Entanglement Temperature and Entanglement Entropy of Excited States

Abstract

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 \emph{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.

17 pages, 3 figures. Typo fixed in eq. (III.12) and constraint equation (VI.18) added generalizing previous result to higher dimensions.Some references added

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• Dynamics of Entanglement Entropy from Einstein Equation
• Entropic Counterpart of Perturbative Einstein Equation
• Entanglement Thermodynamics
• Entanglement entropy from the holographic stress tensor
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