A general class of holographic superconductors

TL;DR

This work generalizes holographic superconductors by introducing a Stückelberg-type bulk action with a general function $\mathcal{F}$ that governs spontaneous $U(1)$ breaking, enabling tunable phase transition order. Using monomial and more general forms of $\mathcal{F}$, the authors show that $n=2$ yields a second-order transition while $n>2$ yields first-order behavior, with metastable regimes and multiple hair-like solutions; free-energy analysis identifies the physical branch and clarifies the phase structure. Transport properties reveal a gapped conductivity and, for $n\ge3$, additional resonances indicating internal structure of the condensate. The study suggests that the mapping between $\mathcal{F}$ and dual CFT data can realize non mean-field critical exponents and richer dynamics, offering a flexible holographic platform for modeling diverse strongly coupled phase transitions.

Abstract

We introduce a simple generalization of the basic holographic superconductor model in which the spontaneous breaking of a global U(1) symmetry occurs via the Stueckelberg mechanism. This more general setting allows tuning features such as the order of the transition. The physical vacuum of the condensed phase and the order of the transition are determined by a detailed analysis of the free energy of the system. For first order transitions, we identify a metastable phase above the critical temperature. In this case, the conductivity shows additional poles, thus suggesting that the condensate has internal structure. We comment on the possibility of obtaining second order phase transitions with non mean-field critical exponents.

Figures (14)

• Figure 1: Normalized condensate $\langle\mathcal{O}_1\rangle$ as a function of temperature at fixed $\rho=-1$. The blue and magenta curves correspond to $n=2$ and $n=3$, respectively. For $n=2$, $T_c \sim 0.268$ while for the $n=3$, $T_c\sim 0.187$.
• Figure 2: Normalized condensate $\sqrt{\langle\mathcal{O}_2\rangle}$ as a function of temperature at fixed $\rho=-1$. The blue and magenta curves correspond to $n=2$ and $n=3$, respectively. For $n=2$, $T_c \sim 0.141$; while for $n=3$, $T_c\sim 0.113$.
• Figure 3: Normalized condensate $\langle\mathcal{O}_1\rangle$ as a function of temperature at fixed $\rho=-1$ for $n=3$. The thicker curve indicates the physical piece of the full solution.
• Figure 4: $(\mathcal{O}_1,\mathcal{O}_2)$ plane at $T=0.30$ (Region I) and at $T=0.20$ (Region II).
• Figure 5: $(\mathcal{O}_1,\mathcal{O}_2)$ plane for $T=0.075$, $0.065$ and $0.050$, well inside Region III.