Volume Law for the Entanglement Entropy in Non-local QFTs

TL;DR

This work demonstrates that simple non-local, translationally invariant free scalar field theories can exhibit a volume-law entanglement entropy for small subsystems, with $S_\Omega(L) \approx (w/2) A L$ when $L\ll A$ and saturation $S_\Omega(L) \sim c A^2$ when $L\gg A$. The authors develop a real-time formalism for free bosons, apply it to 2D lattices with $w=1$ and $w=2$, and confirm the volume-law behavior through numerical results and analytical asymptotics. They generalize to higher dimensions, showing $S_\Omega(L) \sim C_1 A L R^{d-2}$ for small $L$ and $S_\Omega(L) \sim C_2 A^2 R^{d-2}$ for large $L$, indicating a robust volume-law regime beyond 2D. A holographic interpretation via RT/ cMERA considerations yields consistent volume-law scaling in flat-like spacetimes and predicts the same behavior for general $w>0$, linking non-local QFTs to flat-spacetime holography. These results provide a concrete continuum model where non-locality drives a departure from the area law and align field-theoretic findings with holographic expectations.

Abstract

In this paper, we present a simple class of non-local field theories whose ground state entanglement entropy follows a volume law as long as the size of subsystem is smaller than a certain scale. We will confirm this volume law both from numerical calculations and from analytical estimation. This behavior fits nicely with holographic results for spacetimes whose curvatures are much smaller than AdS spaces such as those in the flat spacetime.

Figures (3)

• Figure 1: The entanglement entropy $S_{\Omega}(L)$ of one interval whose length is $L$ for $w=1$ as a function of $L$. In the left picture, the blue, red, yellow and green points correspond to $A=400, 600, 800, 1000$. In the right picture, the blue, red and yellow points correspond to $A=40, 60, 80$.
• Figure 2: The entanglement entropy $S_{\Omega}(L)$ of one interval whose length is $L$ for $w=2$ as a function of $L$. In the left picture, the blue, red, yellow and green points correspond to $A=400, 600, 800, 1000$. In the right picture, the blue, red and yellow points correspond to $A=40, 60, 80$.
• Figure 3: The matrices $\Lambda(L,A)$ for $w=2$ (left) and $w=1$ (right). In the left picture we show $\Lambda(L=100,A=20)$ for $w=2$ and the magnitude of the matrix elements in the white region is smaller than $10^{-15}$ times the maximum of the matrix elements. The matrix elements in the orange (blue) region is positive (negative). In the right picture we show $\Lambda(L=80,A=50)$ for $w=1$ and the magnitude of the matrix elements in the white region is smaller than $10^{-18}$ times the maximum of the matrix elements. The matrix elements in the orange (blue) region are positive (negative) again.