Pseudogap and time reversal breaking in a holographic superconductor

TL;DR

Roberts and Hartnoll explore a holographic superconductor modeled by a classical $SU(2)$ Yang–Mills theory in AdS$_4$ to capture strongly coupled $2+1$-dimensional superconductivity. They demonstrate that the isotropic superconducting phase exhibits a pseudogap in the dissipative conductivity and a Hall response arising from spontaneous time-reversal symmetry breaking, without any external magnetic field. The bulk construction yields a hairy black hole below $T_c \\approx 0.125\\sqrt{\\rho}$ with a zero-temperature gap satisfying $\\frac{2\\Delta}{T_c}\\approx 8$, and linear-response calculations reveal a nonzero $\\sigma_{xy}$ and a delta function in Re$\\,\\sigma_{xx}$ from the Goldstone mode. A stability analysis shows the isotropic phase is dynamically unstable to a rotationally anisotropic background with lower grand potential, implying the true ground state is anisotropic; the work highlights robust strong-coupling signatures of pseudogap physics and spontaneous Hall response in holographic superconductors.

Abstract

Classical SU(2) Yang-Mills theory in 3+1 dimensional anti-de Sitter space is known to provide a holographic dual to a 2+1 system that undergoes a superconducting phase transition. We study the electrical conductivity and spectral density of an isotropic superconducting phase. We show that the theory exhibits a pseudogap at low temperatures and a nonzero Hall conductivity. The Hall conductivity is possible because of spontaneous breaking of time reversal symmetry.

Figures (4)

• Figure 1: The gap as a function of temperature.
• Figure 2: Spectral functions for the currents $J_z$ and $J_{\bar{z}}$. Each plot is in fact five curves, at temperatures $T/T_c=.08,.11,.15,.19,.23$. We see that we have effectively reached the zero temperature limit. There is a clear pseudogap, and a delta function at $\omega=0$. There is also a delta function at $\omega = \mu$.
• Figure 3: Standard and Hall conductivities at low temperatures as a function of frequency. The solid lines are the real part whereas the dashed lines are the imaginary parts. A pole in the imaginary part at $\omega = 0$ indicates that the real part will contain a delta function at $\omega = 0$. Similarly with the pole at $\omega = \mu$
• Figure 4: Comparison of free energies for temperatures below $T_c$. The anisotropic phase is a dashed line. It always has a lower free energy than the isotropic phase.