Insights on Entanglement Entropy in $1+1$ Dimensional Causal Sets

Abstract

Entanglement entropy in causal sets offers a fundamentally covariant characterisation of quantum field degrees of freedom. A known result in this context is that the degrees of freedom consist of a number of contributions that have continuumlike analogues, in addition to a number of contributions that do not. The latter exhibit features below the discreteness scale and are excluded from the entanglement entropy using a "truncation scheme". This truncation is necessary to recover the standard spatial area law of entanglement entropy. In this paper we build on previous work on the entanglement entropy of a massless scalar field on a causal set approximated by a 1 + 1D causal diamond in Minkowski spacetime. We present new insights into the truncated contributions, including evidence that they behave as fluctuations and encode features specific to a particular causal set sprinkling. We extend previous results in the massless theory to include Rényi entropies and include new results for the massive theory. We also discuss the implications of our work for the treatment of entanglement entropy in causal sets in more general settings.

Spacetime entanglement entropy: covariance and discreteness

“…(1.48). The equivalence between the von Neumann entropy 10 A derivation of the same can also be found in [36,37].…”

“…These violations are "small" and become negligible when suitably averaged over the ensemble. Adding to this is the work of [27] where the SJ eigenfunctions in the d = 2 causal diamond were examined in more detailed. They found that the eigenfunctions beyond the knee were highly fluctuating at and "below" the discreteness scale.…”

“…This formula has been applied in the continuum to the d = 2 nested causal diamonds and shown to give the expected Calabrese-Cardy logarithmic behaviour [39]. It has also been calculated in the discrete setting, i.e., for causal sets approximated by causal diamonds in Minkowski and de Sitter spacetimes: in d = 2 where it shows the expected logarithmic behaviour and in d = 4 where it shows the expected area behaviour [40,41,42,43,37].…”