Condensation for non-relativistic matter in Ho\v{r}ava-Lifshitz gravity

We study condensation for non-relativistic matter in a Ho\v{r}ava-Lifshitz black hole without the condition of the detailed balance. We show that, for the fixed non-relativistic parameter $\alpha_2$ (or the detailed balance parameter $\epsilon$), it is easier for the scalar hair to form as the parameter $\epsilon$ (or $\alpha_2$) becomes larger, but the condensation is not affected by the non-relativistic parameter $\beta_2$. We also find that the ratio of the gap frequency in conductivity to the critical temperature decreases with the increase of $\epsilon$ and $\alpha_2$, but increases with the increase of $\beta_2$. The ratio can reduce to the Horowitz-Roberts relation $\omega_g/T_c\approx 8$ obtained in the Einstein gravity and Cai's result $\omega_g/T_c\approx 13$ found in a Ho\v{r}ava-Lifshitz gravity with the condition of the detailed balance for the relativistic matter. Especially, we note that the ratio can arrive at the value of the BCS theory $\omega_g/T_c\approx 3.5$ by taking proper values of $\epsilon$, $\alpha_2$, $\beta_2$ and $m$.

In the Hořava-Lifshitz gravity, Kiritsis and Kofinas [36], Kimpton and Padilla [37] proposed the non-relativistic matter. They constructed the most general action of matter coupled to gravity with the foliation-preserving diffeomorphism. The action obeys the usual power-counting renormalisability conditions used in Hořava-Lifshitz gravity and assuming the temporal derivatives are as in the relativistic theory.
Recently, in order to see what difference will appear for the holographic superconductivity in the Hořava-Lifshitz theory, comparing with the case of the relativistic general relativity, Cai et al. [38] studied the phase transition of planar black holes in the Hořava-Lifshitz gravity with the condition of the detailed balance in which the metric function is described by f (r) = x 2 − √ c 0 x. They argued that the holographic superconductivity is a robust phenomenon associated with asymptotic AdS black holes. And they also got a relation connecting the gap frequency in conductivity with the critical temperature, which is given by ωg Tc ≈ 13, with the accuracy more than 93% for a range of scalar masses. More recently, Lin, Abdalla and Wang [39] generalized the investigation to the holographic superconductors related to the non-relativistic matter in the Schwarzschild black hole in the low energy limit of Hořava-Lifshitz spacetime.
Note that the Hořava-Lifshitz black hole without the condition of the detailed balance has rich physics [40,41,42], i.e., changing the parameter of the detailed balance ǫ from 0 to 1 it can produce the different black holes for the Hořava-Lifshitz theory and Einstein's gravity, and the non-relativistic matter in Hořava-Lifshitz gravity has new properties. In this paper we will extend the study to case of the nonrelativistic matter in a Hořava-Lifshitz black hole without the condition of the detailed balance, and investigate how the parameter of the detailed balance and nonrelativistic parameters influence on the scalar condensation formation, the electrical conductivity, and the ratio ω g /T c which connects the gap frequency in conductivity with the critical temperature.
The paper is organized as follows. In Sec. II we present black hole with hyperbolic horizons in Hořava-Lifshitz gravity in which the action without the condition of the detailed balance. In Sec. III we explore the condensation of the relativistic matter in the Hořava-Lifshitz black hole background by numerical approach. In Sec. IV we study the electrical conductivity and find ratio of the gap frequency in conductivity to the critical temperature. We summarize and discuss our conclusions in the last section.

black hole with hyperbolic horizon in z = 3 Hořava-Lifshitz gravity
In non-relativistic field theory, space and time have different scalings, which is called anisotropic scaling, x i → bx i , t → b z t, i = 1, 2, 3, where z is called dynamical critical exponent. In order for a theory to be power counting renormalizable, the critical exponent has at least z = 3 in four spacetime dimensions. For z = 3, the action without the condition of the detailed balance for the Hořava-Lifshitz theory can be expressed as [40,41] where κ 2 , µ, Λ, and ω are constant parameters, ǫ is parameter of the detailed balance (0 < ǫ ≤ 1), N i is the shift vector, K ij is the extrinsic curvature and C ij the Cotten tensor. It is interesting to note that the action (2.1) reduces to the action in Ref. [41] if ǫ = 0, and it becomes the action for the Einstein's gravity if ǫ = 1.
From the action (2.1), Cai et al. [42] found a static black hole with hyperbolic horizon whose horizon has an arbitrary constant scalar curvature 2k with λ = 1. The line element of the black hole can be expressed as with x + is the horizon radius of the black hole, i.e., the largest root of f (r) = 0. Comparing with the standard AdS 4 spacetime, we may set −Λ 1+ǫ = 1 L 2

AdS
, where L AdS is the radius of AdS 4 . The authors in Ref. [42] also found that the solution has a finite mass M = κ 2 µ 2 Ω k √ −Λc 0 /16. For ǫ = 0, the solution goes back to the solution in Ref. [41].
The Hawking temperature of the black hole is which is always a monotonically increasing function of horizon radius x + in the physical regime. This implies that the black holes with hyperbolic horizons in the Hořava-Lifshitz theory are thermodynamically stable.

Condensation for Non-relativistic matter in Hořava-Lifshitz gravity
We now study the condensation for non-relativistic matter in the Hořava-Lifshitz gravity. For the Arnowitt-Deser-Misner metric the Lagrangian of complex scalar and electromagnetic fields for the non-relativistic matter in the Hořava-Lifshitz gravity can be expressed as [36] L with where G E and H S are the Hořava-Lifshitz higher order corrections, α i and β i can be taken as constants, In this paper, we just consider the lower order terms of above equations.
The coupling between electromagnetic field and scalar field can be constructed and then the Lagrangian L S H should be rewritten as [39] where H S is replaced byH S with ∂ i → ∂ i − iqA i . Therefore, the action of coupling between complex scalar and electromagnetic fields for the non-relativistic matter in the Hořava-Lifshitz gravity can be taken as which will reduce into the model in general relativity when α i = β i = 0.
In the background of the black hole described by Eq. (2.3) with k = 0, we focus our attention on the case that these fields are weakly coupled to gravity, i.e., they do not backreact on the metric of the spacetime. Thus, we can take the ansatz This ansatz implies that the phase factor of the complex scalar field is a constant. Therefore, we may take ψ to be real. In the background of the black hole described by Eqs. (2.2) and (2.3) with k = 0, the equations of the scalar field ψ(r) and the scalar potential φ(r) are given by where a prime denotes the derivative with respect to r.
At the event horizon r = r + , we must have because their norms are required to be finite, where L 2 = L 2 AdS /(1 + ǫ) . And at the asymptotic region (r → ∞), the solutions behave like where µ and ρ are interpreted as the chemical potential and charge density in the dual field theory, respectively. Because the boundary is a (2+1)-dimensional field theory, µ is of mass dimension one and ρ is of mass dimension two. We can read off the expectation values of operator O dual to the field ψ. From Ref. [43], we know that for ψ, both of these falloffs are normalizable, and in order to keep the theory stable, we should either impose Note that the dimension of temperature T is of mass dimension one, the ratio T 2 /ρ is dimensionless. Therefore increasing ρ while T is fixed, is equivalent to decrease T while ρ is fixed. In our calculation, we find that when ρ > ρ c , the operator condensate will appear; this means when T < T c there will be an operator condensate, that is to say, the superconducting phase occurs. We will impose boundary condition ψ − = 0 in the following discussion. Eqs. (3.8) and (3.9) can be solved numerically by doing integration from the horizon out to the infinity with the boundary conditions mentioned above. Changing the values of the balance parameter ǫ and non-relativistic parameter α 2 , we present in Fig. 1 In table 1 we present the critical temperature obtained by the numerical method. We know from the figures and the table that as the parameter of the detailed balance increases with fixed non-relativistic parameter α 2 and effective mass of the scalar field, the condensation gap becomes smaller, corresponding to larger the critical temperature, which means that the scalar hair can be formed easier for the larger ǫ. Similarly, the scalar hair can be formed easier as the non-relativistic parameter α 2 becomes larger with fixed balance parameter and effective mass of the scalar field. And the figures and table also show that, for the same ǫ or α 2 , the condensation gap becomes larger if m 2 ef f becomes less negative, which means that it is harder for the scalar hair to form as the effective mass of the scalar field becomes larger. We should point out that the parameter β 2 dose not affect the condensation in this model.

Electrical Conductivity in Hořava-Lifshitz black-hole background
In the study of (2+1) and (3+1)-dimensional superconductors in Einstein gravity, Horowitz et al. [8] got a universal relation connecting the gap frequency in conductivity with the critical temperature T c , which is described by with deviations of less than 8%. This is roughly twice the BCS value 3.5 indicating that the holographic superconductors are strongly coupled. The authors in Refs.  with the accuracy more than 93% for a planar Hořava-Lifshitz black hole with the condition of the detailed balance for the relativistic matter.
We now study this relation for the non-relativistic matter in the Hořava-Lifshitz gravity. In order to compute the electrical conductivity, we should study the electromagnetic perturbation in this Hořava-Lifshitz black hole background, and then calculate the linear response to the perturbation. In the probe approximation, the effect of the perturbation of metric can be ignored. Assuming that the perturbation of the vector potential is translational symmetric and has a time dependence as δA x = A x (r)e −iωt , we find that the equation of motion for A x in the Hořava-Lifshitz black hole background reads where a prime denotes the derivative with respect to r. An ingoing wave boundary condition near the horizon is given by In the asymptotic AdS region (r → ∞), the general behavior should be (4.5) By using AdS/CFT correspondence and the Ohm's law, we know that the conductivity can be expressed as [8]  We find that, for the same value of m 2 ef f L 2 AdS , the gap frequency ω g decreases with the increase of the parameters ǫ or α 2 . In each plot, the real part of the conductivity, Re[σ], approaches to a limit when the frequency grows. The limit for the case ǫ = 0 and α 2 = 0 is one, but generally it increases as parameters ǫ or α 2 increases. The imaginary part of conductivity Im[σ] becomes zero when ω → ∞, but it goes to infinity when the frequency approaches zero. In Fig. 5 we plot the frequency dependent conductivity for β 2 = 0, 1 and 2 with α 2 = 0.1, ǫ = 0.1 and m 2 ef f L 2 AdS = 0, − 1 , −2. We note that, for the same values of ǫ, α 2 and m 2 ef f L 2 AdS , the gap frequency ω g increases with the increase of the parameters β 2 . That is to say, the ratio of the gap frequency in conductivity ω g to the critical temperature T c increases as the parameters β 2 increases with fixed α 2 , ǫ and m 2 ef f . In table 2 we also present how the ratio ω g /T c relate to the balance parameter and non-relativistic parameter with fixed values m 2 ef f L 2 AdS = 0, −1 and −2, which shows that the ratio ω g /T c decreases with the increase of the balance parameter or the non-relativistic parameter α 2 , but increases with the increase of the parameter β 2 . From Figs. 3, 4 and 5 and table 2, we find that the ratio of the gap frequency in conductivity ω g to the critical temperature T c in this black hole reduces to Cai's result ω g /T c ≈ 13 [38] found in the Hořava-Lifshitz gravity with the condition of the detailed balance for the relativistic matter when ǫ = 0, β 2 = 0 and α 2 = 0, while it tends to the Horowitz-Roberts relation ω g /T c ≈ 8 obtained in the Einstein gravity as ǫ → 1 with α 2 = 0 and β 2 = 0. Especially, the ratio can arrive at the value of the BCS theory ω g /T c ≈ 3.5 if we take right value for ǫ, α 2 , β 2 and m 2 ef f , say ǫ = 0.99, α 2 = 0.47, β 2 = 0 and m 2 ef f L 2 AdS = −2.

conclusions
The behavior of the holographic superconductors in the Hořava-Lifshitz gravity has been investigated in this manuscript by introducing the non-relativistic scalar and electromagnetic fields in a planar black-hole background. We first present a detailed analysis of the condensation of the operator O + by the numerical method for the Hořava-Lifshitz black hole without the condition of the detailed balance. It is found that, as the parameter of the detailed balance ǫ increases with fixed the nonrelativistic parameter α 2 and effective mass of the scalar field m 2 ef f , the condensation gap becomes smaller, corresponding to the larger critical temperature, which means that the scalar hair can be formed easier for the larger ǫ. Similarly, the scalar hair can be formed easier as the non-relativistic parameter α 2 becomes larger with fixed detailed balance ǫ and effective mass of the scalar field. And it is also shown that, for the same ǫ or α 2 , the condensation gap becomes larger if m 2 ef f becomes less negative, which means that it is harder for the scalar hair to form as the effective mass of the scalar field becomes larger. It is interesting to note that the parameter β 2 does not affect the condensation. We then studied the electrical conductivity for the non-relativistic matter in the Hořava-Lifshitz black-hole background and find that the ratio of the gap frequency in conductivity to the critical temperature, ω g /T c , decreases with the increase of the balance parameter ǫ or the non-relativistic parameter  Table 2: The ratio ω g /T c for different values of the parameters ǫ, α 2 and β 2 with m 2 ef f L 2 AdS = 0, − 1 and −2.
α 2 , but increases with the increase of the parameter β 2 . The ratio reduces to Cai's result ω g /T c ≈ 13 [38] found in a Hořava-Lifshitz gravity with the condition of the detailed balance for the relativistic matter when ǫ = 0, α 2 = 0 and β 2 = 0, while it tends to the Horowitz-Roberts relation ω g /T c ≈ 8 [8] obtained in the Einstein gravity if we take ǫ → 1, α 2 = 0 and β 2 = 0. Especially, the ratio can arrive at the value of the BCS theory ω g /T c ≈ 3.5 if we take right values of ǫ, α 2 , β 2 and m.