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 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.

Entanglement entropy: a perturbative calculation

“…We have, however, suppressed such terms as for our applications these terms are contact terms, and in cases where it is relevant it will be implicitly assumed that one accounts for contact terms appearing in correlation functions. To first order in δρ the above expansion reveals a 'first law' of entanglement entropy [33][34][35] …”

“…It is convenient to think of the path integral in (2.12), (2.14) and (2.15) as an effective evolution from the slice C + to the slice C − [35]. In particular, based on these equations one can write…”

Holographic geometry of cMERA for quantum quenches and finite temperature

“…This will be related to the local quenches [42] and there have been examples of their gravity duals [43][44][45][46]. One more intriguing future problem is to explore universal properties of excited states from the viewpoint of cMERA (refer to recent results from holography and CFT calculations [47][48][49][50][51][52][53][54]). …”

Left-right entanglement entropy of Dp-branes

“…Perhaps more speculatively, it is worth mentioning that there have been some other contexts where the entanglement entropy seems to be connected to a thermodynamic entropy as evidenced in first-law like relations discussed in [20,21]. Such thermodynamic properties of the entanglement entropy were, in a sense, anticipated in particular cases like [22] where by considering varying the length of the entanglement region a confinement/deconfinement transition was suggested.…”

“…Such relations have been obtained purely in field theoretic terms [20,21] but in rather symmetry restricted setups. Equivalently, it would be interesting to understand the structure of the modular Hamiltonian underlying our computation of the reduced density matrix.…”