Logarithmically enhanced area-laws for fermions in vanishing magnetic fields in dimension two

Abstract

We consider fermionic ground states of the Landau Hamiltonian, $H_B$, in a constant magnetic field of strength $B>0$ in $\mathbb R^2$ at some fixed Fermi energy $\mu>0$, described by the Fermi projection $P_B:= 1(H_B\le \mu)$. For some fixed bounded domain $\Lambda\subset \mathbb{R}^2$ with boundary set $\partial\Lambda$ and an $L>0$ we restrict these ground states spatially to the scaled domain $L \Lambda$ and denote the corresponding localised Fermi projection by $P_B(L\Lambda)$. Then we study the scaling of the Hilbert-space trace, $\mathrm{tr} f(P_B(L\Lambda))$, for polynomials $f$ with $f(0)=f(1)=0$ of these localised ground states in the joint limit $L\to\infty$ and $B\to0$. We obtain to leading order logarithmically enhanced area-laws depending on the size of $LB$. Roughly speaking, if $1/B$ tends to infinity faster than $L$, then we obtain the known enhanced area-law (by the Widom--Sobolev formula) of the form $L \ln(L) a(f,\mu) |\partial\Lambda|$ as $L\to\infty$ for the (two-dimensional) Laplacian with Fermi projection $1(H_0\le \mu)$. On the other hand, if $L$ tends to infinity faster than $1/B$, then we get an area law with an $L \ln(\mu/B) a(f,\mu) |\partial\Lambda|$ asymptotic expansion as $B\to0$. The numerical coefficient $a(f,\mu)$ in both cases is the same and depends solely on the function $f$ and on $\mu$. The asymptotic result in the latter case is based upon the recent joint work of Leschke, Sobolev and the second named author for fixed $B$, a proof of the sine-kernel asymptotics on a global scale, and on the enhanced area-law in dimension one by Landau and Widom. In the special but important case of a quadratic function $f$ we are able to cover the full range of parameters $B$ and $L$. In general, we have a smaller region of parameters $(B,L)$ where we can prove the two-scale asymptotic expansion $\mathrm{tr} f(P_B(L\Lambda))$ as $L\to\infty$ and $B\to0$.