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 $\mathbb R^2$ and whose strength $B(x)$ at $x\in\mathbb R^2$ converges to some $B_0>0$ as $\|x\|\to\infty$. Furthermore, we allow for an electric potential $V_\varepsilon$ 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 $\mu\ge B_0$ we study the asymptotic growth of its bipartite entanglement entropy associated to $L\Lambda$ as $L\to\infty$ for some fixed bounded region $\Lambda\subset\mathbb R^2$. We show that its leading order in $L$ does not depend on the perturbations $B_\varepsilon := B_0 - B$ and $V_\varepsilon$ if they satisfy some mild decay assumptions. Our result holds for all $\alpha$-R\' enyi entropies $\alpha>1/3$; for $\alpha\le 1/3$, we have to assume in addition some differentiability of the perturbations $B_\varepsilon$ and $V_\varepsilon$. The case of a constant magnetic field $B_\varepsilon = 0$ and with $V_\varepsilon= 0$ was treated recently for general $\mu$ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary $\partial \Lambda$.