A Striped Holographic Superconductor

TL;DR

The paper develops a striped holographic superconductor by imposing a modulated chemical potential, sourcing a CDW background in a 3+1D Einstein-Maxwell-scalar theory within the probe limit. It analyzes the normal CDW state, establishes the onset of superconductivity at a critical temperature $T_c(Q)$, and constructs the full inhomogeneous superconducting solution with stripes and higher harmonics. The study finds that the superconducting stripes are energetically favored below $T_c$, and that $T_c(Q)$ qualitatively resembles weakly coupled BCS behavior but exhibits important quantitative differences, especially at large $Q$. It also computes the conductivity along the stripes, revealing a superconducting gap structure and pronounced anisotropy, thereby providing a concrete holographic realization of inhomogeneous superconductivity and insights into how CDW inhomogeneity affects $T_c$ and transport in strongly coupled systems.

Abstract

We study inhomogeneous solutions of a 3+1-dimensional Einstein-Maxwell-scalar theory. Our results provide a holographic model of superconductivity in the presence of a charge density wave sourced by a modulated chemical potential. We find that below a critical temperature superconducting stripes develop. We show that they are thermodynamically favored over the normal state by computing the grand canonical potential. We investigate the dependence of the critical temperature on the modulation's wave vector, which characterizes the inhomogeneity. We find that it is qualitatively similar to that expected for a weakly coupled BCS theory, but we point out a quantitative difference. Finally, we use our solutions to compute the conductivity along the direction of the stripes.

Figures (11)

• Figure 1: The plot shows the numerical exact (continuous red line) and analytically approximate (dashed black line) solutions of \ref{['simple']} for various values of $Q$.
• Figure 2: We show the $z$ profiles of $\psi_n$ for the first few $n$ and two different values of $Q$ at the phase transition ($T/(q\mu)=0.0269$ for $Q=4$ and $T/(q\mu)=0.0575$ for $Q=1/10$). Dashed lines with longer dashes correspond to larger $n$. It is evident that $\psi_n$ are suppressed for larger $n$ and more so for larger $Q$. This results are obtained solving the (truncated) linear system \ref{['sys']}, therefore the normalization of $\psi_n$ is arbitrary.
• Figure 3: The plots show the $z$ profiles of $\psi_n$ and $A_n$ obtained from the (truncated) full system \ref{['sys2']}, for the first few $n$ and $\delta=3/5$, $Q=1/4$ and $T/(q\mu)=0.0556$ which is very close to the critical temperature. Again, dashed lines with longer dashes correspond to larger $n$. The $A_n$ profiles show that close to $T_c$ only the zeroth and first harmonics of $A$ are excited, which confirms that the backreaction of $\psi$ is negligible.
• Figure 4: The plot shows the critical temperature as function of Q for $\delta=0.2,0.4,0.6,0.8$.
• Figure 5: The plots show the $z,x$ profile of $A$ and $\psi$ for $Q=.25$ and $Q=2$. The striped pattern of condensation of the bulk scalar field $\psi$ is evident in both cases, but more pronounced for lower $Q$'s.