Phase Diagram of a Holographic Superconductor Model with s-wave and d-wave

We consider a holographic model with a scalar field, a tensor field and a direct coupling between them as a superconductor with an s-wave and a d-wave. We find a rich phase structure in the model. The model exhibits a phase of coexistence of the s-wave and the d-wave, or a phase of an order competition. Furthermore, it has a triple point.


Introduction
Recently, AdS/CFT correspondence [1,2,3] is studied actively in superstring theory. Its application to other physics is expected as an innovative method. For example, it is applied to nuclear physics and condensed matter physics. See, for example, [4,5]. Holographic superconductor is one of its applications. Motivation for the holographic superconductor is to better understand physics from the relation between gauge theory and gravity theory. After being shown that a simple scalar model has a property like superconductor or superfluid [6], various models have been studied.
One example is an anisotropic superconductor [7]. For condensed matter physics, rotational symmetry of an angular momentum of a cooper pair is important symmetry. A cooper pair of some superconductors, such as a copper oxide [8], has nonzero angular momentum and they are called as anisotropic superconductor. Motivated by them, holographic models of a vector field or a tensor field were studied [9,10,11,12,13].
In this paper, we consider a holographic model with a scalar field, a tensor field and a direct coupling between them as a superconductor with an s-wave and a d-wave.
We find a rich phase structure in our model. Depending on the direct coupling, the model exhibits coexistence of the s-wave and the d-wave, and/or order competition, and has a triple point.
The organization of the paper is as follows. In section 2, we explain our holographic model with the s-wave and the d-wave and its equations of motion. In section 3, we calculate solutions of the model and their free energy densities and study their properties. In section 4, we study the properties of the phases for a range of values of the direct coupling and see that our model has the rich phase structure as Fig. 6. Section 5 is a summary and a discussion.

Our gravity model
In this section, we explain our holographic superconductor model with an s-wave and a d-wave. Some holographic models with a scalar field or a tensor field are studied previously [6,11,12,13]. These fields are interpreted as order parameters. To study coexistence and competition of the two order parameters, we consider a scalar field, a tensor field and a direct coupling between them.
For exotic superconductivity, temperature is essential and two-dimensional space is considered to be indispensable. To accommodate them in holography, we usually use a four-dimensional AdS planar black hole metric where z = 0 is the AdS boundary and z = z h is the black hole horizon. Temperature of a superconductor corresponds to the Hawking temperature T of this black hole Lagrangians with a scalar field or a tensor field were proposed as an s-wave or a d-wave superconductor [6,11,12,13]. To combine them, we consider a Lagrangian with a Maxwell field A µ , a scalar field ψ, a symmetric tensor field Φ µν and a direct coupling constant η between ψ and Φ µν as where L d is an effective action of a spin two field for the d-wave part [12] 1 . ψ and Φ µν correspond to the s-wave and the d-wave.
Our main purpose to observe variety of the phase diagram, we consider the direct coupling η only and analyze our model in the probe limit 2 . This simplifies our calculations, although we lose the generality of the model.
We set the mass and the charge of the fields [17] as The mass of each field corresponds to the dimension of the order parameters. m 2 s is negative, but it does not lead to an instability because it is above the Breitenlohner-Freedman bound [29]. e s and e d are interpreted as effective charge of the cooper pairs. Our choice of the parameters (2.10) and (2.11) is for the solutions of our model to have a rich phase structure. In fact, e s and e d contribute to the effective mass squared through the covariant derivative (2.7) in (2.5) and (2.6).
We suppose that the superconductor is homogeneous and there are isotropic and anisotropic cooper pairs. Our ansatz for the fields corresponded to them naturally is 12) and the other components are zero. We also set ψ, ϕ and φ to be real for simplicity. Under this ansatz, our model comes down to a two-scalar model with a direct coupling only. The equations of motion are As noted later, our model is similar to a two-scalar model and comes down to it under the ansatz (2.12). In two-scalar models, one can consider also the following Josephson coupling between two scalar fields [24] In our model, correspondingly we can consider a coupling between the scalar field and the tensor field However, we do not consider this coupling because (2.9) is zero under the anzats (2.12). 2 In this paper, we do not consider consistency of the direct coupling and the probe limit. and asymptotic solutions of the equations of motion around the boundary z = 0 are In order to solve the equations of motion, we need boundary conditions. We set them as In holographic superconductor, coefficients of the asymptotic solutions (2.16), (2.17) and (2.18) correspond to external fields and their responses. We regard µ and ρ as a chemical potential and a charge density. Moreover, we regard ψ (2) and ϕ (2) as expected values of the order parameters Fixing µ, we will calculate behavior of O s and O d by changing T .
In the next section, we will find four types of solutions of the model: • solution of the normal conducting phase (the normal conducting solution) 3 , • solution of the s-wave superconducting phase (the s-wave single solution), O s = 0, ϕ = 0.
• solution in which the s-wave superconductivity and the d-wave superconductivity coexist (the s+d coexistent solution),

Solutions of the model
In this section, we calculate the solutions of the equations of motion by a numerical method using Mathematica. By the symmetry of the metric (2.1), we can fix µ = 1. Moreover, we set L = 1 for a numerical calculation.

Single solutions
First, we calculate the single solutions. The single solutions do not depend on η, since ψ = 0 or ϕ = 0. A numerical result is shown in figure 1. The left blue curve is the s-wave single solution, the right red curve is the d-wave single solution and T d is the phase transition temperature at which the d-wave condensation begins. One can see that the d-wave single solution begins to condense at higher temperature than that of the s-wave single solution. Generally, if mass squared of fields m 2 is small, it condenses at a high temperature. In our model, we set e d larger than e s and the effective mass squared of the tensor field is smaller than that of the scalar field at high temperature.

s+d coexsistent solutions
Second, we calculate the s+d coexistent solutions. Figure 2

Free energy density
In order to see which solution is favored, we compare the free energy densities of the solutions. In holographic superconductor, the free energy corresponds to temperature times the on-shell Euclidean action by assuming the GKP-W relation [2,3]. Thus, we will calculate the on-shell Euclidean action by substituting the solutions. Usually, the Euclidean action includes the Gibbons-Hawking term [30] and a counter term. They are needed for a well-defined variational principle and dealing   with divergences. In our calculation, the contribution of them is same for each solution since we consider the probe limit and (2.19). Therefore, in order to compare the magnitude of the on-shell Euclidean action, it is sufficient to consider the Euclidean (2.4) only. Substituting (2.12), the Euclidean (2.4) is where we use t as the imaginary time. By defining β = dt and V 2 = dxdy, the on-shell Euclidean action S on-shell is written by where we have used a partial integration and the equations of motion. (3.2) corresponds to the free energy density because a period of the imaginary time can be interpreted as the thermodynamic β. By using the formula (3.2), we compare the free energy densities of the normal conducting solution F n , the s-wave solution F s , the d-wave solution F d and the s+d coexistent solution F s+d . Then we find following facts: • F n is larger than F s and F d . Hence, the normal conducting phase is favored at T > T d only.
• F d is smaller than F s at high temperature and F s is smaller than F d at low temperature. Near T /T d = 0.716, the magnitude correlation of F s and F d is reversed and the s+d coexistence solution exists.
• F s+d of η = 0 is smaller than F s and F d . However, F s+d of η = 1/10 is larger than F s and F d . Therefore, the s+d coexistence phase of η = 0 is favored, but that of η = 1/10 is not favored.
The relation among the free energy densities for each solution is explained again in figure 5.

Phase diagram
In this section, we study a phase diagram of our model by using the property of the free energy density. Figure 4 is the order parameters in the favored phases at T . Fig. 4 (left) is for the phase of η = 0. Lowering the temperature, the phase is changed in the following order: the normal conducting phase, the d-wave phase, the s+d coexistence phase and the s-wave phase. In these phase transitions, the order parameters change continuously. Fig. 4 (right) is for the phase of η = 1/10. Lowering the temperature, the phase changes in the following order: the normal conducting phase, the d-wave phase  Figure 4: Left figure is the order parameters for the favored phase of η = 0 and right figure is that of η = 1/10. The s+d coexistent phase is found for η = 0, but that of η = 1/10 is not favored. and the s-wave phase. In the latter phase transition, the order parameters change discontinuously since the s+d coexistence phase is not favored. The physics of fig. 4 can be explained as follows. Since a large e d makes the effective mass of the d-wave small, the d-wave phase condenses first. Lowering the temperature, the effect of m 2 s becomes important and the s-wave phase appears. There is a case that the s+d coexistence phase exists to connect the two phases continuously.
In the Lagrangian, ηψ 2 ϕ 2 /2 is the term corresponding to the potential energy. If η is small enough, the s+d coexistence solution is advantageous about energy. Thus, the range of T in which the s+d coexistence solution of η = −1/10 exists is larger than that of η = 0. If η is large enough, the s+d coexistence phase cannot exist, and one example shown in Fig. 5 is for η = 1/10. Figure 5 is the free energy of each solution at T . This figure is a rough sketch and the scale is not correct. The blue line is F s , the red line is F d , the purple line is F s+d of η = −1/10, the orange line is that of η = 0 and the green line is that of η = 1/10. The solution which is favored at a given T corresponds to the lowest line since the free energy is the smallest. The s+d coexistence solution of small η (purple) exists at the lower left. Increasing η, the solution (orange) moves to the upper right. If η is large enough, the solution (green) has a free energy larger than that of the single solutions, and the s+d coexistence phase cannot exist.
By summarizing the above results, we can draw a phase diagram. Figure 6 is the η-T phase diagram of favored states. The green line is T d . The red curve is the temperature at which the s+d coexistence phase starts to appear as lowering T , and the blue curve is that at which the s+d coexistence phase ends. In these curves, the black dots are our numerical results, and we simply connect them by lines. The purple line, the magnitude correlation of the free energy of the single  Figure 6: η-T phase diagram of favored states. η is the direct coupling between the s-wave and the d-wave. The green line is T d . The red curve is the temperature at which the s+d coexistence phase starts to appear as lowering T , and the blue curve is that at which the s+d coexistence phase ends. The purple line, the magnitude correlation of the free energy of the single solutions is reversed.
solutions is reversed. There are four phases corresponding to four solutions in figure  5, the normal conducting phase, the s-wave single phase, the d-wave single phase and the s+d coexistence phase. When the temperature passes the green, red or blue line, a phase transition at which O s and O d change continuously occurs. Otherwise, when the temperature passes the purple line, the phase transition at which they change discontinuously occurs.

Summary and discussion
In this paper, we have calculated the solutions of the holographic superconductor model with an s-wave and a d-wave (2.4) in the probe limit. This model cannot have the Josephson coupling existing in two-scalar models [24] and we consider the direct coupling only. We found that the model has a rich phase structure: a phase transition of the order parameters which is continuous or discontinuous, coexistence of the s-wave and the d-wave, order competition and a triple point.
There are some future issues: • Calculating the model with different values of the parameters. For the effective action of a spin-two field, large m 2 d is better. However, we expect that there is not large difference about m 2 d because our model comes down to a two-scalar model.
• Calculating with back reaction and other coupling. For this calculation, it is important to study a higher spin theory on curved spacetime.
• Interpreting e s and e d as a superconductor. In our model, we choose different values as e s and e d . It is not clear that this set up is valid.
• Estimating an error of the numerical calculation. Especially, there is a range of the temperature that F s+d of η = 1/10 is smaller than F s . It is important to check whether this result is caused by an error of the numerical calculation or not. However, this effect does not change the phase diagram because F s+d of η = 1/10 is larger than F d .
Other future work is, for example, calculation of conductivity, detailed calculation of the blue and red curve of figure 6, analysis of instability of the model about η. Furthermore, we expect that a holographic model with bosonic and fermionic degrees of freedom and a Yukawa coupling [23] has the property similar to our model.