Remarks on the entanglement entropy for disconnected regions

TL;DR

This work probes whether the mutual information for disconnected regions in quantum field theory can be extensive. It develops a general framework showing that, if extensivity held, the mutual information would be expressible as a boundary integral with a monotone RG-related function, and then tests this across several theories. The main findings are that extensivity holds in 2D for massless Dirac fermions but fails for holographic entropies, free scalars, and massive modes in higher dimensions, with only partial, sign-varying deviations in some cases. The results tightly constrain the existence of extensive theories and connect entanglement structure to RG irreversibility and conformal anomalies, outlining a path toward identifying interacting models that may realize extensivity.

Abstract

Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures.

Figures (4)

• Figure 1: Two spatial sets $A$ and $B$ in two dimensional Minkowski space-time. Light rays are plotted at $\pm 45^\circ$. The spatial sets $A$, $A^\prime$, and $B$, $B^\prime$, have the same causal domain of dependence $\hat{A}$ and $\hat{B}$ respectively. Then we have $I(A,B)=I(A^\prime,B^\prime)$.
• Figure 2: In order to obtain the entropy corresponding to $A$, we can evaluate the mutual information between $A$ and $-A_\epsilon$, which is the set formed by the points separated from $A$ more than a short distance cutoff $\epsilon$.
• Figure 3: The tripartite information function $I(A,B,C)$ for a two dimensional Dirac field with different masses. The involved sets are an interval $A$ of size $a$ separated by a distance $d$ from the two adjacent intervals $B$ of length $b$ and $C$ of length $c$. The examples shown are for $a=b=c$, $d/a=3$ (the curve with the smaller well), and $a=d$, $b=c=2/3 a$ (the curve at the bottom, with the larger well). The $x$ axis is the product of the mass with the total length $a+d+b+c$.
• Figure 4: Comparison between the coefficients of the logarithmic term in three dimensions for different theories. The picture shows, from top to bottom, the functions $s_S/s_E$, $s_D/s_E$ and $s_H/s_E$. The differences increase toward the origin where these ratios are $s_S/s_E=1.078$, $s_D/s_E=.980$ and $s_H/s_E=.956$ (not shown). The functions $s_H$ and $s_E$ are normalized such that the coefficient of $(x-\pi)^2$ in the Taylor series expansion around $x=\pi$ coincide with the ones of $s_D$ and $s_S$.