Excited states of holographic superconductors

In this paper we re-investigate the model of the anti-de Sitter gravity coupled to Maxwell and charged scalar fields, which has been studied as the gravitational dual to a superconductor for a long time since the famous work [Phys.\ Rev.\ Lett.\ {\bf 101}, 031601 (2008)]. By numerical method, we present a novel family of solutions of holographical superconductor with excited states, and find there exists a lower critical temperature in the corresponding excited state. Moreover, we study the condensate and conductivity in the excited states. It is very interesting that the conductivity $\sigma$ of each excited state has an additional pole in $\text{Im}[\sigma]$ and a delta function in $\text{Re}[\sigma]$ arising at the low temperature inside the gap, which is just the evidence of the existence of excited states.


Introduction
In condensed matter physics, there is still no consensus on the mechanism of high temperature superconductivity. Over the past decade, the AdS/CFT correspondence [1,2,3] is one of the most important results from string theory, which establishes the relationship between the strongly correlated field on the boundary and a weak gravity theory in one higher-dimensional bulk spacetime. In the past ten years, the AdS/CFT correspondence has provided a novel way to study condensed matter theory, and received a great deal of attention. In the seminal papers [4,5,6], the authors investigated the model of a complex scalar field coupled to a U (1) gauge field in a (3 + 1)-dimensional Schwarzschild-AdS black hole and found that due to the U (1) symmetry breaking below the critical temperature T c , the condensate of the scalar field could be interpreted as the Cooper pair-like superconductor condensate. Moreover, there exists a gap in the optical conductivity of the superconducting state, and the value of the gap is close to the value of a high temperature superconductor. Thus, this model can be regarded as the dual explanation to the high temperature superconductor. When replacing the scalar field with other matter fields, one can also obtain the condensate of matter fields corresponding to various kinds of holographic superconductors.
For example, holographic d-wave model was constructed by introducing a symmetric, traceless second-rank tensor coupled to a U (1) gauge field in the bulk [7,8,9]. With the SU (2) Yang-Mills field coupled to gravity, the holographic p-wave model has also been discussed in [10]. Two alternative holographic realizations of p-wave superconductivity could arise from the condensate of a two-form field [11] and a complex, massive vector field with U (1) charge [12,13], respectively. The model of holographic superconductor can also be extended to study the holographic Josephson junction [14,15,16,17,18,19,20,21,22,23,24,25,26], which is made up of two superconductor materials with weak link barrier [27]. For reviews of holographic superconductors, see [28,29,30,31].
Until now, lots of work of holographic superconductor were investigated in the ground state 6 , that is, the scalar field can keep sign along the radial direction. It is well-known that the excited states could have some nodes along the radial direction, where the value of the scalar field could change the sign. So, it is interesting to see whether there exist the solutions of holographic superconductor with excited states of the scalar field. In the present paper, we would like to numerically solve the Maxwell-Klein-Gordon system with the background of a four-dimensional Schwarzschild-AdS black hole and give a family of excited 6 In [32,33] the ground state represents the zero-temperature limit of holographic superconductor while in this paper the ground state refers to the scalar field without nodes. states of holographic superconductors with different critical temperatures. Moreover, we will also study the condensate and optical conductivity in the excited states of holographic superconductors.
The paper is organized as follows: in Sect. 2, we review the model of a U (1) gauge field coupled with a charged scalar field in (3 + 1)-dimensional AdS spacetime and show a gravity dual model of holographic superconductor. We show the numerical results of the excited states and study the characteristics of the condensate and conductivity in Sect. 3.
The conclusion and discussion are given in the last section.

Review of holographic superconductors
Let us introduce the model of a Maxwell field and a charged complex scalar field in the four-dimensional Einstein gravity spacetime with a negative cosmological constant. The bulk action reads with the constant κ ≡ 1/q 2 . From the above, we can see that the value of the parameter κ will decrease when q increases, and the backreaction of matter fields on the spacetime metric could be ignored in the large q → ∞ limit, which is also called as the probe limit.
Here we adopt to the probe approximation. Thus, the following equations of the scalar and Maxwell fields can be derived from the Lagrangian density (2.2): In addition, with neglect of the backreaction of matter fields on the metric, the solution of Einstein equations is the well-known Schwarzschild anti-de Sitter black hole. The solution with a planar symmetric horizon can be written as follows , and r h is the radius of the black hole's event horizon. The Hawking temperature is given by which can be regarded as the temperature of holographic superconductors.
In order to build a holographic model of holographic superconductors, one could introduce the following ansatz of matter fields: With the above ansatz, the equations of motion for the scalar field ψ(r) and electrical scalar potential φ(r) in the background of the Schwarzschild-AdS black hole are At the AdS boundary, the asymptotic behaviors of the functions ψ(r) and φ(r) take the following forms where the constants µ and ρ are the chemical potential and charge density in the dual field theory, respectively. According to AdS/CFT duality, ψ (i) (i = 1, 2) are the corresponding expectation values of the dual scalar operators O i , respectively. Note that in the fourdimensional spacetime the values of mass need to satisfy the Breitenlohner-Freedman (BF) bound of m 2 ≥ −9/4 [34]. In this paper, we will set m 2 = −2 for simplify.

Numerical results
In this section, we will solve the above coupled equations (2.8) and (2.9) numerically. It is convenient to change the radial coordinate r to the new radial coordinate z = r h /r.
For simplify, we set ℓ = 1 and r h = 1. Thus the inner and outer boundaries of the shell are fixed at z = 0 and z = 1, respectively. All numerical calculations are based on the spectral methods. Typical grids used have the sizes from 50 to 300 in the integration region 0 ≤ z ≤ 1 . Our iterative process is the Newton-Raphson method, and the relative error for the numerical solutions in this work is estimated to be below 10 −5 .
It is well-known that there exists a critical temperature T c , above which there is no scalar condensate in the Schwarzschild-AdS black hole background, while for T ≤ T c , the scalar condensate begins to appear due to the spontaneously broken U (1) gauge symmetry. Until now, only the ground state of holographic superconductor with a fixed critical temperature was found out, that is, the scalar field ψ could keep sign along the radial direction. By numerically solving the same equations of motion with boundary conditions as the case of ground state, we could also obtain the solutions of the excited states, in which the value of the scalar field can change sign along the radial direction.

Condensates
According to AdS/CFT duality, the expectation values of the condensate operator O i is dual to the scalar field ψ (i) : In Fig. 2  and O 2 (right) of excited states. In both plots, the black, red and blue lines denote ground state, first and second states, respectively.
When the temperature drops below the critical temperature T c , the model of holographic superconductor has a second order phase transition from normal state to superconductor state. Moreover, there exists a square root behaviour near the critical point, which could be expected from mean field theory. We fit these condensation curves for the two operators where the critical temperatures T

Conductivity
In this subsection we study the conductivity in the excited states of holographic superconductors. We turn on perturbations of the vector potential A x in the bulk geometry of Schwarzschild-AdS black hole. Considering the Maxwell equation with a time dependence of e −iωt , the linearized equations are given as (3.14) When imposing ingoing boundary conditions at the horizon, we give the asymptotic behaviour of the Maxwell field at the boundary: According to the AdS/CFT duality and Ohm's law we can obtain the conductivity Because the value of the scalar ψ in the excited stats is different from the ground state at the same temperature, the value of the conductivity in Eq. (3.14) could be changed with the different excited states.
In Fig. 3, we plot the conductivity as a function of the frequency at the low temperature c , a new branch of solution with a node in radial direction begins to develop, which is the first excited state solution. As the temperature continues to decrease to lower values, we could obtain a series of excited states of holographic superconductor with differential critical temperatures. It is well-known that the condensate of ground state for the operator O 1 appears to diverge as the temperature T → 0. However, the condensate of excited states does not appear to diverge and goes to a constant near zero temperature.
Moreover, the condensate of the excited state is smaller than the ground state. Note that the value of the condensate for the second excited state near zero temperature is very closed to that of the first excited state. We calculated the condensate O 1 near zero temperature until the sixth excited state, and found that this value is about 4.4. For the operator O 2 , the condensate of an excited state is larger than the ground state. We also studied the conductivity in the excited states of holographic superconductors. It is very interesting to mention that each excited state has an additional pole in Im[σ] and a delta function in Re[σ] arising at the low temperature inside the gap. These behaviors are just the evidence of the existence of the excited states.
There are many interesting extensions of our work. First, we have noticed that the value of the condensate O 1 near zero temperature with higher excited state is about 4.4. In addition, the difference of the critical chemical potential µ c between the consecutive states is about 5 for both of the condensates O 1 and O 2 , but the reason of these configurations is not clear. It would be very interesting to study these cases with the semi-analytical method [36] to see how these values are related to excited states. The second extension of our study is that away from the large charge limit, the backreaction of the matter fields on the bulk metric needs to be included. Thus, one must solve the coupled differential equations including Einstein equations. Finally, we are planning to study the excited states of the p-wave and d-wave holographic superconductor and construct the excited vector and tensor condensates in the background of black hole in future work.