Universality class of holographic superconductors

TL;DR

The paper investigates the static and dynamic universality of holographic superconductors by analyzing bulk scalar perturbations around high-temperature RN-AdS backgrounds. Using a double-series expansion in dimensionless frequency and momentum, combined with numerical probe-limit computations, it shows that static critical exponents assume mean-field values, with $\gamma=1$, $\nu=1/2$, and $\eta=0$, while the dynamic universality class is model A with $z=2$ and critical slowing down, $\tau \propto \xi^z$. The results hold across dimensions and are supported by numerical evidence, including a calculable critical point $\sigma_c$, correlation length $\xi$, and quasinormal mode spectra. The work reinforces the AdS/CFT correspondence in the critical regime and highlights how gravity-side degrees of freedom can realize conventional mean-field behavior in the dual field theory, offering a framework to classify gravity duals by their universality class.

Abstract

We study "holographic superconductors" in various spacetime dimensions. We compute most of the static critical exponents in the linear perturbations and show that they take the standard mean-field values. We also consider the dynamic universality class for these models and show that they belong to model A with dynamic critical exponent z=2.

Figures (5)

• Figure 1: (color online). The value of $\varphi_0^{(-)}$ ("$+$", (red)) and $(\varphi_0^{(+)} - 4.585)/100$ ("$\times$", (blue)) as a function of $\sigma$. As $\sigma$ approaches the critical value $\sigma_c \sim 4.15686$, $\varphi_0^{(-)}$ goes to zero and $\varphi_0^{(+)}$ goes to a finite constant.
• Figure 2: (color online). The thermodynamic susceptibility $\chi$ as a function of $\epsilon_\sigma := 1 - \sigma/\sigma_c$. Plotted are $1/\chi$ for $\epsilon_\sigma = 10^{-5} n$ ($n = 0, 1, \cdots, 20$).
• Figure 3: (color online). The correlation length $\xi^2$ defined by $\xi^2 := - 1/\mathfrak{q}_*^2$. Plotted are $1/\xi^2$ for $\epsilon_\sigma = 10^{-5} n$ ($n = 0, 1, \cdots, 20$).
• Figure 4: (color online). The static susceptibility at the critical point $\chi_{\mathfrak{w}=0,\mathfrak{q}} \vert_{T_c}$. Plotted are $1/\chi_{\mathfrak{w}=0,\mathfrak{q}} \vert_{T_c}$ for $\mathfrak{q}^2 = 10^{-5} n$ ($n = 0, 1, \cdots, 20$).
• Figure 5: (color online). The trajectory of the lowest quasinormal frequency $\mathfrak{w}_{\text{QNM}}$ for $\epsilon_\sigma = 10^{-5} n$ ($n = 0, 1, \cdots, 20$). As $\sigma \to \sigma_c$, $\mathfrak{w}_{\text{QNM}}$ approaches the origin with equal spacing, and it vanishes at $\sigma = \sigma_c$.