Reduced density matrix and internal dynamics for multicomponent regions

TL;DR

The paper constructs and diagonalizes the reduced density matrix for a massless Dirac field in 1+1D reduced to multiple disjoint intervals, revealing a nonlocal yet quasilocal modular flow that mixes fields along a finite set of trajectories. By solving the resolvent of the Cauchy-type kernel D on a union of intervals, the authors obtain explicit eigenvectors and decompose the modular Hamiltonian into local and nonlocal parts, showing that the nonlocal mixing occurs at a finite number of points per interval. Entropy and mutual information are computed in the massless case and extended to the small-mass regime, where leading log terms and extensive mutual information persist, with long-distance behavior governed by Bessel functions. The work provides an exactly solvable toy model for multipartite entanglement and the emergence of internal time in quantum field theory, with potential extensions to other CFTs and curved geometries.

Abstract

We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary powers of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval this evolution is non-local, producing both, advance in the causal structure and "teleportation" between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.

Figures (3)

• Figure 1: Two spatial sets $V$ and $V^\prime$ in two dimensional Minkowski space. They have the same causal domain of dependence which is the diamond shaped set in the figure. Light rays are shown at $\pm 45^\circ$.
• Figure 2: Three related trajectories for a three interval case. The null coordinates in the three diamonds are $u_\pm^l(z_\pm)$, $l=1$, $2$, $3$, and $z_\pm\in (-\infty,\infty)$, with $z_+-z_-=$constant. The related points do not show any geometric symmetry in general.
• Figure 3: Two intervals of size $d$ separated by: (a) $(1/4) d$ (top pair of diamonds), (b) $(1/70) d$, and (c) $(1/1000) d$ (bottom pair of diamonds). The trajectories in (a) approach the ones corresponding to single independent diamonds. In (b) the curves get distorted by the proximity of the diamonds. In (c) the curves are very similar to the ones of a single diamond of size $2 d$ (shown with dashed lines), excepting when they hit the null boundaries and run along them afterward. The trajectories shown for the two diamonds are equal spaced in the $t=0$ plane, and they are not related to each other.