Analytical holographic superconductors in $AdS_N$ Lifshitz topological black holes

TL;DR

This paper analytically explores Lifshitz black hole solutions with scalar and gauge fields in higher-dimensional AdS spacetime, examining their thermodynamics and phase transitions relevant to holographic superconductors.

Contribution

It provides explicit Lifshitz solutions in N dimensions with thermodynamic analysis and studies the phase transition to superconductivity in a holographic context.

Findings

Existence of a critical temperature depending on Lifshitz parameter z

Below critical temperature, the system exhibits superconductivity

Critical exponent matches that of standard holographic superconductors

Abstract

We present the analytic Lifshitz solutions for a scalar field model minimally coupled with the abelian gauge field in $N$ dimensions. We also consider the presence of cosmological constant $\Lambda$. The Lifshitz parameter $z$ appearing in the solution plays the role of the Lorentz breaking parameter of the model. We investigate the thermodynamical properties of the solutions and discuss the energy issue. Furthermore, we study the hairy black hole solutions in which the abelian gauge field breaks the symmetry near the horizon. In the holographic picture, it is equivalent to a second order phase transition. Explicitly we show that there exists a critical temperature which is a function of the Lifshitz parameter $z$. The system below the critical temperature becomes superconductor, but the critical exponent of the model remains the same of the usual holographic superconductors without the higher order gravitational corrections, in agreement with Ginzburg-Landau theories.