Entanglement Entropy and Quantum Field Theory
Pasquale Calabrese


Abstract
Recently there has been considerable interest in formulating measures of quantum entanglement and applying them to extended quantum systems with many degrees of freedom, such as quantum spin chains. One of these measures is entanglement entropy. We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy $S_A=-{\\rm Tr}\\,\\rho_A\\log\\rho_A$ 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 $S_A\\sim(c/3)\\log\\ell$ of Holzhey et al. when $A$ is a finite interval of length $\\ell$ 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.

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