Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field - Leschke, Hajo; Sobolev, Alexander; Spitzer, Wolfgang (Communications in Mathematical Physics, 2020)
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 mu = B (in suitable physical units). For this (pure) state we define its local entropy S(Lambda) associated with a bounded (sub)region Lambda subset of 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 vertical bar Lambda vertical bar. 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 root B vertical bar partial derivative Lambda vertical bar 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 derivative Lambda of Lambda to be sufficiently smooth which, in particular, ensures that its arc length vertical bar partial derivative Lambda vertical bar 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 L-2(R-2) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Renyi entropies. As opposed to the case B = 0, the corresponding asymptotic coefficients depend on the Renyi index in a non-trivial way.
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