Entanglement Temperature and Entanglement Entropy of Excited States

TL;DR

This work develops a path-integral framework that ties the ground-state entanglement Hamiltonian to the physical stress tensor, yielding a local entanglement temperature for spherical entangling surfaces in conformal field theories. It extends the formalism to excited states with conserved charges, deriving a generalized first-law-like relation for entropy and providing a CFT derivation of entanglement entropy in these settings, including constraint equations in any dimension. The authors also connect to holographic results, offering first-order holographic computations and highlighting discrepancies for nonuniform energy densities, while deriving dynamical equations for the evolution of entanglement entropy and introducing the entanglement density as a local diagnostic. Overall, the paper deepens the field-theoretic understanding of entanglement, clarifies its relationship to holography, and supplies practical tools for computing entanglement in 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.

Figures (3)

• Figure 1.1: Evaluating $\rho_{A}$ along Euclidean time s
• Figure 3.1: Foliation of the Euclidean plane corresponding to angular quantization
• Figure 3.2: A rotation on the z plane (represented as a Riemann sphere) is mapped to a conformal rotation on the w plane