[Paper Review] Entanglement Wedge Minimum Cross-Section in Holographic Axion Gravity Theories

Fang-Jing Cheng, Zhe Yang+3 more|arXiv (Cornell University)|Sep 8, 2021

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates mixed-state entanglement in holographic axion gravity models using the entanglement wedge minimum cross-section (EWCS), comparing it with holographic entanglement entropy (HEE) and mutual information (MI). It finds that EWCS monotonically increases with the axion-Maxwell coupling constant κ, while HEE and MI exhibit non-monotonic behavior, demonstrating that EWCS captures distinct quantum correlations beyond thermal noise, offering a unique probe of mixed-state entanglement in strongly correlated systems with momentum dissipation.

ABSTRACT

We study the mixed state entanglement properties in two holographic axion models by examining the behavior of the entanglement wedge minimum cross section (EWCS), and comparing it with the holographic entanglement entropy (HEE) and mutual information (MI). We find that the behavior of HEE, MI and EWCS with Hawking temperature is monotonic, while the behavior with the axion parameter $k$ is more rich, which depends on the size of the configuration and the values of the other two parameters. Interestingly, the EWCS monotonically increases with the coupling constant $κ$ between the axion field and the Maxwell field, while HEE and MI can be non-monotonic. It suggests that the EWCS, as a mixed state entanglement measure, captures distinct degrees of freedom from the HEE and MI indeed. We also provide analytical understandings for most of the numerical results.

Motivation & Objective

• To investigate the behavior of entanglement wedge minimum cross-section (EWCS) in holographic axion gravity models as a probe of mixed-state entanglement.
• To compare EWCS with holographic entanglement entropy (HEE) and mutual information (MI) in capturing quantum correlations under varying system parameters.
• To understand the physical origin of non-monotonic and monotonic behaviors in entanglement measures with respect to temperature, axion parameter k, and coupling constant κ.
• To provide analytical explanations for numerically observed behaviors in EWCS, HEE, and MI across two distinct linear axion models.

Proposed method

• The study employs two holographic axion gravity models with linear axion fields coupled to the Maxwell field, solving for black brane backgrounds numerically.
• HEE is computed via the Ryu-Takayanagi formula, requiring minimization of bulk surface area extending into the AdS spacetime.
• Mutual information (MI) is calculated as the sum of HEE for subsystems minus the HEE of their union, quantifying total correlations.
• EWCS is computed as the minimal area of a codimension-2 surface in the entanglement wedge, representing the entanglement of purification.
• Numerical analysis is performed across various configurations, temperatures, axion parameters k, and coupling constants κ to map entanglement behavior.
• Analytical expansions are used to understand the asymptotic behavior of EWCS and its derivatives with respect to κ and k, particularly in the small-k and large-κ limits.

Experimental results

Research questions

• RQ1 How does the entanglement wedge minimum cross-section (EWCS) behave with respect to the Hawking temperature T in holographic axion models?
• RQ2 How does EWCS respond to changes in the axion parameter k, and how does this compare to HEE and MI?
• RQ3 What is the dependence of EWCS on the coupling constant κ between the axion and Maxwell fields, and how does it differ from HEE and MI?
• RQ4 Why does EWCS show monotonic behavior with κ while HEE and MI show non-monotonicity, and what does this imply about the nature of entanglement captured?
• RQ5 Can analytical approximations explain the numerical results for EWCS, particularly in the small-k and large-κ regimes?

Key findings

• EWCS decreases monotonically with increasing Hawking temperature T, independent of configuration, k, or κ, consistent with thermal entanglement degradation.
• EWCS increases monotonically with the coupling constant κ between the axion and Maxwell fields, a behavior that is universal across configurations, T, and k.
• HEE and MI show non-monotonic behavior with κ, while EWCS remains monotonic, indicating that EWCS captures entanglement degrees of freedom distinct from those probed by HEE and MI.
• The behavior of EWCS with the axion parameter k is non-monotonic and configuration-dependent, revealing richer structure than HEE and MI, which show monotonic dependence on k.
• Analytical expansions confirm that ∂κf ∼ 1/κ at large κ, explaining the flattening of EWCS at large coupling, and ∂kf ∼ k at small k, explaining the flat behavior of EWCS in small-k regimes.
• The monotonic decrease of EWCS with T is physically interpreted as thermalization causing entanglement to vanish at high temperatures, with the minimum surface approaching the horizon.