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 continuum-like 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 $1+1$D 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 R\'enyi 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.

On the (Non)Hadamard property of the SJ state in a 1+1 D causal diamond

“…Based on standard results, a spatial area-law scaling (logarithmic scaling of S with N in 1 + 1D) was anticipated [31,44,45], but instead a surprising spacetime volume-law (linear scaling of S with N in 1 + 1D) was obtained [32]. The excess entropy was found to be related to numerous unexpected fluctuation-like components [33], in the small eigenvalue regime of the SJ eigendecomposition. These eigenvalues also have another marked difference from those expected to contribute to the entropy: they do not follow a power law when sorted from largest to smallest.…”

“…We then proceed to study both the coincidence limit and non-coincidence behavior of the SJ Wightman function, to probe where it may not be Hadamard. Second, recent results on entanglement entropy in causal set theory indicate the presence of numerous unexpected ultraviolet contributions to the entropy [32][33][34][35]. Since the SJ vacuum is used as the pure state in these calculations, and since the Hadamard condition is often linked to short-distance behavior, we investigate whether the two may be linked.…”