Entanglement entropy and quantum field theory
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 rgr_{A}logrgr_{A} corresponding to the reduced density matrix rgr_{A} of a subsystem A. For the case of a 1+1dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we rederive the result S_{\mathrm {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 xgr is large but finite, we show that S_{\mathrm {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 offcritical 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.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 June 2004
 DOI:
 10.1088/17425468/2004/06/P06002
 arXiv:
 arXiv:hepth/0405152
 Bibcode:
 2004JSMTE..06..002C
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter  Statistical Mechanics;
 Quantum Physics
 EPrint:
 33 pages, 2 figures. Our results for more than one interval are in general incorrect. A note had been added discussing this