Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2 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 μ≥B (in suitable physical units). For this (pure) state we define its local entropy S(Λ) associated with a bounded (sub)region Λ⊂R2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Λ of finite area |Λ|. In this setting we prove that the leading asymptotic growth of S(LΛ), as the dimensionless scaling parameter L>0 tends to infinity, has the form LB−−√|∂Λ| up to a precisely given (positive multiplicative) coefficient which is independent of Λ and dependent on B and μ only through the integer part of (μ/B−1)/2. Here we have assumed the boundary curve ∂Λ of Λ to be sufficiently smooth which, in particular, ensures that its arc length |∂Λ| 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 L2(R2) 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