Kink Crystalline Condensate and Multi-kink Solution in Holographic Superconductor

We explore inhomogeneous solutions of a holographic superconductor. At lower T/μ, we find kink crystalline condensates in which condensates are well-fitted by the Jacobi elliptic functions. We present the relation between T/μ and the elliptic modulus ν, which determines the profile of the functions. We also find multi-kink solutions, which is not realized as a solution within the conventional Ginzburg-Landau theory with secondorder derivative and φ4 potential. This shows that the holographic superconductor model captures the physics beyond the conventional Ginzburg-Landau theory.

model or the chiral Gross-Neveu model [41][42][43][44][45]. Recently, the connection between the Nambu-Jona-Lasinio model and the non-linear sigma model is also of particular interest to the inhomogeneous condensate [46]. Moreover, the generalized Ginzburg-Landau approach with higher derivatives of the order parameter has been developed and it has been revealed that the LOFF phase of the chiral condensate is energetically preferred in Ref. [47].
In this study, we investigate inhomogeneous solutions of the holographic superconductor. Our work is an extension of the previous reports on holographic dark solitons [16,17,19] and we focus on inhomogeneous solutions with constant chemical potential in space. In particular, we explore inhomogeneous solutions with multiple nodes, which can be identified as the LO-like phase or a multi-kink solution. The emergence of the multi-kink solution shows that the holographic superconductor model captures phenomena beyond the conventional Ginzburg-Landau model. We also compute the thermodynamic potential of these solutions. This paper is organized as follows. In Sec. 2, we briefly review the holographic superconductor. We introduce the holographic superconductor model and review the derivation of homogeneous solutions and its extension to the inhomogeneous case. We obtain the solution with kink crystalline condensate and the multi-kink solution. We find that the charge density becomes minimum at the location of condensate's node. In Sec. 3, we compute the thermodynamic potential of solutions. We find that inhomogeneous solutions have lower value of the thermodynamic potential than that of the normal state and higher values than that of the homogeneous solution. In Sec. 4, we present conclusion and discussion.

Holographic Setup
To begin with, we review the basic setup of the holographic superconductor. We consider the Einstain-Maxwell theory with a negative cosmological constant and a charged complex scalar field Ψ in (3+1) dimensional spacetime [8][9][10]. The action of this model is given by The covariant derivative is defined by D µ Ψ = (∇ µ − iqA µ ) Ψ, where A µ is the U (1) gauge field and q is the charge of the scalar field. Here, the AdS radius is set to be 1. µ and ν are indices of the four dimensional bulk spacetime. This bulk system with a U (1) gauge symmetry is dual to the strongly coupled field theory with a global U (1) symmetry in (2+1) dimensions [48]. If one rescales the scalar field and the gauge field by q and takes the limit q → ∞ with the amplitude of fields fixed, the matter sector is decoupled from the gravity sector. In this limit, the so-called probe limit, one can ignore the backreaction of the matter sector to the background geometry. Then the Einstein equations yield the (3+1) dimensional planar AdS black hole geometry Here, z H is the location of the black hole horizon. The Hawking temperature is given by T H = 3/(4πz H ), which corresponds to the temperature of the dual field theory. The dynamics of this system is determined by the action of the matter sector We assume that the potential is given by V = −m 2 |Ψ| 2 . For simplicity, we set q = z H = 1.
The equation of motion for the complex scalar field is given by The Maxwell equations are obtained as

Homogeneous Solutions
In this subsection, we review the homogeneous case of the holographic solution by following Ref. [8]. First, we take the ansatz that the scalar field and the gauge field, respectively, have the following form where we employed the gauge A x = A y = A z = 0. Since the phase of Ψ must be constant in space in this gauge from the Maxwell equations (2.6), we take Ψ to be real. Then, the equation of motion for the scalar field becomes

8) and the Maxwell equation becomes
Hereafter, we use the notation y = dy dz and y = d 2 y dz 2 . We choose the mass of the scalar field to be m 2 = −2, which is above the Breitenlohner-Freedman (BF) bound m 2 BF = −9/4 for stability of AdS spacetime [49].
Let us consider boundary conditions in order to solve these equations of motion. The asymptotic forms of each field are obtained from these equations of motion at the vicinity of the AdS boundary, (2.10) According to the AdS/CFT dictionary, µ and ρ correspond to the chemical potential and the charge density, respectively. The condensate is defined by O 2 = √ 2Ψ (2) . For the scalar field, both of these two leading terms are normalizable since we choose a scalar mass close to the BF bound. There are two different possibilities to identify the source and the condensate of operators in the dual field theory. Since we are interested in the non-zero vacuum expectation value for the condensate without the corresponding source, there are two possible boundary conditions, Ψ (1) = 0 or Ψ (2) = 0. In this study, we focus on the condition Ψ (1) = 0. Thus, we obtain boundary conditions at the AdS boundary: In addition, we impose regularity conditions at the black hole horizon as boundary conditions: One can obtain these conditions by expanding Eqs. (2.8) and (2.9) with respect to z near black hole horizon and assuming the regularity at the black hole horizon (z = 1). Note that one of the solutions which satisfy these boundary conditions is Ψ = 0 and A t = µ(1 − z). In this trivial solution, there is no scalar hair and thus this solution is referred to as the normal state. On the other hand, a non-trivial solution Ψ = 0 is referred to as the superconducting state. In order to find a non-trivial solution, we have to solve Eqs. (2.8) and (2.9) numerically with above the boundary conditions (2.11) and (2.12). As an example of non-trivial solutions, we plot the scalar field Ψ(z) and the gauge field A t (z) with µ = 6.5 in Fig. 1. The condensate O 2 and the charge density ρ can be computed from asymptotic form of these solutions. A finite condensate implies that the black hole in the bulk has developed scalar hair.

Inhomogeneous Solutions
In this subsection, since we are interested in the inhomogeneous solutions, we assume that the fields depend on the coordinates z and x. The only non-zero component of the gauge field is A t = A t (z, x). In the same gauge choice as the homogeneous case, one can set Ψ to be real. For convenience, we define the field Ψ(z,   real function. Thus, equations of motion can be rewritten as Though these non-linear partial differential equations are difficult to solve analytically, one can solve them numerically. Note that these equations can give homogeneous solutions of both the normal state and the superconducting state. First we consider boundary conditions. The gauge field must vanish at the horizon as in the homogeneous case: A t (1, x) = 0. The boundary condition of the scalar field is obtained by imposing the regularity condition to equations of motion at the horizon. We consider the case that the chemical potential is constant in space at the AdS boundary: For the scalar field, we assume the same condition Ψ (1) = 0 as in the homogeneous case. In terms of φ(z, x), this condition is corresponding to φ(0, x) = 0.
In addition, we need to impose the boundary conditions for the x coordinate. First, we employ the Neumann boundary condition at x = ±L/2: for technical reason. Then we will see the behavior of the solution with taking L larger. If the solution asymptotes to a fixed profile and is stable against the variation of L at a sufficiently large L, we adopt the solution. In summary, our boundary conditions are following In order to perform numerical calculations, we employ the Newton-Raphson relaxation method. In this scheme, we first prepare a seed configuration with L = 4 and iterate this until it relaxes to a stable solution within a small numerical error. After we obtain the solution, we examine whether it survives when L becomes sufficiently large. In our study, we confirm this up to L = 40. We also introduce a cutoff ε = 10 −10 at the AdS boundary in order to avoid the numerical divergence.
We have to prepare an appropriate seed configuration in order to find inhomogeneous solutions. One candidate is an odd function of the spatial coordinate x for the scalar field as discussed in [16,17], which has a single node on the x-axis. After we iterate the relaxation process, we obtain inhomogeneous solutions of φ(z, x) and A t (z, x) as shown in Fig. 2. In addition, if we control the number of nodes of an initial seed along the x-axis and the value of the chemical potential, we also find other solutions as shown in Figs. 3 and 4.
We obtain the condensate O 2 and the charge density ρ from the asymptotic form of each field in the vicinity of the AdS boundary, (2.19)  Here, it is convenient to normalize the condensate and the charge density by using the chemical potential. In Fig. 5, the condensate O 2 /µ 2 and the charge density ρ/µ 2 profiles obtained from above bulk solutions are shown. We find that solutions with higher number of nodes appear for larger chemical potentials. We also find that dips of the charge density are located at the nodes of the condensate. Furthermore, we study the T /µ dependence of inhomogeneous solutions with the number of nodes fixed. Fig. 6 shows the condensate and the charge density profiles with three nodes for each T /µ. These numerical results imply that our inhomogeneous condensates can be characterized by the Jacobi elliptic functions, which is defined by M (x) = a sn(bx, ν). Here, a and b are constants and ν is the so-called elliptic modulus, which determines the profile of functions (0 ≤ ν ≤ 1). It has been known that the Jacobi elliptic functions is an analytic solution of the (1+1) dimensional Gross-Neveu model, which exhibits the solitonic inhomogeneous behavior (see Refs. [43,44] and references therein). Assuming that the condensate profile follows the Jacobi elliptic functions, we determine the T /µ dependence of ν by fitting as shown in Fig. 7.
Here, we confirm whether the inhomogeneous solutions survive when L becomes large. We find two categories of solutions. The first category of solutions has a property that the number of nodes increases when L becomes large. These solutions have a periodic structure as already shown in Fig. 6. Thus, we regard this category of solutions as the kink crystalline condensate (LO-like phase). The second category of solutions shows a property that the number of nodes is unchanged when L becomes large. This category of solutions can be further classified into two types. The first type of solutions has a feature that the distance between the nodes can be larger when we increase L. Each kink and anti-kink is considered to be independent at sufficiently large L, and we regard these solutions as a dilute gas of kinks and anti-kinks. An example of this type of solutions for L = 40 is shown in Fig. 8. The other type of solutions has a notable property that the distance between the nodes is almost unchanged when L becomes large. We show the condensate and the charge density profiles with two or three nodes for L = 40 in Fig. 9. This type of solution corresponds to the multi-kink solution. For example, the condensate with two nodes corresponds to the kink-anti-kink solution. Surprisingly, we find that the holographic superconductor model achieves to yield the multi-kink solution, which is not the solution of the conventional nonlinear Schrödinger equation (conventional Ginzburg-Landau theory) [50,51]. It is known that the multi-kink solution is obtained from the Gross-Neveu model or the Ginzburg-Landau theory with higher derivatives and with the higher order potential terms. Thus, the holographic superconductor model produces phenomena beyond the conventional Ginzburg-Landau model.

Thermodynamic Potential
In this section, we study the thermodynamic potential for inhomogeneous superconducting states in order to analyze their stability. According to the AdS/CFT dictionary, the thermodynamic potential of the field theory on the boundary corresponds to −L os , where L os  is the on-shell Lagrangian. We derive the thermodynamic potential for our inhomogeneous superconducting state following Ref. [20]. We rewrite the action (2.4) as the explicit form The on-shell action is obtained by integrating by parts and substituting equations of motion (2.13) and (2.14) into Eq. (3.1).
where we have also used boundary conditions given in the previous section and f (1) = 0. In this study, furthermore, since we assume that Ψ (1) = 0 in (2.10), ΨΨ /z 2 in Eq. (3.2) vanishes. Hence, the thermodynamic potential is obtained as where we change the notation of the scalar field: Ψ → zφ/ √ 2. The thermodynamic potential per unit volume is given by where Vol = L dy. In Fig. 10, we show ∆Ω/Vol as a function of T /µ for kink crystalline solutions whose number of nodes is denoted by N , where ∆Ω is the difference of the thermodynamic potential between the normal state and the superconducting state. We find that kink crystalline solutions that have higher number of nodes are realized in lower T /µ, namely lower T and/or larger µ. Moreover, we find the inhomogeneous solutions have lower ∆Ω than the normal state, but higher than homogeneous solutions. In other words, the inhomogeneous solutions that we found in this study are metastable. We confirm that this qualitative behavior remains unchanged when L becomes sufficiently large.

Conclusion and Discussion
In this paper, we have investigated the inhomogeneous solutions of the holographic superconductor for finite temperatures. We find that the three types of solutions of the condensate: the kink crystalline condensate (LO-like phase), the dilute gas of kinks and anti-kinks, and the multi-kink solution. It is known that the LO phase and the dilute gas of kinks and anti-kinks are realized in the conventional Ginzburg-Landau model with φ 4 potential. On the other hand, this conventional Ginzburg-Landau model does not include the multi-kink solution, for instance, the kink-anti-kink solution. Therefore, our results show that the holographic superconductor model exhibits the physics of superconductor, which appears in the BCS model, the Nambu-Jona-Lasinio model, or the chiral Gross-Neveu model, beyond the conventional Ginzburg-Landau model.
We confirm that the condensate with higher number of nodes emerges at lower T /µ. In the LO phase, the profile of the condensate can be characterized by the Jacobi elliptic functions. A similar discussion is given in QCD [39]. Assuming that the condensate profile with multiple nodes follows the Jacobi elliptic functions, we determine the T /µ dependence of ν by fitting. We reveal that the value of ν approaches asymptotically to ν = 1 at smaller T /µ. We also perform the stability analysis by evaluating the thermodynamic potential. The computation of the thermodynamic potential shows that the inhomogeneous solutions that we found in this study are metastable.
Before closing the conclusion, we have some remarks. First, we are interested in the connection between the holographic kink crystalline condensate found in this work and the LOFF phase. In the LOFF phase, the spin degree of freedom plays central role for the stabilization of the LOFF phase. Therefore, it would be interesting to apply an external magnetic field so that we can access the nature of the possible spin degree of freedom in the model. Moreover, it is also known that Aharonov-Bohm flux increases the critical temperature of the LOFF phase [25,52]. It will be straightforward to apply our method to these setups. These perspectives are related to another interesting question of whether the inhomogeneous solution in holography can become the ground state. It can be expected that inhomogeneous solutions are energetically favored with a different form of the potential, for example, which includes higher derivative terms as found in [47]. Since we have restricted our analysis to the probe limit, we ignore the backreaction of the matter sector into the gravity sector, which becomes important at low temperature. It is straightforward to consider the backreaction in our case by following recent work [53]. This study partially reveals that the holographic superconductor model has a richer structure than the conventional Ginzburg-Landau theory. It would be interesting to investigate how the highly non-trivial inhomogeneous solutions, such as the multi-kink solution realize instead of the simple setup in the holographic superconductor model and that would lead us to reveal a cornucopia of new physics.