Holographic Magnetic Susceptibility

The (2+1)-dimensional static magnetic susceptibility in strong-coupling is studied via a Reissner-Nordstr\"{o}m-AdS geometry. The analyticity of the susceptibility on the complex momentum $\mathfrak{q}$-plane in relation to the Friedel-like oscillation in coordinate space is explored. In contrast to the branch-cuts crossing the real momentum-axis for a Fermi liquid, we prove that the holographic magnetic susceptibility remains an analytic function of the complex momentum around the real axis in the limit of zero temperature, At zero temperature, we located analytically two pairs of branch-cuts that are parallel to the imaginary momentum-axis for large $|\text{Im}\ \mathfrak{q}|$ but become warped with the end-points keeping away from the real and imaginary momentum-axes. We conclude that these branch-cuts give rise to the exponential decay behaviour of Friedel-like oscillation of magnetic susceptibility in coordinate space. We also derived the analytical forms of the susceptibility in large and small-momentum, respectively.


I. INTRODUCTION
Strongly correlated electronic systems, such as the high temperature superconductors or graphene, are characterized by a spectrum of novel static and transport phenomena that cannot be explained by the traditional Fermi liquid theory of Landau and are difficult to explore with ordinary field theoretic techniques. The perturbative expansion or mean field approximation becomes unreliable, especially in lower dimensions, and the first principle numerical simulation is hindered by the fermion sign problem. The holographic theory [1][2][3][4][5][6] built on the conjectured gauge/gravity duality is expected to shed some lights on the nonperturbative physics and to reveal some generic properties pertaining to a strongly-coupled system [7][8][9], such as a non-Fermi liquid [10][11][12][13][14]. According to the holographic dictionary, the classical solution of the gravity-matter system in an asymptotically AdS space-time with a black hole is linked to the thermodynamics of a strongly coupled quantum field theory on the AdS boundary [15]. In particular, the linearized solutions of the former generate various two-point correlation functions of the latter [1,6,16], and the photon polarization tensor to be investigated in this work is one of them.
The general structure of the polarization tensor in energy-momentum representation, dictated by the current conservation, is given by Π ij ( q, ω) = χ(ω, q)(q 2 δ ij − q i q j ) + ω 2 α(ω, q) q i q j q 2 Π 0j ( q, ω) = Π j0 ( q, ω) = ωα(ω, q)q j Π 00 ( q, ω) = q 2 α(ω, q) , with the transverse and longitudinal form factors, χ(ω, q) and α(ω, q), representing the magnetic susceptibility and electric polarizability, respectively. Both variables ω and q in χ(q, ω) and α(q, ω) can be continuated to the complex planes. The singularities on the ωplane reflect the excitation spectrum, while the singularities on the complex q-plane give rise to the Debye-like screening and Friedel-like oscillation in coordinate space. The analyticity of χ(ω, q) and α(ω, q) in weak coupling is well known. For the (2+1)-dimensional static polarization tensor considered in this paper, the one-loop calculation of χ(0, q) and α(0, q) for a spinor QED reveals two lines of square root branch points located at [17] q = ±2 µ + iπT (2n + 1) with T the temperature and µ the chemical potential. In the zero temperature limit T → 0, these singularities merge into two cuts with Re q = ±2µ across the real axis, which results in a discontinuity in the derivative of χ(0, q) and α(0, q) at q = ±2µ and the Friedel oscillation in coordinate space with the amplitude decaying according to a power law.
In strong coupling, the holographic χ(ω, q) and α(ω, q) extracted from different bulk geometries along with their analyticity have been discussed extensively in the literature, such as Ref. [18][19][20][21][22] for q = 0, and Ref. [23,24] for ω = 0. (For more details, see Ref. [8] and the references therein). In this paper, we shall focus on the momentum analyticity of the holographic polarization tensor from a Reissner-Nordström-AdS geometry. In the same system, Ref. [25] studied the conductivity via the small frequency expansion in the IR limit of CFT, finding that the conductivity at zero-momentum scales as ω 2 in ω → 0. For the χ(0, q) and α(0, q) extracted from the Schwarzschild-AdS geometry (corresponding to zero chemical potential), it was shown in Ref. [26] that all of singularities on the q-plane are poles located along the imaginary momentum-axis. A similar result was obtained by a study on the probe D3/D5 system at a nonzero density in Ref. [23] and the authors revealed that such poles at the purely imaginary momentum screen exponentially a point charge in the medium and do not cause Friedel-like oscillation. Then came the work by Blake et.al. [27], who solved the Einstein-Maxwell equations numerically for the gauge field and metric tensor fluctuations in the Reissner-Nordström-AdS background with a complex momentum and found two lines of poles of α(0, q) whose locations tend to be parallel to the imaginary q-axis for large |Imq| and bend toward the imaginary axis at lower |Imq|. Their numerical solution also indicates an exponentially decaying Friedel-like oscillation behavior even at zero temperature. In our previous works [17] and [28], we were able to prove that both χ(0, q) and α(0, q) extracted from the non-extremal Reissner-Nordström-AdS geometry are meromorphic functions and to locate their poles analytically for large |Im q| via WKB solution of the Einstein-Maxwell equations. The asymptotic distribution of the poles is given by with the integer n 1, where L 1 and L 2 are two elliptic integrals dependent of the temperature T , defined in Eqn.(A7) of Appendix A. As the temperature T → 0, the distance between adjacent poles, π QL 1 ∼ log T µ −1 → 0 and the poles merge into two pairs of cuts, parallel to the imaginary axis but at much slower rate than the weak coupling case. For α(0, q), the asymptotic locations (3) match well with the numerical result in Ref. [27] even with a moderate Im q. Unfortunately, the condition for the WKB prevented us from making any rigorous statements regarding the distribution of these poles near the real momentum axis, which may be more relevant to experimental observations. This work is a continuation of Refs. [17]. Different strategies are employed here to explore the analyticity of the holographic magnetic susceptibility χ(0, q) in the complex q-plane, especially at zero temperature where the RN black hole becomes extremal. Through the series solution of the Heun equation involved, we show that the complex poles of χ(0, q) discussed in [17] merge into four branch cuts of square root type at zero temperature, whose trajectories are located analytically. Coming from the infinity, these cuts are nearly parallel to the imaginary axis for large |Im q|, in agreement with the WKB approximation, bending towards the imaginary axis for lower |Im q| and terminating at respective branch points with |Im q| = 0 and |Re q| = 0, without crossing either the real or imaginary axes on their paths. Through a relation between the Einstein-Maxwell equations and the eigenvalue problem of an one-dimensional Schrödinger equation, we prove that χ(0, q) is an analytic function for any finite real q at any temperature, which excludes any oscillatory behavior caused by singularities on the real axis. Consequently,the magnetic susceptibility manifests a Friedel-like oscillation in coordinate space which decays exponentially even down to zero temperature.
The analytic technique employed in this work is not yet to be generalized to the case of electrical polarization, α(0, q) in order to extend the result of Ref. [28] to zero temperature, in which case the Einstein-Maxwell equations involved are far more complicated. We hope to report our progress along this line in near future.
The paper is organized as follows: In Sec. II, we formulate the holographic magnetic susceptibility dual to an Einstein-Maxwell system in the background of a Reissner-Nordström blackhole with an asymptotically Anti-de Sitter boundary. The analyticity of the magnetic susceptibility is explored in Sec. III. The asymptotic forms for small and large complex momenta are derived in Sec. IV, Sec. V concludes the paper.

A. Background Solutions and Fluctuations in D= 2+1 space-time
According to the holographic principle, the generating functional of correlators of a strongly-coupled quantum field theory(QFT) defined in space-time S is associated with the partition function of a classical gravity-matter theory in a bulk bounded by S. This relation, as was formulated by Gubser-Klebanov-Polyakov and Witten (GKPW) [1][2][3][4][5][6], is: where with φ i the bulk fields of the gravity-matter system and φ i the classical solutions whose boundary valueφ i representing the source for Z QFT , conjugate to the operators O i .
The bulk action of the gravity-matter system considered in this work reads where R is the scalar curvature corresponding to the metric tensor g µν , Λ is the negative cosmological constant, in D = 3 + 1 dimensional AdS space-time, Λ = − 3 L 2 , L is the AdS radius, and F µν is the electromagnetic tensor, F µν = ∂ µ A ν − ∂ ν A µ , corresponding to the gauge potential A µ . The mass dimension of the gauge potential is [A µ ] = 1 and that of the coupling constant G 4 is [G 4 ] = 2. The coupling constant K 4 is thereby dimensionless, The background solution (Ā µ ,ḡ µν ) of the Einstein-Maxwell equation dictated by the action (6) consists of the Reissner-Nordström-AdS metric and the gauge potentialĀ where the metric function with the horizon of this Reissner-Nordström black hole at u = 1 and the AdS boundary at u = 0. The chemical potential µ of the boundary field theory is related to the dimensionless charge of the black hole Q via µ = Q/z + . The Hawking temperature T in terms of the chemical potential and the charges reads which corresponds to the background temperature of the boundary field theory, and Q ∈ [0, 3] and L G 4 /K 4 is re-scaled to 1. At Q 2 = 3, it represents the zero-temperature limit with the extremal metric function f 0 = (u − 1) 2 (3u 2 + 2u + 1) holding a double zero at the horizon.
Introducing the metric and gauge potential fluctuations (h µν , a µ ) via Owing to the homogeneity with respect to the boundary coordinates, (x, y, t), the linearized Einstein-Maxwell equations can be Fourier transformed into the frequencymomentum space for with P = (ω, q x , q y ). Aligning the spatial momentum q along the x-axis, the linearized equations can be decomposed into two decoupled subsets according to the parity under the transformation y → −y [29], i. e.
Even Parity : {h t t , h x x , h y y , a t } and {h x t , a x }; Odd Parity : {h y t , a y } and {h x y } In the static limit (ω = 0), each group of Einstein-Maxwell equations are further decoupled into the two subsets, denoted by the curly brackets above. The electric component is extracted from the even-parity group, while the magnetic component from the odd-parity one, respectively. The two coupled equations responsible to the static magnetic susceptibility read where Z = 3 4 (1 + Q −2 ), and we have introduced a dimensionless momentum q ≡ q/µ and the dimensionless modified momentum k is defined by For the full set of Einstein-Maxwell equations in terms of our notations, see [17,28]. The static magnetic susceptibility at a temperature T is given by where C yy is the coefficient of |a y (P ; 0)| 2 in the on-shell action with P = (0, q, 0), following the GKPW formulation (4) and Ref. [16], and it is dependent of the momentum q, temperature T and chemical potential µ of the system.
In terms of the static solution of the linearized Einstein-Maxwell equations for (a y , h t y ) that are regular at the horizon u = 1 and subject to the boundary condition h t y = 0 at u = 0 (in order for extracting the polarization tensor only), the on-shell action becomes where the prime refers to the derivative with respect to the dimensionless radical variable u. Consequently, we have The following sections will elucidate the solution of the Einstein-Maxwell equations specified above along with the properties of the temperature-dependent function C yy (q). B.

Master-fields and decoupled equations of motion
The linearized Einstein-Maxwell equations in the odd parity sector, eqs. (15), can be transformed into a pair of decoupled differential equations for the so-called master-field Φ ± , Ref.
[29] [30]: with from which the fluctuations a y and h y t can be extracted according to Eliminating h y t from (22), we obtain that Moreover, as was discussed in [17], the solution for h y t under the homogeneous boundary condition at u = 0 vanishes as h y t u→0 = O(u 3 ), and we obtain the relation from (22): The notations in Eqn.(22)-(24) emphasize that a y and Φ ± are functions of k or q as well and the analyticity with respect to q is the main theme of this paper.
Because of the complexity of the Eqn. (20), it's impossible to find out explicit solutions for arbitrary momentum q. Asymptotic solutions can be obtained, however, for small or large magnitude of q, and can shed lights on the analyticity. For a small momentum q, the master-field equations turn into where the leading order equations are exactly soluble and the subsequent corrections can be figured out perturbatively. For a large momentum q, it is convenient to transform the master equation into a Schrödinger-like where the potential energy is dominated by the first term inside the bracket as q → ∞ and the WKB approximation becomes handy then. This approximation is particularly useful to locate the Friedel-like singularities of C yy (q) for a large imaginary-part of the complex-momentum q ∼ k. The details of the solutions of both cases, small q and large q, will be presented in Sec. IV.

BILITY
In this section, we shall explore the analyticity of the correlator C yy (q) on the complex q-plane. It follows from Eqn. (19) and (23) that the magnetic susceptibility can be singular in two ways: (1) The boundary value of the master field Φ ± (0|q) itself is singular. (2) a y vanishes on the AdS-boundary. The former possibility will be ruled out on the entire physical Riemann sheet (defined below) in q-plane besides four branch points at zero temperature and on the entire q-plane at nonzero temperature in the subsection III A below. The latter possibility will be ruled out along the real axis at an arbitrary temperatures in the subsection III B.
A. The analyticity of the solutions of the master field equations Considering different singularity structures of the master field equations at zero and nonzero temperatures, we treat the two cases separately.

Zero temperature case
At T = 0, Q 2 = 3, the RN-AdS black hole becomes extremal, and the metric function and each master field equation of (20) becomes a Fuchs equation with four regular points, and ∞, which can be transformed into the standard Heun equation.
The indices at the horizon (u = 1) read with +,(±) for Φ + and −,(±) for Φ − 1 , and produce the asymptotic solutions near the For a real k, the indices ±,(−) < −1 give rise to a divergent solutions at the horizon, which in turn generates divergent on-shell actions through the F 2 term in the integrand of Eqn. (6): Consequently, only the positive indices, ±,(+) , should be retained, which give rise to a finite on-shell action. For an arbitrary complex k, we may replace (31) with supplemented with the requirement Re{α ± } > −1/2, i.e.,Re{ 1 + 2(k ± 1) 2 } > 0 for a finite action solution. This defines the physical Riemann sheet of the square root on the complex k-plane, being cut along the lines where 1 + 2(k ± 1) 2 becomes imaginary, i.e.
Consider Φ + (u|q) first. Introducing a new variable v = 1 − u, and writing the master-field equation Φ + in Eqn. (20) is transformed into a Heun-type equation for with the coefficients given by This equation can be solved by an infinite series with the recurrence relation among successive coefficients where we have set G 0 = 1. Evidently, the denominators in (41)  On the other hand, the distance from the regular point v = 0 to the nearby regular points ) is greater than one, implying that the AdS-boundary, v = 1, is inside the convergence circle of the power series (42). It follows that the infinite series P + (1|k) and its derivative with respect v at the boundary converge uniformly with respect to a finite k and thereby share the same analyticity with their coefficients G n (k). To demonstrate this point, the infinite series (42) is splitted into the sum of its first N terms, that implies asymptotic expression of G N +n from Eqn. (41) 2 , for n = 1, 2, 3, · · · in terms of G N and G N −1 . Since |v −1 ± | < 1, we have For a finite k, say, |k| < K, there is always a k-independent N such that (43) approximates to a specified accuracy. In addition, we can always find k-independent upper bound of |G N | and |G N −1 | and thereby a k-independent upper bound of the remainder for a given |v| < |v ± |.
Consequently, we end up with two uniformly convergent series with respect to k, which is thereby analytic on the physical Riemann sheet of the complex k-plane.
It follows from (21) and (34), and the analysis given above that and both P − (1|k) and P − (1|k) are also analytic on physical Riemann sheet of the complex k-plane.
Now we construct the correlation function C yy (q) at zero temperature. It follows from (23) and (24) that and a y (0|k) = C − 6(k − 1) 2 The derivation is left in Appendix C hence, where we have removed the subscript "+" of α + (k) and P + (1|k), and write α − (k) = α(−k) and P − (1|k) = P (1| − k) with the aid of (47). As the RHS of (50) is an even function of k, the transformation k = √ q 2 + 1 will not introduce new branch points on the physical Riemann sheet of the complex q-plane, which remains characterized by the branch cuts (35).
The trajectories of the branch cuts (35) on the complex k-plane and the relation q 2 = k 2 − 1 implies the following trajectories of the same set of branch cuts on the complex q-plane or explicitly with η = [1, ∞). The end-points of the branch-cuts (the branch points) are given by η = 1, which gives rise to To rule out such a possibility, we start with the modified master-field equations (27), and find that the nontrivial solutions φ ± of the modified master-field equations which contribute to the poles of C yy (q) correspond to the solutions φ ± = √ f Φ ± of (27) where the AdS-boundary conditions (54) follow from eq.(22) and the horizon conditions (55) result from the regularity requirement of Φ ± there.
The solutions φ ± (u|k, Q 2 ) of (27) together with the boundary conditions (54) and (55) correspond to the zero energy eigenstate of the one-dimensional Hamiltonian: defined between two infinitely repulsive barriers for u < 0 and u > 1. Because of the following two properties of the potential (28), an eigenstate of H − at k = Z(q = 0) with a negative eigenvalue might be escalated to zero eigenvalue of H ± at some k > Z(q = 0). If we could rule out the former, we would rule out the zero energy eigenstate in both H ± when q = 0. It follows from (22) that the zero eigenstate of H − at k = Z itself, however, does not imply a vanishing a y (0|q) and thereby does not imply a singularity of C yy (q) at q = 0. The perturbation theory in the next section shows that C yy (q) ∼ q 2 as q → 0.

It is easy to find an explicit solution of the master-field equation for Φ
The second possibility for the singularity of C yy (q) is thereby ruled out for a real q.
Summarizing this section, we have analytically located the branch cuts of the holographic magnetic susceptibility on the physical sheet of the complex-q plane at zero temperature and proved rigorously the absence of poles on the real axis at any temperature. What we have not succeeded is to rule out poles on the physical Riemann sheet away from the real axis at zero temperature. The "inhomogeneous" equations in Eqn. (25) facilitate an iterative procedure to find the perturbative solutions for small q, provided that their homogeneous parts, q = 0 case,

IV. MAGNETIC SUSCEPTIBILITY AT A SMALL MOMENTUM AND
are explicitly solvable, which is indeed the case. It is easy to verify the following particular for Φ The other linearly-independent particular solutions of (60), denoted as η (0) ± , can be obtained from the Wronskians of (60), i.e.
where the arbitrary multiplicative constant is set as 1. Solving the first-order differential equations in Eqn. (63), i.e., we find that η (0) ± are cumbersome and both of them are singular at the horizon.

Fluctuation a y and magnetic susceptibility up to q 2 -order
Employing the method of variation of parameters, we obtain a pair of particular solutions of (25): which is regular at the horizon and serves the next order correction to (62). Combining (62) and (66), we find the solutions of the master-field equations (25) in small momentum approximation: where χ (1) The two coefficients in Eqn. (67) are not arbitrary and are constrained by the behaviour of the perturbed metric fields h y t as u → 0, which implies (24). Substituting (67) and (68) into (24), we obtain the ratio of the two coefficients Following eqs. (67) and (23), the fluctuation a y (u|q) and its derivative w.r.t u, a y (u|q), take the form: and The overall constant a − drops in the correlation function C yy in accordance with Eqn. (19) and we obtain that where q is the unscaled momentum. The dimension of the holographic polarization tensor is [C yy ] = [q 2 /µ] = 1, as expected. Following Eqn. (17), the magnetic susceptibility at zero momentum reads which becomes at zero temperature.

B. WKB approximation at a large momentum
The region far away from the real momentum-axis can be explored by the WKBapproximation of the modified master-fields φ ± = √ f Φ ± , and the fluctuation a y in the WKB-approximation can be obtained from the solutions of the Schrödinger-like equations (29) via the relation (23). The nonzero temperature case has been worked out in [17] and we include the key steps in Appendix A for self-containedness. There we also derived the asymptotic form of the magnetic susceptibility which was missing in [17]. In what follows, we shall focus on the zero temperature case. Unlike the non-extremal blackhole, the validity of the WKB-approximation extends all the way from the boundary to the horizon because the condition of the approximation, |V ± | |V ± | 3/2 [32] holds for 0 ≤ u ≤ 1.
The general WKB solutions of (29) at T = 0 read with f 0 = (1 − u) 2 (1 +2u + 3u 2 ). The integrals in the exponents can be carried out explicitly, where A(u) = 1 √ 6 ln 2 + 4u + 6(1 + 2u + 3u 2 ) and Because of the divergence of A(u) as u → 1, one of the terms inside the bracket of Eqn. (75) blows up at the horizon for Re k = ±1 and has to be dropped for a finite on-shell action.
We have and As expected, the discontinuity of the master-field φ − at Rek = 1 and the discontinuity of the master field φ + at Rek = −1 match the asymptotic trajectories of the branch-cuts (35), which correspond to the condensation of the poles discussed in [17] as T → 0 .

V. DISCUSSION AND CONCLUSION
In this work, we explored the analyticity of the static transverse component of the holographic polarization tensor, C yy (q), in 2+1 dimensions with respect to the complex spatial momentum. The dimensionless magnetic susceptibility is given by C yy (q)/q 2 . The zero temperature features of the static holographic susceptibility are not determined by the nearhorizon IR data, but by the analyticity in the complex momentum plan. We provided a rigorous proof that C yy (q) is analytic in the neighbourhood of a real q even at zero temperature. In addition, we located analytically four branch cuts on the complex q-plane at zero temperature, which terminated at the branch points ±1/ √ 2 ± i, staying away from the real and imaginary axes. We also worked out the asymptotic form of C yy (q) for small q and large q. The momentum analyticity of the transverse holographic polarization appears similar to that of the longitudinal one, as was demonstrated by the numerical solution [27] and WKB approximation [28] of the Einstein-Maxwell equations in the sector of even parity 4 . What we have not achieved is to rule out analytically the poles of C yy (q)/q 2 away from the real axis and the branch cuts at zero temperature.
As the holographic polarization tensor may reflect certain strong-coupling properties, it is instructive to compare our results with the polarization tensor in weak coupling at zero temperature to find their difference. The transverse component of the static polarization tensor in the massless spinor QED at one loop order reads and σ tr (q) is the weak-coupling counterpart of the holographic C yy (q). such that K4 G4 = L 2 = 1. An arbitrary ratio K4 G4 ≡ η 2 amounts to the transformations a t → ηa t and µ → ηµ in the Einstein-Maxwell equations, and thereby the transform of the asymptotic branch cuts from k µ to k µη. The notation in [27] corresponds to η = 1 2 which gives rise to the asymptotic cuts at q ±1/2k F , while Ref. [33] that starts with ABJM theory gives rise to "q ±1k F " result, as oppose to the result in weak-coupled system.
The comparison between the momentum analyticity of C yy (q) in strong coupling and that of σ tr (q) in weak-coupling at T = 0 is depicted in Fig.1.
The left panel in Fig. 1 shows discontinuities in the derivative of σ tr (q) at q = ±2µ because of the condensation of the branch-points (2), which form a pair of branch-cuts crossing the real momentum axis and cause the Friedel oscillation in coordinate space whose amplitude decays with distance according to a power law. In contrast, the right panel in Fig. 1 for C yy (q) shows that the real axis is free from singularities and is spared by the bending branch- To the first order in the (3+1)-dimensional electromagnetic coupling e 2 , we have where the first term D 0 ij ( q) is the static transverse component of the (3+1)-dimensional free propagator and the magnetization comes from the polarization in the second term that is our main consideration. Also, in the static case, D ij (q|0, 0) is always contracted with the Fourier component of a stationary electric current and the factor q b in (84) does not contribute because of the current conservation. Hence, effectively, D ij ( q|0, 0) = D(q)δ ij with a scalar form factor where the first term on RHS comes from the free propagator and the second term reflects the polarization of the medium with e the electric charge in 3+1 dimensions, σ tr. (q) is the 2D polarization, proportional to C yy (q) for the holographic polarization in our case. Assuming that C yy (q) has no poles on the entire physical Riemann sheet (not just around the real axis) and only focusing on the second term of (84), Its Fourier transform takes the asymptotic form at large | x|, i.e.
where the integral in (86) is calculated via a contour integration, going along the branchcuts in the upper half plane of Fig. 1(b) and φ is a phase constant. The details behind (84) and (86) can be found in Appendix D. The exponential factor on the right hand side of (86) is explicitly in contrast to the case in a weakly-coupled field theory, such type of Friedel-like oscillation with faster than power-law decay behavior is observed in the densitydensity correlation in other holographic strongly-coupled systems [27,33,34] and in the zero fermionic flavor limit: N f → 0 [35].
The Friedel-like oscillation caused by the transverse component of the polarization tensor is responsible for the RKKY effect [36], where the local magnetic field acting on a nuclear magnetic moment is generated by other nuclear magnetic moments and is polarized by the Fermi sea of electrons. Therefore, the effect discussed in this work may find its application in the RKKY effect in some 2D metals, whose low-lying excitations are Dirac like, such as a doped graphene.
On the other hand, one may associate the absence of the cuts crossing the real momentum axis to the bosonic degrees of freedom which may dominate in the boundary field theory.
Even in weak coupling, the singularities for scalar QED at one-loop order at [28] with m the mass of the charged bosons will not condense towards the real axis for |µ| < m, which is required for the positivity of the quadratic action underlying the perturbation theory in the absence of a Bose condensate, while a singularity as |µ| → m will show up in thermodynamic functions. But in the holographic model considered in this work, the chemical potential appears unconstrained. Therefore it is likely that the analyticity of the static polarization tensor around the real momentum axis reflects a generic feature of the strongly coupling of a fermionic system if the gauge/gravity duality holds and it would be interesting to observe the exponentially decayed oscillation at zero temperature in some strongly correlated electronic systems.
At the moment, we are unable to generalize the analytical works presented above to the case of the longitudinal component of the polarization tensor because of the technical complexity and hope to report our progress along this line in near future. ACKNOWLEDGEMENT We are grateful to the anonymous referee for his (her) criticism and suggestions which Substituting (A4) and (A5) into the expressions of a y , (23), and C yy , (19), with the ratio (A3), the following WKB approximation of C yy (q) is obtained after some algebra: with L 1 and L 2 the two temperature-dependent elliptic integrals: which is of mass dimension the first sign switching zero-point away from u = 0, and, without loss of generality, assume φ > 0 for 0 < u < ζ. Obviously It follows from the eigenvalue equations, Then the positivity of the integral on LHS, ζ 0 φ 0 φ du > 0, implies that E > 0. The proof is completed.
that R (N ) Solving Eqn. (C2) for R (N ) + , we obtain that: where v ± = 1 3 [4 ± i from which the asymptotic expression of G n for large indexes n → ∞ can be extracted as Eqn. (44) follows then.
where the first term on RHS comes from the free propagator and the second term reflect the polarization of the medium. For the holographic polarization tensor considered in this work, σ tr. (q) ∝ C yy (q) . Transforming the second term of Eqn. (D6) into coordinate space and denoting the result by P ( x), we have where we have used polar coordinates for the momentum integral and J 0 (z) is the zeroth order Bessel function. We employ the technique of contour integral to calculate the radial integral on RHS of Eqn. (D7), starting with where H (1) 0 (z) = J 0 (z) + iY 0 (z) is the zeroth order Hankel function of the first kind and Y 0 (z) is the zeroth order Neumann function, Y 0 (z) = 2 π J 0 (z) ln z 2 + an analytic function even w.r.t z .
The integration path of (D8) is chosen to run just above the logarithmic cut along the negative real axis. Since J 0 (q · | x|) and σ tr. (q) are even with respect to q, the non-zero result of I is given by the non-even part of H 0 (q · | x|) with respect to q, and reads To calculate the integral in (D8), we assume that there are no poles on the entire physical sheet and deform the contour on the upper-half q-plane to wrap up the pair of branch-cuts on the right panel of Fig. 1. We obtain that P ( x) = Re I = Re (I + + I − ) with where C ± denotes a contour wrapping up the cut originated from the branch point q ± = ± 1 2 + i µ and turning around the branch point counterclockwisely. For large | x| H (1) 0 (q · | x|) ∼ 2 πq · | x| e i(q·| x|− π 4 ) , we have I ± = 2 π| x| e −i π 4 C ± σ tr. (q) q 3/2 e iq·| x| dq .
According to the analysis in section III A, the branch points are of square root type and σ tr. (q) does not diverging at the branch points. Hence we may write σ tr. (q) q 3/2 = f (q) + g(q) and the functions f (q) and g(q) can be expanded according to integer powers of t ≡ q − q ± .
The term that dominates the large | x| behavior is the leading term of the power series of g(q), which predominately contributes to the contour integral, i.e.
I ± 2 π| x| e −i π 4 g(q ± )e ±iq ± | x| Consequently, where the phase φ depends on the phases of I + and I − . The large-| x| behavior (86) is thereby obtained.