Holographic entanglement entropy in imbalanced superconductors

We study the behavior of holographic entanglement entropy (HEE) for imbalanced holographic superconductors. We employ a numerical approach to consider the robust case of fully back-reacted gravity system. The hairy black hole solution is found by using our numerical scheme. Then it is used to compute the HEE for the superconducting case. The cases we study show that in presence of a mismatch between two chemical potentials, below the critical temperature, superconducting phase has a lower HEE in comparison to the AdS-Reissner-Nordstrom black hole phase. Interestingly, the effects of chemical imbalance are different in the contexts of black hole and superconducting phases. For black hole, HEE increases with increasing imbalance parameter while it behaves oppositely for the superconducting phase. The implications of these results are discussed.


Introduction
Holography is a remarkable concept that plays vital role to understand many features in modern physics-starting from black holes and cosmology to AdS/CFT correspondence. Historically it was first realized through the expression of black hole entropy [1,2] S BH = Area(Σ H ) 4G N (1.1) which was found surprisingly proportional to the horizon area and not the volume. It motivates one to think that the bulk degrees of freedom somehow "holographically" mapped to the surface/horizon degrees of freedom which results this non-extensive behavior in entropy. Later on this enabled 't Hooft, Susskind and others [3]- [6] to explain our Universe using the concept of holography. Most recent additions to this list are AdS/CFT correspondence and entanglement entropy. AdS/CFT correspondence, first conjectured by Maldacena [8], is a realization of much discussed proposition of 't Hooft [7] on the large N limit of strong interactions. AdS/CFT correspondence states that a supergravity theory in AdS 5 × S 5 is a "dual" description of strongly coupled N = 4, SU (N ) SYM theory "residing" in its boundary in the limit of N → ∞. Here S 5 is compactified to a radius L >> l s (l s = string length) which is also the radius of curvature of AdS spacetime. Therefore effectively a five dimensional gravity theory is "holographically" reduced to a four dimensional conformal field theory. This "duality" in two theories was quantified by Witten [9], by identifying the bulk field with boundary operator and n point correlation functions in terms of derivatives of the gravitational partition function with respect to the boundary value of that field. In support of this yet unproven AdS-CFT correspondence, there exist many direct and indirect evidences, for example-(i) the isometry group SO(4, 2) of AdS 5 being isomorphic to the conformal group of the SYM theory, (ii) matching of correlation functions calculated separately from CFT and that using AdS/CFT tool and many others (for more see reviews [10]- [12]), which makes it robust. It is true that the exact reason/s why such two apparently different theories should behave so cohesively is/are not known, but the role of holography is undeniable, and therefore it needs further attention. The major applications of this correspondence can be broadly classified in two parts: one which are in the context of QCD (for a review [13]) and the other in the context of condensed matter physics [14,15]. It is the second case which is our interest in this paper.
The role of holography in the much focussed issue of entanglement entropy has been recently highlighted by Ryu and Takayanagi [16,17]. If a system described by certain quantum field theory or some quantum many body theory is divided into two parts say A and B, then entanglement entropy S A of the subsystem A is a non-local quantity which measures how the above systems are correlated, quantum mechanically, with each other. In defining S A one traces out the degrees of freedom of the space-like submanifold B which is not accessible to an observer in A. Anyone familiar with the concept of black hole entropy would find this definition very much analogous to the case where an observer outside the black hole event horizon has no access to the information inside. Indeed this is one of the motivation for the authors of [16,17] to heuristically propose an "holographic" formula of entanglement entropy, given by where γ A is the d dimensional surface whose d − 1 dimensional boundary ∂ γ A matches with the boundary ∂ A of the field theory subsystem A. Of course the choice for such a surface is not unique. In this context it is suggested that, this surface, among various choices, should be the minimal. This minimal surface is found by extremizing the area functional and finding out the solution (in case there are more than one) whose area takes the minimum value. At the present status the HEE formula (1.2) is not conclusively proven 1 . Nevertheless there is a list of evidences which bolsters the robustness of this formula. One direct evidence comes from the AdS 3 /CFT 2 context where the CFT result of the entanglement entropy S A = c 3 log l a , matches with the holographic calculation, in which l is the width of the subsystem A and c = 3R 2G N relates the central charge c with the radius of curvature R of the AdS 3 spacetime. Although this evidence has not been explicitly seen in higher dimensional cases (AdS d+1 /CF T d with d > 2), there are more compelling arguments which put confidence on (1.2) ( for details see reviews [21,22] and references therein). The major usefulness of the HEE is the same as the basic principle of AdS/CFT: overcoming the computational difficulties of complex many body field theoretic calculations in terms of much more simpler classical gravity calculations.
Our work in this paper is motivated by a recent study by Albash and Johnson [23,24], where it is argued that HEE might be an useful physical quantity for characterizing holographic superconductors. They found that the finite part of the HEE (S f ) of superconducting and non-superconducting phases follow a pattern which enables one to identify the phase of a system. For a given system size and for all temperatures below the critical value T c , S f takes a lower value for the superconducting phase compared to its value for the corresponding non-superconducting (black hole) phase. Whereas for temperature higher than T c , where no superconducting state appears, S f only exist for the latter phase. The reason behind the smaller value of HEE for the superconducting state is explained in terms of number of the degrees of freedom that the system possesses. This number is higher in the black hole phase but as the superconductor forms some of them are condensed and results into a lower HEE. Further works in this direction are also reported in [25]- [27]. It should also be mentioned that apart from the finite value of HEE given by S f there is also a diverging part. However, such a divergence is not the characteristic of the holographic calculation only, it also appears in the continuum limit of the conformal field theory calculations. One can avoid such diverging terms by introducing a UV cut off through the introduction of a lattice spacing in the expression of entanglement entropy. In the holographic calculation, the divergences can be avoided if the boundary of the minimal surface is chosen slightly away from the asymptotic infinity by choosing the appropriate limit of the radial coordinate.
In this paper we explore the behavior of HEE in an imbalanced mixture of two fermionic systems with a mismatch in their chemical potential [28]. One motivation of choosing this system is that the imbalanced superconducting systems are quite interesting in the condensed matter framework (discussed in more detail in section (2.1)). We organize this paper in the following manner. In the next section we set the platform by introducing the imbalanced system from condensed matter and holographic perspectives. Section 3 is devoted for providing the equations of motion whose solutions are discussed in next two sections (4,5). In Section 4 we consider the case where only RN-AdS black hole solution exists and compute its HEE for various values of the imbalanced parameter β. In Section 5 we consider the case where superconducting state appears and compute its HEE for different β. In both cases we plot HEE with respect to the system strip width l of the field theory subsystem. Here we also compare the values of HEE for the normal (black hole) and superconducting states. Finally we conclude in Section 6.

Condensed matter description
Imbalance in the population of spin-up and spin-down fermions leads to exotic superconducting states. In the context of solid-state superconductors the existence of these exotic superconducting states were theoretically proposed in 1960s by Sarma [31] and Maki [32] in high magnetic field and low temperatures. Soon after Fulde and Ferrell [33] and Larkin and Ovchhinikov [34] extended this proposal and predicted a spatially inhomogeneous superconducting state which is presently known as FFLO state. This exotic imbalanced supercon-ducting state is unique as it has a spatially-modulated order parameter, while the standard Bardeen-Cooper-Schrieffer(BCS) superconducting state has a spatially-homogeneous order parameter. The existence of the FFLO state is surprising in the sense that it retains superconductivity overcoming the orbital and Pauli-paramagnetic pair-breaking effects, even at very high magnetic fields. For this reasons the imbalanced systems has been studied vigorously -both theoretically and experimentally. Theoretical studies on imbalanced systems often focus on the possibility of exploring imbalanced superconducting states in different physical systems, for example, in population-imbalanced Ultracold atomic gases [38,39,40,36], optical lattices [41], heavy-fermionic superconductor CeCoIn 5 [42], twodimensional organic superconductors [43,44,35,37] and quark matter core of the neutron stars [45,46]. The experimental search is a topic of vigorous research till date as it is very hard to pinpoint this state in the phase diagram. In an experiment involving an imbalanced system, one can find the imbalanced state if: (i) the superconductor is in the clean limit and (ii) the value of Maki parameter is greater than 1.8. The most promising experimental systems in this context are the heavy-fermionic superconductor CeCoIn 5 [47,48,49,50] and quasi two-dimensional(2D), organic superconductors like κ-(BEDT-TTF) 2 Cu(NCS) 2 , in which BEDT-TTF is bisethylenedithio-tetrathiafulvalene [51,52]. So, even 50 years after its prediction, this field of imbalanced superconductivity remains an active field full of surprises (For a review see [53] and references therein.).

Holographic description
More recently there has been a lot of effort [54,28,55,56,57] to understand the imbalanced systems using holography and AdS/CMT. Generally, the bulk gravitational Lagrangian which holographically describe an imbalanced system is given by which is comprised of the AdS gravity with Λ = − 6 L 2 , two U (1) gauge fields with field strengths and one scalar field (φ) with potential which is charged under U A (1) but uncharged with respect to the other. As known from the AdS/CFT correspondence mass of the above bulk scalar field dictates the conformal dimension (∆) of the dual field in the following manner This relation is particularly helpful to capture the physics of an field theory operator with a conformal dimension of interest. For example to describe a Cooper pair type condensate which has ∆ = 2, one fixes the mass of the bulk scalar field to be m 2 = − 2 L 2 . Note that this choice does not violate the Brietelhoner-Freedman bound which for this case is m 2 ≥ − 9 4L 2 . Since our interest lies in this theoretical aspect, in this paper, we will fix the above mass value for the bulk scalar field in all our computations. For completeness it should be mentioned that other than mass, the scalar field also has a charge q, and different values of charge lead to different physical properties in the dual field theory.
The above description of the gravitational system has the minimal ingredients needed to describe the superconductivity in the imbalanced systems. Starting from the equations of motion which include Einstein equations, Maxwell equations and a scalar field equation, one looks for the cases where the scalar field is zero and non-zero. The vanishing of the scalar field gives a normal Reissner-Nordström black hole phase. On the other hand, if one finds a non-zero scalar field it is understood that a condensate has been formed in the dual field theory. Of course this situation has a serious contradiction with the black hole no-hair theorem that supports the vanishing scalar field, but the fact of getting non-zero scalar field in the context of holographic superconductors hints that one needs to re-examine the no-hair theorem itself [29,30]. The above statement is true for any holographic superconductor. For the imbalanced case, with two U (1) gauge fields with unequal chemical potential, we have the following additional advantage.
In imbalanced superconducting systems Cooper pair forms between two fermionic species with unequal chemical potentials (say µ 1 and µ 2 ). Now to capture this behavior in the dual gravitational theory, one needs two U (1) bulk fields (say U A (1) and U B (1)) with field strengths A a which accounts total chemical potential 2µ = µ 1 + µ 2 and B a which accounts the mismatch 2δµ(= βµ) = µ 1 − µ 2 of those fermionic species in boundary theory.
With these preliminaries we now move to the next sections to deal with the equations of motion and to compute the HEE separately for black hole and superconducting phases.

Equations of motion
Extemizing the Lagrangian (2.1) with respect to various fields one has the following set of equations: Einstein equation, where the energy-momentum tensor of the matter field is defined as T ab = 2 √ −g δLmatter δg ab . Maxwell equations for A a and B a fields reads where the scalar/gauge coupling takes place only in U A (1) sector.
In addition there is also a scalar field equation given by In order to proceed further we consider the follwing background metric where χ(r) accounts for the backreaction due to matter fields. For a case where backreaction is negligible one sets χ = 0. For all matter fields, the anstaz is assumed to be homogeneous Now one finally unwind all field equations by substituting the ansatz. The final set of equations now have two independent Einstein equations two Maxwell equations for ψ and v fields and a scalar field equation In the remaining part of our work we will look for the simultaneous solution of the above set of equations to compute the HEE. From now on we set 2k 2 4 = 1, L = 1.

HEE for the normal (black hole) phase with varying β = δµ µ
At high temperature (above T c ), when no superconductivity appears, one has a vanishing bulk scalar field. For such a case the right hand side of the Maxwell equation (3.2) vanishes and the resulting solution of the set of field equations is a doubly charged Reissner-Nordström black hole given by the metric where the gauge fields are Hawking temperature of this RN-AdS spacetime is given by To compute HEE we change the radial coordinate from r to z = r H r . This redefinition is not necessary as one can also use r as a radial coordinate and carry out the calculation, but since we find z to be more helpful in the context of superconducting phase, to be discussed on the next section, here we make such a transformation. In the t, z, x, y system the metric looks like where 'h' is the determinant of the induced metric of the codimension 2 hypersurface and in the second equality prime denotes derivative with respect to x. Equation (4.12) also tells us that the system is equivalent to one defined by the Lagrangian L = 1 r H z 2 r 2 H + z 2 . In order take into accout that the surface is minimal, we extremize the Lagrangian. This extremization problem has a constant of motion which is nothing but the canonical Hamiltonian. In this way we obtain a measure of how the entangling surface is extended within the bulk (towards the horizon) and gives an infrared cut-off (z 0 ) on the integrating variable, given by Then converting the integrating variable from x to z the final expression of HEE reads as (4.14) = S f + S div. On the other hand the width of the subsystem 'A' is expressed as In order to integrate (4.14) and (4.17) one has to substitute the metric function g(z), set the UV cut-off to a small value and consider z 0 near to the horizon. By changing z 0 it is possible to study the behavior of S f as a function of the strip width l, which we have plotted in Figure 1 in which each of the curves corresponds to different value of β. The larger l corresponds to the infra-red limit [23]. In addition to conforming the earlier results [23], from this set of plots we find that, with increasing imbalance of the chemical potential β, HEE of the non-superconducting phase of same width gradually increases. If one considers HEE as a measure of the number of degrees of freedom of a system, the plots in figure 1 tell us that a system of given width with larger β has more degrees of freedom. Now we move to the next section where we examine the case where superconducting state appears.

HEE for the superconducting phase with varying β = δµ µ
We now proceed with the calculation of the HEE when the black hole has developed a scalar hair, in other sense, a superconducting state has been formed in the boundary field theory. For that we first express g(z) = r 2 H z 2 +h(z) which is helpful for further computations. The equations of motion (3.7 to 3.11) now reads, To proceed further we solve the above equations using Taylor series expansion. The Taylor series expansion of the fields at the horizon z H = 1 reads In the Taylor expansion of h H (z), we set the first term as −r 2 H to fulfil the requirement that the metric coefficient g(z) vanishes at the horizon. Also, in order to prevent the gauge fields from acquiring infinite norm at the horizon one needs ψ H (z = 1) = 0 = v H (z = 1). Therefore upto a second order expansion one has twelve unknown coefficients in the Taylor expansions. However not all of them are independent, they are related by five equations (5.1 to 5.5). This reduces to the number from twelve to seven. Shortly we will also see that in order to match the above near horizon expansion of fields to corresponding boundary expansions, there will be additional five equations coming. As a result the full parameter space would have two independent coefficients, defined in the near horizon region and whose variation actually set the dynamics of the boundary theory. These two independent variables are chosen to be χ H0 and φ H0 . In our analysis we assign a value to χ H0 = 2 (which provides an estimate of the backreaction on the metric due to fields), and vary φ H0 to generate the dynamics at the boundary. Particularly our interest lies in the calculation of HEE by using its expression (4.14). The only difference from the RN-AdS case is arrested in the expression of g(z) which should be modified when the superconductor forms. In other words we need to find h(z) which is defined from the near horizon region to the boundary and numerically integrate (4.14) to calculate S f . The temperature of this superconducting state is given by [28] The critical temperature corresponds to the vanishing of the scalar field φ, which is achieved when all the Taylor oefficients in (5.10) vanishes identically. above two cases it is clear that the effect of β on superconducting state is milder-plots in figure (2) are much more closer to each other than in figure 1, though the range of β differs slightly for the cases. In Figure 3 we compare the relative values of the HEE between the black hole and superconducting states for a given mismatch parameter β. For different values β = 0, 0.5, 1 S f vs. l plot for the superconducting state remains below the RN-AdS case which is reassuring.

Conclusions and Discussions
In this paper we computed Holographic entanglement entropy (HEE) starting from a gravitational theory which describes imbalanced superconductivity below the critical temperature and and RN-AdS black hole at high enough temperature (more than critical temperature). It is found that HEE for the superconducting state is always lower than the black hole/normal phase. This is true for any value of the imbalance parameter (β), for which the superconducting phase exists. While for increasing β, HEE for black hole increases, it slowly decreases when superconductor forms. The fact that HEE for imbalanced holographic superconductor (also for other cases reported earlier [23]- [27]) is less than the black hole might insist one to consider this as a good physical parameter to identify the preferable state below T c . Usually for a condensed matter system one uses free energy in order to say anything about the preferable state and further studies are required to confirm whether HEE is able to serve the same purpose or not.