Entanglement entropy and quantum field theory

TLDR

Entanglement entropy is the von Neumann entropy of a subsystem’s reduced density matrix. The authors perform a systematic study of entanglement entropy in relativistic quantum field theory. They verify their results in a free massive field theory and in integrable lattice models such as Ising and XXZ, using corner transfer matrix methods to confirm a scaling ansatz. They recover the Holzhey–Latorre result for 1+1‑dimensional critical systems, extend it to finite size, finite temperature, and multiple intervals, show that away from criticality the entropy scales with the number of boundary points, and generalize the free‑field findings to higher dimensions, suggesting a scaling form for the singular part near quantum phase transitions.

Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.