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, \(\textrm{tr} f(P_B(L\Lambda ))\), for polynomials f with \(f(0)=f(1)=0\) of these localised ground states in the joint limit \(L\rightarrow \infty \) and \(B\rightarrow 0\). 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 \(L\rightarrow \infty \) 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 \(B\rightarrow 0\). 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 [7] 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 \(\textrm{tr} f(P_B(L\Lambda ))\) as \(L\rightarrow \infty \) and \(B\rightarrow 0\).