Diagnosing Critical Behavior in AdS Einstein-Maxwell-Scalar Theory via Holographic Entanglement Measures

TL;DR

The paper analyzes holographic mixed-state entanglement measures in the EMS model to diagnose phase transitions between normal and scalarized black branes. Using HEE, MI, EWCS, and butterfly velocity $v_B$, it shows MI and EWCS rise while HEE falls across the transition, and $v_B$ exhibits non-monotonic dependence on the coupling $b$ due to competing entanglement and thermal contributions. A universal set of critical exponents is found: $\alpha_{v_B}=\alpha_{S_E}=\alpha_{E_w}=1$, with the scalar-field exponent $\alpha_\phi=1/2$, yielding $\alpha$ for geometry-related quantities twice that of $\alpha_\phi$. An inequality is established where the growth rate of MI always exceeds that of EWCS during phase transitions, suggesting a fundamental ordering of mixed-state correlations in holographic systems. The results point to universal behavior of holographic quantum information across thermodynamic transitions and motivate extending the analysis to other phase transitions and gravity theories.

Abstract

We investigate the holographic mixed-state entanglement measures in the Einstein-Maxwell-Scalar (EMS) theory. Several quantities are computed, including the holographic entanglement entropy (HEE), mutual information (MI), entanglement wedge cross-section (EWCS), and butterfly velocity ($v_B$). Our findings demonstrate that these measures can effectively diagnose phase transitions. Notably, EWCS and MI, as mixed-state entanglement measures, exhibit behavior opposite to that of the HEE. Additionally, we study the butterfly velocity, a dynamic quantum information measure, and observe that it behaves differently from the static quantum information measures. We propose that the butterfly velocity is initially dominated by entanglement and subsequently by thermal entropy as the coupling constant increases. Moreover, we examine the scaling behavior of the holographic entanglement measures and find that all the critical exponents are equal to $1$, which is twice that of the scalar field. We also explore the inequality between EWCS and MI, noting that the growth rate of MI consistently exceeds that of EWCS during phase transitions. These features are expected to be universal across thermodynamic phase transitions, with the inequalities becoming more significant as one moves away from the critical point.

Figures (11)

• Figure 1: Phase diagram of thermal phase transition in the EMS model.
• Figure 2: The left plot: The minimum surface for a given width $w$. The right plot: The minimum cross-section (green surface) of the entanglement wedge.
• Figure 3: The Illustration of holographic mutual information.
• Figure 4: The illustration depicting the calculation of EWCS, and the red surface representing the minimum cross-section.
• Figure 5: The behavior of HEE and MI versus temperature $T$ and coupling constant $b$ when the phase transition occurs. The red dashed line represents the critical point. Left plot: the behavior of HEE $S_E$ when we set the width $w=1$. Right plot: the behavior of MI when we set the configurations $(a,p,c)=(0.3,0.1,0.2)$.