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 $\partial \Lambda$.

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