Zero Temperature Limit of Holographic Superconductors

TL;DR

The paper analyzes the zero-temperature limit of holographic superconductors described by gravity with a Maxwell field and a charged scalar of potential $V(|\psi|)$. By mapping the conductivity to a one-dimensional Schrödinger problem, it proves that the horizon causes the potential to vanish, ensuring a nonzero low-frequency conductivity even at $T=0$, i.e., there is no hard gap. It constructs the extremal hairy black holes for $V(\psi)=m^2|\psi|^2$ and shows the ground state horizon shrinks to zero, with IR behavior ranging from emergent conformal symmetry (when $m^2=0$) to emergent Poincaré symmetry (when $m^2<0$ with sufficiently large $q$), including cases with null singular horizons or complete horizon smoothness at special parameters. Overall, the work clarifies the ground-state structure and transport properties of holographic superconductors, highlighting the role of IR geometry in determining the low-$\omega$ behavior of $\mathrm{Re}\,\sigma(\omega)$ and the absence of a hard energy gap in these models.

Abstract

We consider holographic superconductors whose bulk description consists of gravity minimally coupled to a Maxwell field and charged scalar field with general potential. We give an analytic argument that there is no "hard gap": the real part of the conductivity at low frequency remains nonzero (although typically exponentially small) even at zero temperature. We also numerically construct the gravitational dual of the ground state of some holographic superconductors. Depending on the charge and dimension of the condensate, the infrared theory can have emergent conformal or just Poincare symmetry. In all cases studied, the area of the horizon of the dual black hole goes to zero in the extremal limit, consistent with a nondegenerate ground state.

Figures (6)

• Figure 1: Schrödinger potential for $\lambda=2, q=10$. The potential increases as $T/T_c$ is lowered from one to zero. See section 4.2 for more details on this solution.
• Figure 2: Schrödinger potential for $\lambda=3, q=1$ at $T=0$. Note that the potential does not rise as high as it does in figure 1. See section 4.1 for more details on this solution.
• Figure 3: Zero temperature, $\lambda = 3$ and $q=1$ solution (dashed blue), compared to successively lower temperature hairy black holes (solid black.) Note that $g$ almost has a double zero at $r/\mu=1/2\sqrt{3}=r_+(T=0)$ for RN-AdS and all of the scalar hair is behind it. In the limit $q\rightarrow \sqrt{3}/2$ the solution becomes extremal Reissner-Nördstrom with all hair behind the horizon.
• Figure 4: Values of $\alpha$ for various charges. Note that it approaches a constant as $q\rightarrow \infty$.
• Figure 5: Zero temperature solution with $m^2=-2,~\lambda=2,~q=10$ (dashed blue) compared to successively lower temperature hairy black holes (solid black.)