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

arXiv2102.07287MaRDI QIDQ6360595

Paul Pfeiffer

Publication date: 14 February 2021

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 $xinmathbb{R}^2$ converges to some $B_0>0$ as $|x|oinfty$. Furthermore, we allow for an electric potential $V_{v}arepsilon$ 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 $muge{B}_0$ we study the asymptotic growth of its bipartite entanglement entropy associated to $LLambda$ as $Loinfty$ for some fixed bounded region $Lambdasubsetmathbb{R}^2$. We show that its leading order in $L$ does not depend on the perturbations $B_{v}arepsilon:=B_0-B$ and $V_{v}arepsilon$ if they satisfy some mild decay assumptions. Our result holds for all $alpha$-R' enyi entropies $alpha>1/3$; for $alphale1/3$, we have to assume in addition some differentiability of the perturbations $B_{v}arepsilon$ and $V_{v}arepsilon$. The case of a constant magnetic field $B_{v}arepsilon=0$ and with $V_{v}arepsilon=0$ 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 $partialLambda$.

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