Asymptotic growth of the local ground-state entropy of the ideal Fermi gas in a constant magnetic field

Abstract

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane ${\mathbb{R}}^2$ perpendicular to an external constant magnetic field of strength $B>0$. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $\mu \ge B$ (in suitable physical units). For this (pure) state we define its local entropy $S(\Lambda )$ associated with a bounded (sub)region $\Lambda \subset {\mathbb{R}}^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $\Lambda$ of finite area $|\Lambda |$. In this setting we prove that the leading asymptotic growth of $S(L\Lambda )$, as the dimensionless scaling parameter $L>0$ tends to infinity, has the form $L\sqrt{B}|\partial \Lambda |$ up to a precisely given (positive multiplicative) coefficient which is independent of $\Lambda$ and dependent on $B$ and $\mu$ only through the integer part of $(\mu /B-1)/2$. Here we have assumed the boundary curve $\partial \Lambda$ of $\Lambda$ to be sufficiently smooth which, in particular, ensures that its arc length $|\partial \Lambda |$ is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln (L)$ is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space ${\text{L}}^2({\mathbb{R}}^2)$ to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies.

References in corpus (2)

• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement entropy of integer Quantum Hall states in polygonal domains

Cited by in corpus (7)

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• Stability of the enhanced area law of the entanglement entropy
• Stability of a Szegő-type asymptotics
• The Widom-Sobolev formula for discontinuous matrix-valued symbols
• Enhanced area law in the Widom-Sobolev formula for the free Dirac operator in arbitrary dimension
• An enhanced term in the Szegő-type asymptotics for the free massless Dirac operator