Investigation of the deformed fermi surfaces mechanism for pairing of two species of fermions with mismatched fermi surfaces

Variational method is used to investigate, at zero temperature, the deformed- Fermi-surfaces mechanism for solving the problem of superconducting pairing of two species of fermions (i.e., spin-up and -down) of mismatched Fermi surfaces due to the existence of a uniform exchange or Zeeman field. After analyzing the depairing regions in the whole three-dimensional parameter space, we obtain a trial groundstate wave-function as a function of the three variational parameters, one of which is the gap function. Then within the frame work of the weak-coupling BCS theory, the expectation value of the Hamiltonian of a conductor under an exchange or Zeeman field is derived, from which a gap equation is derived by differentiation. The influence of deformed Fermi surfaces on the chemical potential is then calculated. Computer programing is finally used to solve the gap equation, and find the minimum-energy state with respect to the remaining two variational parameters (?? and z). These two parameters are better than the original parameters used in the trial Hamiltonian when compared with the FF state. And we also found if we keep the total number of electrons fixed, the system prefers an unchanged chemical potential and the ground state energy of the deformed-Fermi-surfaces state, which is found to be an angle dependent case of Sarma's solution III, is no better than that of the unpolarized BCS state.