Logarithmically Enhanced Area-Laws for Fermions in Vanishing Magnetic Fields in Dimension Two
Year: 2024
Type: article
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 tend
Authors: Paul Pfeiffer, Wolfgang Spitzer
Institution: FernUniversität in Hagen
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4402543115