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 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 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, 5 ψ$_{+}$ = 0 , and 〈O − 〉 = $^{ψ}$−. (3.10) $^{ψ}$− = 0 , and 〈O + 〉 = ψ$_{+}$, (3.9) $^{λ}$± = 1 2 [ 3 ± √ 9 + 4 m $^{2}$L 2 AdS ] , (3.8) ψ = $^{ψ}$− r $^{λ}$− + ψ$_{+}$ r λ$_{+}$ , φ = µ − ρ r , (3.7) ψ ( r$_{+}$ ) = − 3 ψ ′ ( r$_{+}$ ) 2 m $^{2}$L2 , φ ( r$_{+}$ ) = 0 , (3.6) φ ′′ + 2 r φ ′ − 2 ψ 2 f φ = 0 , (3.5) ψ ′′ + ( f ′ f + 2 r ) ψ ′ + ( φ 2 f 2 − m 2 f ) ψ = 0 , (3.4)