content_image/355960.jpg content_image/355959.jpg content_image/355958.jpg content_image/355956.jpg content_image/355955.jpg content_image/355954.jpg content_image/355957.jpg
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 (3.1), 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 

<img src='content_image/355960.jpg'>

<img src='content_image/355959.jpg'>

<img src='content_image/355958.jpg'>

because their norms are required to be finite, where L 2 = L 2 $_{AdS}$/ (1 + ϵ ) . And at the asymptotic region ( r → ∞ ), the solutions behave like 

with 

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 o ff the expectation values of operator O dual to the field ψ . From Ref. [40], we know that for ψ , both of these fallo ff s are normalizable, and in order to keep the theory stable, we should either impose 

or 

<img src='content_image/355956.jpg'>

<img src='content_image/355955.jpg'>

<img src='content_image/355954.jpg'>

<img src='content_image/355957.jpg'>

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, 

