Kohn proved in 1961 that interactions between electrons did not change the de Haas–van Alphen (dHvA) oscillation frequency for single electrons in the nondegenerate ground-state [Phys. Rev.123(4), 1242 (1961)]. It was proved recently that the pure-state Wigner function for an electron in a magnetic field carries this quantum and physical oscillation, and a quantum dielectric function, so the conductance can be calculated from the Wigner function [Int. J. Mod. Phys. B17(25), 4555 (2003)], [Int. j. Mod. Phys. B17(26), 4683 (2003)]. We present the first complete proof that at a finite temperature, the mixed-state Wigner function also shows dHvA oscillations with the same frequency. The Wigner function is a fundamental quantity, the fact that it carries observable physical information shows a great potential in the design of new quantum materials at the nanoscale. The definition of the mixed-state Wigner function involves a grand canonical partition function (GCPF). Although dHvA is a well-known phenomenon, we present the first complete proof of it happening in degenerate mixed-states, based on a GCPF, which requires reconciliation between the dHvA experimental condition of a fixed number of particles and the GCPF's sum over number of particles. The GCPF is applied to one of the two spin species, while both the spin and spin-magnetic moment interaction are considered. We show that the contour integration in ω(ε) leads to a non-oscillatory term that is much larger than an oscillatory term, in the dHvA experimental conditions of high fields and low temperatures. This dominance of the non-oscillatory term explains the constancy of the chemical potential, allowing it to reduce to the Fermi energy in the limit of zero temperature. The obtained mixed-state Wigner function shows a fundamental period of oscillation with respect to B-1 that reduces to the Onsager's period for dHvA oscillations. This indicates that in mixed-states, dHvA oscillations depend on electrons of one spin species, this means the population of electrons of each spin species oscillates with the magnetic field. The temperature dependence in the Wigner function will allow a combination of phase-space and thermodynamics information for mesoscopic structures, and the study of phase-space density holes such as BGK modes in the quantum domain.
Harmonically trapped one-dimensional Fermi gas using the static fluctuation approximation
