Entanglement entropy of ground states of the three-dimensional ideal Fermi gas in a magnetic field

TL;DR

This paper investigates the asymptotic behavior of entanglement entropy in three-dimensional free fermion gases under a magnetic field, revealing a logarithmically enhanced area law and deriving explicit formulas for the entropy coefficient.

Contribution

It provides the first detailed analysis of entanglement entropy growth in 3D fermion gases with magnetic fields, including explicit coefficient expressions and improved asymptotic trace estimates.

Findings

Entanglement entropy scales as L^2 log(L) in 3D with magnetic field.

Derived explicit surface integral formula for the entropy coefficient.

Established improved asymptotic expansion for Wiener--Hopf operator traces.

Abstract

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in $\mathbb R^3$ subject to a non-zero, constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order $L^2\ln(L)$, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region $L\Lambda\subset \mathbb R^3$ as the scaling parameter $L$ tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra $\ln(L)$ factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom formula in the non-magnetic case. It differs however in the sense that the dependence on the boundary is not solely on its area but on the "area perpendicular to the direction of the magnetic field". On the way we prove an improved two-term asymptotic expansion (up to an error term of order one) of certain traces of one-dimensional Wiener--Hopf operators with a discontinuous symbol. This is of independent interest and leads to an improved error term of the order $L^2$ of the relevant trace for piecewise $\mathsf{C}^{1,\alpha}$ smooth surfaces $\partial \Lambda$.