Statistical Mechanics of an Electron Gas in a Magnetic Field

The statistical mechanics of an interacting electron gas in a magnetic field is developed based on a propagator method for weak fields and low temperatures. Detailed analytical considerations are given of the eigenvalues of relevant propagators, through which the grand partition function and the correlation energy are evaluated. The energy and the susceptibility are expressed in powers of $r_s$ and the magnetic field. Corrections to the Landau susceptibility are obtained taking the finiteness of the density into consideration. In the absence of the magnetic field, the correlation energy reduces to the correct limit. The paramagnetic susceptibility and the polarization energy agree with what Brueckner and Sawada obtained by a different method in which a polarization parameter is assumed. An explicit expression for this parameter is also given. Thus, the evaluation of the correlation energy by Gell-Mann and Brueckner and that of the susceptibility by Brueckner and Sawada, Pines, and others are generalized.