Temporal Entanglement Entropy as a probe of Renormalization Group Flow

TL;DR

This work defines temporal entanglement entropy (tEE) by extending timelike EE to Euclidean space and linking tracing over a Euclidean-time interval $T_0$ to coarse-graining and momentum-space entanglement, interpreted through RG flow. It leverages cutoff holography and the irrelevant $T\bar{T}$ deformation in AdS$_3$/CFT$_2$ to relate the UV cutoff to the smallest resolvable Euclidean time and to examine how integrating out UV modes corresponds to larger $T_0$, including a finite-$\mu$ entanglement formula that parallels finite-temperature CFT behavior. The paper presents explicit tEE expressions in finite-temperature and Lifshitz-like nonrelativistic settings, revealing a Hawking-Page-type transition between bulk geometries and showing that tEE can detect the dynamical exponent $z$ in Lifshitz theories, which spacelike EE at zero density cannot access. Together, these results establish temporal entanglement as a practical probe of RG flow and momentum-space structure, with potential applications to diagnosing critical exponents in nonrelativistic quantum field theories.

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.

Figures (3)

• Figure 1: Relation of AdS cylinder in global coordinates and CFT in radial quantization. Time translations in the bulk correspond to dilations in CFT linking energies in AdS to dimensions in CFT. Figure adapted from KaplanJHU. The long dash dotted lines in the left plot correspond to the cutoff geometry which is dual to a $T\bar{T}$ deformation of the field theory.
• Figure 2: Minimal surfaces \ref{['eq:minimal']} as a function of the radial coordinate $\eta$ for different turning points in the bulk. To probe larger time intervals $T_0$ we require information from deeper in the bulk (the IR), i.e. the turning point moves so smaller $\eta$. Moving the cutoff into the bulk results into losing the minimal surfaces that resolved the smallest time intervals in the original theory.
• Figure 3: Difference in entanglement entropy $\Delta S_\text{EE}\equiv S_\text{EE}(\mu)-S_\text{EE}(\mu=0)$ for a temporal interval $T_0=L/10$ as given by \ref{['eq:timelikeb']} (blue) and a spatial interval $X_0=L/10$ as given by \ref{['eq:spacelikeb']} (red, dashed). The phase transition from thermal AdS to BTZ is indicated by the vertical green line. For simplicity, we set $G_N=L/2$ and $\eta_c=10^7 L$.