Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

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 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–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, \(\partial \Lambda \), is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order \(L^2\) of the relevant trace for piecewise \(\textsf{C}^{1,\alpha }\) smooth surfaces \(\partial \Lambda \).