Entanglement Entropy and Quantum Field Theory

TL;DR

This paper systematically studies entanglement entropy in relativistic quantum field theory, deriving key formulas for various systems and confirming them with models, and extends results to higher dimensions and quantum phase transitions.

Contribution

It provides a comprehensive derivation and extension of entanglement entropy formulas in quantum field theory, including critical, off-critical, and higher-dimensional cases, verified by multiple models.

Findings

Re-derivation of entanglement entropy for 1+1D conformal field theories.

Extension of entropy formulas to finite systems, temperatures, and multiple intervals.

Verification of results using free massive fields and integrable lattice models.

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 rho_A log rho_A corresponding to the reduced density matrix rho_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 S_A\sim(c/3) log(l) of Holzhey et al. when A is a finite interval of length l 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 \xi is large but finite, we show that S_A\sim{\cal A}(c/6)\log\xi, where \cal A 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.