Enhanced area law in the Widom-Sobolev formula for the free Dirac operator in arbitrary dimension

TL;DR

This paper establishes a logarithmically enhanced area law for Rényi entanglement entropies of free relativistic Dirac fermions, revealing how entanglement scales with region size across different dimensions and mass regimes.

Contribution

It extends the Widom-Sobolev formula to matrix-valued symbols and applies it to derive entanglement entropy asymptotics for Dirac fermions in arbitrary dimensions.

Findings

Logarithmic enhancement of area law for certain Fermi energies and mass conditions.

Entanglement entropy growth is bounded by the spatial region's area in other cases.

Generalization of Widom-Sobolev formula to matrix-valued symbols.

Abstract

We prove a logarithmically enhanced area law for all R\'enyi 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 R\'enyi entropies and relies on a recently proved extension of the Widom-Sobolev formula to matrix-valued symbols by the authors.