Towards spacetime entanglement entropy for interacting theories

Abstract

Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.

On the Entanglement Entropy in Gaussian cMERA

“…[13,14,17] Consequently, we conjecture that, for those circuits, the entanglement entropy can be computed through (48). The results so obtained would be compared with recent results in the literature [19,26,27] dealing with entanglement entropy in interacting theories.…”

Entanglement entropy: non-Gaussian states and strong coupling

“…The author showed that a consistent renormalization can be performed, providing finite contributions to the entanglement entropy at one loop. In [7], in a quantum mechanical setting consisting on quartic perturbative perturbations on the harmonic oscillator free case, the replica trick was used to check that to first order in perturbation theory, the entanglement entropy can be computed by means of the perturbatively corrected version of the two point correlation functions of the system.…”

Bosonic and fermionic Gaussian states from Kähler structures

“…where q,p ≡ and a,a † ≡ indicate that the RHS corresponds to the matrix representation with respect to one of the two standard bases (55) or (54). Note that these standard bases are only determined up to an overall group transformation in G that will preserve the respective structures.…”

“…The entanglement entropy of a non-Gaussian state will in general also depend on higher npoint functions, so we cannot use (230) anymore. Interestingly, if we perturb a Gaussian state in a non-Gaussian way, the entanglement entropy will at linear order only feel the Gaussian part of the perturbation [55], so that we can use the linearization of (230) to deduce the linear change…”