Entanglement Entropy of Free Fermions in Timelike Slices

TL;DR

This paper investigates the entanglement entropy of free fermions in arbitrary spacetime slices, revealing volume and logarithmic laws depending on the slice's causal nature and temperature, with connections to known conformal field theory results.

Contribution

It introduces a framework for calculating entanglement entropy in timelike slices of free fermions, extending previous spatial analyses to causal slices and identifying new entropy scaling behaviors.

Findings

Volume law for zero-temperature ground states when time slices are large.

Logarithmic scaling of entanglement entropy at critical time intervals.

Transition from area law to volume law when changing from spacelike to timelike slices.

Abstract

We define the entanglement entropy of free fermion quantum states in an arbitrary spacetime slice of a discrete set of points, and particularly investigate timelike (causal) slices. For 1D lattice free fermions with an energy bandwidth $E_0$, we calculate the time-direction entanglement entropy $S_A$ in a time-direction slice of a set of times $t_n=n\tau$ ($1\le n\le K$) spanning a time length $t$ on the same site. For zero temperature ground states, we find that $S_A$ shows volume law when $\tau\gg\tau_0=2\pi/E_0$; in contrast, $S_A\sim \frac{1}{3}\ln t$ when $\tau=\tau_0$, and $S_A\sim\frac{1}{6}\ln t$ when $\tau<\tau_0$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite temperature thermal states, the mutual information also saturates when $\tau<\tau_0$. For non-eigenstates, volume law in $t$ and signatures of the Lieb-Robinson bound velocity can be observed in $S_A$. For generic spacetime slices with one point per site, the zero temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.