A Soluble Fermi-Gas Model. Validity of Transformations of the Bogoliubov Type

Abstract

A linear soluble Fermi‐gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is $$\mathrm{H(\rho )=-\Sigma (}\cos {\mathrm{k+\rho )c}}_k^{+}{c}_k\mathrm{+\rho \Sigma }\cos {\mathrm{q\; c}}_{\mathrm{k+q}}^{+}{c}_{\mathrm{u-q}}^{+}{c}_u{c}_k$$ . For a half‐filled band, the energies of the low‐lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree‐Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree‐Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.