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 $\Vert x\Vert \to \infty$. Furthermore, we allow for an electric potential $V_{\epsilon }$ 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 $B_{\epsilon }:=B_0-B$ and $V_{\epsilon }$ 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 $B_{\epsilon }$ and $V_{\epsilon }$. The case of a constant magnetic field $B_{\epsilon }=0$ and with $V_{\epsilon }=0$ 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 $\partial \Lambda$.

arXiv comment38 pages; In the previous version, there was a mistake in the proof of Theorem 3.3. The estimate for the final term proved to be a lot harder than anticipated. This version fixes that mistake and got longer in the progress

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Cited by in corpus (1)

• Stability of a Szegő-type asymptotics