On the stability of the area law for the entanglement entropy of the Landau Hamiltonian

Abstract

We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane R 2 and whose strength B(x) at x ∈ R 2 converges to some B 0 > 0 as x → ∞. Furthermore, we allow for an electric potential Vε which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential µ ≥ B 0 we study the asymptotic growth of its bipartite entanglement entropy associated to LΛ as L → ∞ for some fixed bounded region Λ ⊂ R 2 . We show that its leading order in L does not depend on the perturbations Bε := B 0 − B and Vε if they satisfy some mild decay assumptions. Our result holds for all α-Rényi entropies α > 1/3; for α ≤ 1/3, we have to assume in addition some differentiability of the perturbations Bε and Vε. The case of a constant magnetic field Bε = 0 and with Vε = 0 was treated recently for general µ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary ∂Λ.