Holographic Studies of Entanglement Entropy in Superconductors

TL;DR

Addresses how entanglement entropy behaves across holographic superconducting transitions in a fully back-reacted AdS$_4$ gravity model. The authors compute $S_{\mathcal{A}}$ using the Ryu–Takayanagi prescription and reveal distinct signatures for the ${\cal O}_1$ and ${\cal O}_2$ condensates, including a slope discontinuity at $T_c$ for ${\cal O}_1$ and a finite jump plus a new length scale $\tilde{\xi}$ for ${\cal O}_2$. They report multivalued entanglement entropy and a nonmonotonic metric function $f(z)$ that underpins a domain-wall interpretation and large-$\ell$ saturation effects. The results illuminate how entanglement reorganizes during the superconducting transition and toward the $T=0$ ground state, highlighting entanglement entropy as a sharp diagnostic of RG flow and emergent scales in strongly coupled systems.

Abstract

We present the results of our studies of the entanglement entropy of a superconducting system described holographically as a fully back-reacted gravity system, with a stable ground state. We use the holographic prescription for the entanglement entropy. We uncover the behavior of the entropy across the superconducting phase transition, showing the reorganization of the degrees of freedom of the system. We exhibit the behaviour of the entanglement entropy from the superconducting transition all the way down to the ground state at T=0. In some cases, we also observe a novel transition in the entanglement entropy at intermediate temperatures, resulting from the detection of an additional length scale.

Figures (14)

• Figure 1: Diagram of the strip shape we will consider for region $\mathcal{A}$. This is the case of a dual geometry that is asymptotically AdS$_4$, and here, $z$ denotes the radial direction in AdS$_4$. The quantity $\ell$ sets the size of region $\cal A$, and $L$ is a regulator that is understood to be taken to infinity.
• Figure 2: Plots of operator versus temperature for scalar charged black hole solutions with either $\mathcal{O}_1$ or $\mathcal{O}_2$ non--zero. The vertical dotted line on the $\mathcal{O}_2$ plot denotes the transition temperature. See text.
• Figure 3: Free energy density difference. When $\Delta \mathcal{F} > 0$, the superconductor is thermodynamically favored. This occurs at $T_c \approx 0.1199 \frac{\sqrt{\rho} (16 \pi G_4 )^{1/4}}{R^{1/2}}$ for the $\mathcal{O}_1$ case and $100T_c \approx 0.3638 \frac{\sqrt{\rho} (16 \pi G_4 )^{1/4}}{R^{1/2}}$ for the $\mathcal{O}_2$ case.
• Figure 4: Entanglement entropy vs. strip width $\ell$ for the $\mathcal{O}_1$ case. The solid blue curve is the superconductor solution, and the red dashed curve is the Reissner--Nordström solution.
• Figure 5: The entanglement entropy in the $\mathcal{O}_1$ case, as a function of temperature, for fixed $\ell$. (We choose $\sqrt{\rho}(16 \pi G_4)^{1/4}R^{-1/2} \ell/2 = 2.5$) The solid blue curve is from the superconductor solutions, while the red dashed curve is from the Reissner--Nordström solutions. Trace the physical curve by always choosing the lowest entropy at a given $T$. There is a discontinuity in the slope of the decreasing entanglement entropy at the transition temperature $T_c$, indicated by the vertical dotted line. (While we do not plot all the superconductor points, due to lack of numerical control at low temperature, we display the zero temperature solution, since the solution is known exactly there.)