Holographic Superconductors with Various Condensates

TL;DR

Horowitz and Roberts extend holographic superconductors by analyzing a charged scalar in $AdS_{d+1}$ for $d=3$ and $d=4$ with varying operator dimension $oldsymbol{eta}$. Using the probe limit Abelian-Higgs model, they show scalar hair forms below a critical temperature $T_c$, producing a superconducting phase with a delta function at $ ilde extomega=0$ and a gap $ ilde extomega_g$ in the AC conductivity, but with $ ilde extomega_g/T_c eq 2 ilde extDelta$ in general. For $oldsymbol{eta}>oldsymbol{eta}_{BF}$, they find a robust universality $ ilde extomega_g/T_c o ext{≈}8$ across dimensions, while at the BF bound they observe bound states and vector normal modes as $T o 0$, indicating strong coupling and new spectral structure. They also report universal relations between the superfluid density $n_s$, the gap, and a finite-temperature correlation length $oldsymbol{ ilde ext xi_k}$, along with mean-field critical behavior near $T_c$, highlighting rich non-BCS physics in holographic superconductors.

Abstract

We extend earlier treatments of holographic superconductors by studying cases where operators of different dimension condense in both 2+1 and 3+1 superconductors. We also compute a correlation length. We find surprising regularities in quantities such as $ω_g/T_c$ where $ω_g$ is the gap in the frequency dependent conductivity. In special cases, new bound states arise corresponding to vector normal modes of the dual near-extremal black holes.

Figures (6)

• Figure 1: The condensate as a function of temperature. $\lambda$ is the dimension of the operator ${\mathcal{O}}$, and $d$ is the spacetime dimension of the superconductor. $\lambda_{BF} = -3/2$ for $d=3$ and $\lambda_{BF} = -2$ for $d=4$. The condensate tends to increase with $\lambda$.
• Figure 2: Conductivity for $2+1$ dimensional superconductors. Each plot is at low temperatures, about $T/T_c\approx 0.1$. The solid line is the real part, dashed is imaginary. The pole at $\omega =0$ is clearly visible in ${{\frak{Im}}}[\sigma]$. For $\lambda\ge 3/2$, we find the shape of the curve near the edge of the gap largely dictated by a second pole in the lower half complex $\omega$ plane. As $T\rightarrow0$ and $\lambda\rightarrow 3/2$, this pole hits the real axis.
• Figure 3: Conductivity for $3+1$ dimensional superconductors. Each plot is at low temperatures, about $T/T_c\approx 0.1$. The solid line is the real part, dashed is imaginary. The pole at $\omega =0$ is clearly visible in ${{\frak{Im}}}[\sigma]$. For $\lambda= 2$, we find the shape of the curve largely dictated by multiple poles. As $T\rightarrow0$ and $\lambda\rightarrow 2$, these poles approach and hit the real axis.
• Figure 4: The absolute value of the retarded Greens function is shown as a function of temperature and (real) frequency for $\lambda=\lambda_{BF}$ at zero spacial momentum. Lighter shades denote larger values. One can clearly see the complex frequency poles as they move onto the real axis.
• Figure 5: Superfluid density in $d=3$ and $d=4$.