Abstract The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.
We consider the fermionic entanglement entropy for the free Dirac field in a bounded spatial region of Minkowski spacetime. In order to make the system ultraviolet finite, a regularization is introduced. An area law is proven in the limiting cases where the volume tends to infinity and/or the regularization length tends to zero. The technical core of the paper is to generalize a theorem of Harold Widom to pseudo-differential operators whose principal symbols develop a specific discontinuity at a single point.
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 + 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.
Abstract In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2, 3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set’s retarded nonlocal d’Alembertians parametrised by a length scale l k . In both cases we provide numerical evidence that the area law is recovered when the double-cutoff prescription proposed in Sorkin and Yazdi (2016 Entanglement entropy in causal set theory (arXiv:1611.10281)) is imposed. We discuss in detail the need for this double cutoff by studying the effect of two cutoffs on the quantum field and, in particular, on the entanglement entropy, in isolation. In so doing, we get a novel interpretation for why these two cutoff are necessary, and the different roles they play in making the entanglement entropy on causal sets finite.
We define the entanglement entropy of free fermion quantum states in an arbitrary space-time slice of a discrete set of points and particularly investigate timelike (causal) slices. For one-dimensional lattice free fermions with an energy bandwidth
E0, we calculate the time-direction entanglement entropy
SA in a time-direction slice of a set of times ${t}_{n}=n\ensuremath{\tau}$ $(1\ensuremath{\le}n\ensuremath{\le}K)$ spanning a time length
t on the same site. For zero-temperature ground states, we find that
SA shows volume law when $\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\tau}}_{0}=2\ensuremath{\pi}/{E}_{0}$; in contrast, ${S}_{A}\ensuremath{\sim}\frac{1}{3}lnt$ when $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$, and ${S}_{A}\ensuremath{\sim}\frac{1}{6}lnt$ when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite-temperature thermal states, the mutual information also saturates when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$. For noneigenstates, volume law in
t and signatures of the Lieb-Robinson bound velocity can be observed in
SA. For generic space-time slices with one point per site, the zero-temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.
Abstract We study the entanglement properties of non-Hermitian free fermionic models with translation symmetry using the correlation matrix technique. Our results show that the entanglement entropy has a logarithmic correction to the area law in both one-dimensional and two-dimensional systems. For any one-dimensional one-band system, we prove that each Fermi point of the system contributes exactly 1/2 to the coefficient c of the logarithmic correction. Moreover, this relation between c and Fermi point is verified for more general one-dimensional and two-dimensional cases by numerical calculations and finite-size scaling analysis. In addition, we also study the single-particle and density–density correlation functions.