Temporal Entanglement Entropy as a probe of Renormalization Group Flow

TL;DR

This paper extends timelike entanglement entropy to Euclidean space, linking it to RG flow and holography, and demonstrates its ability to detect critical exponents in non-relativistic theories.

Contribution

It introduces temporal entanglement entropy in Euclidean space and connects it to RG flow and holographic cutoff, providing new insights into UV degrees of freedom and critical phenomena.

Findings

Tracing over Euclidean time corresponds to coarse-graining.

UV cutoff relates to the smallest resolvable time interval.

Temporal entanglement entropy detects Lifshitz critical exponent z.

Abstract

The recently introduced concept of timelike entanglement entropy has sparked a lot of interest. Unlike the traditional spacelike entanglement entropy, timelike entanglement entropy involves tracing over a timelike subsystem. In this work, we propose an extension of timelike entanglement entropy to Euclidean space ("temporal entanglement entropy"), and relate it to the renormalization group (RG) flow. Specifically, we show that tracing over a period of Euclidean time corresponds to coarse-graining the system and can be connected to momentum space entanglement. We employ Holography, a framework naturally embedding RG flow, to illustrate our proposal. Within cutoff holography, we establish a direct link between the UV cutoff and the smallest resolvable time interval within the effective theory through the irrelevant $T\bar T$ deformation. Increasing the UV cutoff results in an enhanced capability to resolve finer time intervals, while reducing it has the opposite effect. Moreover, we show that tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom (or lowering the temperature). As an application, we point out that the temporal entanglement entropy can detect the critical Lifshitz exponent $z$ in non-relativistic theories which is not accessible from spatial entanglement at zero temperature and density.