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(LLambda ), as the dimensionless scaling parameter L>0 tends to infinity, has the form Lsqrt{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. As opposed to the case B=0, the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.