Entanglement temperature and entanglement entropy of excited states
We derive a general relation between the ground state entanglement Hamiltonian and the physical stress tensor within the path integral formalism. For spherical entangling surfaces in a CFT, we reproduce the local ground state entanglement Hamiltonian derived by Casini, Huerta and Myers. The resulting reduced density matrix can be characterized by a spatially varying “entanglement temperature”. Using the entanglement Hamiltonian, we calculate the first order change in the entanglement entropy due to changes in conserved charges of the ground state, and find a local first law-like relation for the entanglement entropy. Our approach provides a field theory derivation and generalization of recent results obtained by holographic techniques. However, we note a discrepancy between our field theoretically derived results for the entanglement entropy of excited states with a non-uniform energy density and current holographic results in the literature. Finally, we give a CFT derivation of a set of constraint equations obeyed by the entanglement entropy of excited states in any dimension. Previously, these equations were derived in the context of holography.
KeywordsField Theories in Lower Dimensions AdS-CFT Correspondence Conformal and W Symmetry Field Theories in Higher Dimensions
- L. Susskind and J. Lindesay, An introduction to black holes, information and the string theory revolution: the holographic universe, World Scientific, Hackensack U.S.A. (2005) [INSPIRE].
- J.L. Cardy, Conformal invariance and statistical mechanics, in Les Houches, Session XLIX. Champs, Cordes et Phénomènes Critiques: Fields, Strings and Critical Phenomena, E. Brézin and J. Zinn-Justin eds., Elsevier Science Publishers (1988), pp. 169–245.Google Scholar