Entanglement Entropy of Free Fermions in Timelike Slices

TL;DR

This work defines a spacetime-slice entanglement entropy for free fermions, enabling the entanglement of quantum states restricted to arbitrary spacetime slices. Using the Heisenberg-picture construction and a corresponding path-integral formalism, the authors show how to identify a finite-dimensional sub-Hilbert space h_A on a given slice and compute S_A from two-point correlators. In 1D tight-binding models, zero-temperature time-direction slices exhibit a tunable crossover from a volume law at large time spacing to Calabrese–Cardy-type logarithmic scaling in the continuum-time limit, with a distinct (1/6) coefficient signaling a single energy-space Fermi point in the continuum limit. Finite-temperature states reveal saturation behaviors and a robust transition between spacelike and timelike slices, while linear slices at angle θ interpolate between these regimes, illustrating a broad, controllable landscape for spacetime entanglement in free fermion systems with potential extensions to interacting or bosonic theories.

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 $t_n=nτ$ ($1\le n\le K$) spanning a time length $t$ on the same site. For zero temperature ground states, we find that $S_A$ shows volume law when $τ\ggτ_0=2π/E_0$; in contrast, $S_A\sim \frac{1}{3}\ln t$ when $τ=τ_0$, and $S_A\sim\frac{1}{6}\ln t$ when $τ<τ_0$, 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 $τ<τ_0$. 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.

Figures (18)

• Figure 1: (a) An arbitrary spacetime slice $A$ (yellow) with discrete spacetime points $(\mathbf{r}_n,t_n)$ at lattice sites $\mathbf{r}_n$ and times $t_n$. In a 1D lattice fermion model, (b) shows a time-direction slice (yellow) of length $t$ containing $K$ points at times $t_n$ ($1\le n\le K$) on the same site $m=0$; (c) shows a linear slice (yellow) containing points at $t_n=v_{\text{max}}^{-1}x_n \tan(\theta)$ on the $n$-th site ($0\le n\le \ell-1$), with $v_{\text{max}}$ defined in \ref{['eq:tau0']}.
• Figure 2: Non-constant time slice $A$ is part of a hypersurface $S=A\cup A^c$ that covers the entire space of the system.
• Figure 3: (a)-(d) Entanglement entropy $S_A$ of model \ref{['eq:H-model']} with $L=500$ in time-direction slice of length $t$ with $K$ points separated by $\tau=t/(K-1)$, for (a) different filling $\nu$ given in the legend at temperature $T=0$ and fixed $2\pi t/\tau_0 = 168$; (b) finite temperature $S_A$ with fixed $2\pi t/\tau_0=168$, at fillings $\nu=0.5$ (solid) and $\nu=0.3$ (dashed). For each $\nu$, the four curves from low to high have $T/E_0=0,0.075, 0.15, \infty$, respectively. (c)-(d) $S_A$ versus $\ln t$ for different fillings $\nu$ (see legend) at $T=0$ and fixed (c) $\tau=\tau_0$ and (d) $\tau=\frac{2}{\pi} \tau_0$. (e)-(f) Mutual information $\mathcal{I}$ with $L=200$ at $\nu = 0.5$ and temperatures $T/E_0$ given in the legends while fixing (e) $2\pi t/\tau_0 = 168$ and (f) $\tau = \frac{5}{2\pi} \tau_0$, respectively.
• Figure 4: (a) The ancillary dispersion for explaining $S_A$ when $\tau<\tau_0$ ("anc" for ancillary, "emp" for empty). (b) For 1D gapless lattice fermion, the zero temperature Fermi sea state has two Fermi points at $\pm k_F$ in the momentum space. Accordingly, the Calabrese-Cardy formula for the spatial entanglement entropy is $S_{A'} =\frac{1}{3}\ln\left(\frac{l}{l_{c}}\right)$. (c) In the energy space, the zero temperature state has single-particle states with $E \leq E_F$ are filled, so there is only one Fermi point $E_F$ in the energy domain. Accordingly, the time-direction entanglement entropy in the continuum time limit is $S_{A} =\frac{1}{6}\ln\left(\frac{t}{t_{c}}\right)$.
• Figure 5: Entanglement entropy $S_A$ of time-direction slice of $K$ points at site $m=0$ for non-eigenstates (a)-(b) $|\psi_q\rangle=\prod_{m} c_{q m}^\dag|0\rangle$ ($L=500$) with fixed $2 \pi \tau / \tau_0 = 6$, and (c)-(d) $|\psi\rangle=\prod_{-\frac{N_f}{3}<m<\frac{2N_f}{3}}c_m^\dag|0\rangle$ ($L=1000$) with $N_f=100$. $q$ is given in the legend of (a). In (b), the dots are the fitted $\left(\frac{\partial S_A}{\partial K}\right)_\tau$ and the red line is $-\nu\ln\nu-(1-\nu)\ln(1-\nu)$. (c) is calculated with fixed $2 \pi t / \tau_0$ given in the legend, while (d) has fixed $2 \pi \tau / \tau_0$ given in the legend.