulk is Here the Dirac operator is D M = ∂ M + 1 4 w abM Γ ab − iqA M . The definition of the Gamma matrices in the local Lorentz frame are with γ 0 ≡ iσ 3 , γ 1 ≡ σ 1 , and γ 2 ≡ −σ 2 , where σ 1 , σ 2 , σ 3 are the Pauli matrices. We follow the notation of [43] and [32] except for the assignment of γ µ (this difference is necessary to see the diagonalization as for fermion components, see below). The notation for the indices are: M = 0, 1, 2, z, and µ = 0, 1, 2.
Writing the 4-component fermion as where φ ± is a two-spinor, the Dirac equation is In the Dirac equation, the four spinor components are coupled, while we would like to group them into 2-spinors to derive (2.6). In our convention, among the gamma matrices, γ 0 is a diagonal matrix while γ i is not, so if we can bring the K i γ i to a diagonal form, the decomposition to the 2-spinors is complete. This is nothing but solving the following eigen equation in the x-y space, where α and β are complex constants, andφ ± (x, y) are 2-spinor wave functions representing Landau levels of fermions in magnetic fields. The gauge field configuration (2.4) satisfies [K 1 , K 2 ] = iqB, so using a creation and an annihilation operator we have the harmonic osccillator [a, a † ] = 1. The eigen equation (A.5) becomes A solution can be easily found as We need β = −2qBn/α to have this solution. With this wave function for (x, y), we choose then the Dirac equation (A.4) is written as The lower spinor component of the equation (A.10) yields (A.14) For (A.13) and (A.14) to be consistent with (A.11) and (A.12), we need This can be solved as Then (A.13) and (A.14) become the same as (A.11) and (A.12), and we obtain two independent solutions as (A.16). The resultant equations (A.13) and (A.14) can be re-written as with χ ≡ (f + , f − ) T . This is for the choice α ± = ± √ 2qBn. For the other choice α ± = ∓ √ 2qBn, The operator D n of (2.6) appears here as D (i) n . See also [13,14]. Note that due to the interaction term between spins and magnetic fields, there are two D n 's labeled by i. Generically this eigen-equation allows only a discrete spectrum 10 for E n , so the energy E n is also labeled by l (the Kaluza-Klein modes in z space) in addition to n (the Landau levels) and i (spins) as we can combine these to eliminate φ − , to have This is a 2nd order differential equation for a two-spinor φ − , so we generically have four independent solutions. Without magnetic field, given E, four states are degenerate and they correspond to spin {up and down}, and {normalizable and non-normalizable} modes. In the presence of magnetic fields, this degeneracy splits up by spins, so given spin and energy eigenvalue E, there is a set of a normalizable and a non-normalizable mode.

B. Evaluation of the fermion free energy
In this appendix, we demonstrate the calculation of the fermion one-loop free energy (2.8).

B.1 Reduction to 1 + 0 dimension
In the previous appendix, we demonstrate that given l (the Kaluza-Klein modes in z space), n (the Landau levels) and i (spins), the Dirac equation reduces to a single equation (A.20) which determines E l,n . Therefore the action for the fermion should read as Here Ψ l,n (t) is a single component fermion, since the dependence on the Landau level n is already included with the specific Landau level wave function in the x-y space. The fermion field Ψ l,n (t) is properly normalized to have the action above. Note also that there are two E l,n 's, depending on the choice of the Dirac operator D n in (A. 18) and (A.19) labeled by i. In the following, we omit the index i for simplicity of the notation.
Let us proceed to calculate the free energy. We bring the Tr Log into the following expression, Here, we have used the fact that the momentum integration in the k x -k y space is now replaced by the Landau level summation, as We have used the momentum relation k 2 = 2qBn which we obtained in the last appendix. Now we perform the off-shell w integration. The standard path in the complex w-plane for the integration of w rounds the upper half plane plus the real axis. In the i prescription, the pole contributing in the path integral is the one on the negative real axis of w. This appears only for the negative s so the integral is non-zero only when E l,n < 0, and we obtain This expression is (2.8) in the Euclidean notation.

B.2 Another viewpoint: Reduction to 1 + 2 dimension
We shall present another viewpoint here, to evaluate the fermion free energy, which is a natural dimensional reduction along z. First, we shall decompose the bulk fermion into Kaluza-Klein modes along z labeled by l. The effective action for the l-th mode has a kinetic operator D µ γ µ − m l where m l is the mass for the l-th mode of the decomposed fermion. Then, we compute Tr Log(D µ γ µ − m l ) for each state l and make a summation over l.
Let us work out the Kaluza-Klein decomposition explicitly. The bulk fermion action is (B.5) Using the notation in the previous appendix, this action S fermion can be explicitly written as and D z ≡ ∂ z − iqA z . In view of this, we consider the following matrix equation so that the above matrix is diagonalized, where m l is some eigenvalue. We write 11 φ + φ − ≡ l g (l) + (z)Ψ (l) (t, x, y) g where Ψ is a two-spinor which is a function of (t, x, y), and g ± (z) are scalar functions. These g ± (z) are required to satisfy . (B.10) 11 The reason why we took a common factor g ± (z) for the 2-spinor φ ± is that this z-dependent factor go through the γ matrices inK µ γ µ such that the z integration can be done independently as (B.11), (B.12).
With explicit eigen wave functions, we can reduce the action to a 3-dimensional action. Substituting (B.9) to the action, we obtain The z integration gives a normalization for the fermion which we require as dz g zz −g tt g (l) * + (z)g Furthermore, we calculate the chemical potential for each mode as µ lm ≡ dz g zz −g tt A 0 (z) g Here we assume that this chemical potential is diagonal µ lm = µ l δ lm , (B.14) and also consider only the case below where g ii (z) = −g tt (z) such that dimensional reduction along z ensures the Lorentz invariance in the t-x-y spacetime 12 . Then we finally obtain the decomposition The on-shell condition for the l-th fermion is E l + qµ l = m 2 l + k 2 where k is the magnitude of the momentum in the (x, y) space. 13 The derivative in the Dirac operator has eigenvalues ∂ µ = (−i(w + qµ l ), ik 1 , ik 2 ). In the presence of the magnetic field, the momentum k is replaced by the Landau levels.
Let us proceed to calculate the free energy. We bring the Tr Log into the following expression, In the presence of the magnetic field, the momentum k 2 is replaced by the Landau levels 2qBn with a non-negative integer n. The momentum integral is accordingly normalized as dk 1 dk 2 = 2πkdk = πd(k 2 ) = 2πqBdn, so This expression is (2.8) in the Euclidean notation.