Quantum criticality and mixed-state entanglement in holographic superconductor--insulator transitions

TL;DR

This work analyzes quantum criticality in a holographic EMDA p-wave superconductor with a superconductor–insulator transition (SIT) driven by zero-temperature quantum fluctuations. It shows that EWCS, a mixed-state entanglement measure, provides a robust diagnostic of the QPT, outperforming holographic entanglement entropy (HEE) which becomes thermal-entropy dominated for large subsystems. The superconducting gap Δ exhibits quantum-critical scaling near the QCP with exponent α_T ≈ α_k ≈ 0.85, and the SIT is present only for the p-wave case due to vector-order parameter physics and axion-induced lattice breaking. The results position EWCS as a practical tool for identifying holographic quantum criticality in mixed states and highlight the role of vector order and translation-symmetry breaking in SIT phenomena.

Abstract

We study quantum criticality in a holographic Einstein--Maxwell--Dilaton--Axion (EMDA) p-wave superconductor exhibiting a superconductor--insulator transition (SIT). By tracking the superconducting energy gap, we show that approaching the quantum critical point (QCP) closes the gap and induces incipient insulating features, indicating that enhanced quantum fluctuations suppress superconducting order and trigger the SIT. To probe the transition, we employ two holographic indicators: holographic entanglement entropy (HEE) and the entanglement wedge cross-section (EWCS), the latter being a mixed-state entanglement measure. In contrast to HEE, which for sufficiently large configuration is dominated by the thermal entropy and is therefore largely insensitive to entanglement along the temperature direction, EWCS displays pronounced critical scaling and provides a robust diagnostic of the quantum phase transition (QPT). We attribute this contrast to the fact that HEE at large scales is controlled by the infrared (IR) geometry, whereas EWCS is governed by deformations of the entire bulk. Our results establish EWCS as a robust probe of holographic quantum criticality in mixed states.

Figures (11)

• Figure 1: Both first and second order superconducting phase transitions occur when the temperature is lower than the critical temperature $T_c$. Left panel: for $k=0.5$, the system undergoes a first-order transition at $T_c=0.0433$ (black dashed line). The inset shows the free-energy density $\Omega(T)$ for the superconducting (red) and normal (blue) phase, the thermodynamically stable phase is determined by the lower $\Omega$, and the transition occurs where the two branches exchange. Right panel: for $k=1.5$, a second-order transition occurs at $T_c=0.0863$. The inset demonstrates the scaling behavior of the order parameter near $T_c$, yielding a critical exponent $\alpha=0.13$.
• Figure 2: Phase diagrams in the $(T,k)$ plane showing the SIT as a zero-temperature QPT. The superconducting and insulating phases are separated by a critical line.
• Figure 3: Temperature dependence of the superconducting gap $\Delta(T)$. A finite gap emerges for $T<T_{c1}=0.14$ and is driven to zero upon further cooling at $T_{c2}=1.5\times10^{-5}$, signaling the gap is closed. The red shaded region denotes the parameter region with the open superconducting gap, whereas the blue region corresponds to the gap-closed phase. Momentum-resolved spectral functions $A(\omega_x,k)$ at representative temperatures are presented in panels (A–D). As $T$ is lowered, the superconducting coherence features evolve markedly, consistent with a superconducting–insulator transition.
• Figure 4: Scaling of the energy gap $\Delta$ with temperature $T$ and wave vector $k$. Left panel: $\Delta$ exhibits critical scaling at $T_{c2}$ with exponent $\alpha_T=0.85$. Right panel: $\Delta$ shows analogous scaling at $k_c=0.128$ with exponent $\alpha_k=0.85$.
• Figure 5: The schematic of the numerical solving the asymmetric EWCS.