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

Abstract

AbstractWe study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in $${{\mathbb {R}}}^3$$ 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 $$L^2\ln (L)$$ L 2 ln ( L ) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region $$L\Lambda \subset {{\mathbb {R}}}^3$$ L Λ ⊂ 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 $$\ln (L)$$ 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, $$\partial \Lambda $$ ∂ Λ , 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 $$L^2$$ L 2 of the relevant trace for piecewise $$\textsf{C}^{1,\alpha }$$ C 1 , α smooth surfaces $$\partial \Lambda $$ ∂ Λ .