Logarithmically Enhanced Area-Laws for Fermions in Vanishing Magnetic Fields in Dimension Two

Abstract

We consider fermionic ground states of the Landau Hamiltonian, $$H_B$$ H B , in a constant magnetic field of strength $$B>0$$ B > 0 in $${\mathbb {R}}^2$$ R 2 at some fixed Fermi energy $$\mu >0$$ μ > 0 , described by the Fermi projection $$P_B:=1(H_B\le \mu )$$ P B : = 1 ( H B ≤ μ ) . For some fixed bounded domain $$\Lambda \subset {\mathbb {R}}^2$$ Λ ⊂ R 2 with boundary set $$\partial \Lambda $$ ∂ Λ and an $$L>0$$ L > 0 we restrict these ground states spatially to the scaled domain $$L \Lambda $$ L Λ and denote the corresponding localised Fermi projection by $$P_B(L\Lambda )$$ P B ( L Λ ) . Then we study the scaling of the Hilbert-space trace, $$\textrm{tr} f(P_B(L\Lambda ))$$ tr f ( P B ( L Λ ) ) , for polynomials f with $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 of these localised ground states in the joint limit $$L\rightarrow \infty $$ L → ∞ and $$B\rightarrow 0$$ B → 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 $$L \ln (L) a(f,\mu ) |\partial \Lambda |$$ L ln ( L ) a ( f , μ ) | ∂ Λ | as $$L\rightarrow \infty $$ L → ∞ for the (two-dimensional) Laplacian with Fermi projection $$1(H_0\le \mu )$$ 1 ( H 0 ≤ μ ) . On the other hand, if L tends to infinity faster than 1/B, then we get an area law with an $$L \ln (\mu /B) a(f,\mu ) |\partial \Lambda |$$ L ln ( μ / B ) a ( f , μ ) | ∂ Λ | asymptotic expansion as $$B\rightarrow 0$$ B → 0 . The numerical coefficient $$a(f,\mu )$$ a ( f , μ ) in both cases is the same and depends solely on the function f and on $$\mu $$ μ . 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 $$\textrm{tr} f(P_B(L\Lambda ))$$ tr f ( P B ( L Λ ) ) as $$L\rightarrow \infty $$ L → ∞ and $$B\rightarrow 0$$ B → 0 .