We prove a logarithmically enhanced area law for all Rényi entanglement entropies of the ground state of a free gas of relativistic Dirac fermions. Such asymptotics occur in any dimension if the modulus of the Fermi energy is larger than the mass of the particles and in the massless case at Fermi energy zero in one space dimension. In all other cases of mass, Fermi energy and dimension, the entanglement entropy grows no faster than the area of the involved spatial region. The result is established for a general class of test functions which includes the ones corresponding to Rényi entropies and relies on a recently proved extension of the Widom-Sobolev formula to matrix-valued symbols by the authors.
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 two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane
R2 and whose strength
B(x) at
x∈R2 converges to some
B0>0 as
∥x∥→∞. Furthermore, we allow for an electric potential
Vε which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential
μ≥B0 we study the asymptotic growth of its bipartite entanglement entropy associated to
LΛ as
L→∞ for some fixed bounded region
Λ⊂R2. We show that its leading order in
L does not depend on the perturbations
Bε:=B0−B and
Vε if they satisfy some mild decay assumptions. Our result holds for all
α-R\' enyi entropies
α>1/3; for
α≤1/3, we have to assume in addition some differentiability of the perturbations
Bε and
Vε. The case of a constant magnetic field
Bε=0 and with
Vε=0 was treated recently for general
μ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary
∂Λ.
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.