Fermi Gas with 4-Fermion BCS-type Interaction

A gas of spin 1/2 fermions with an interaction V + W = - Σ$\text{}_{k,k'}$g$\text{}_{kk'}$b*$\text{}_{k}$ b*$\text{}_{-k}$b$\text{}_{k'}$b$\text{}_{-k'}$ + Σ$\text{}_{k}$ γ$\text{}_{k}$b*$\text{}_{k}$b$\text{}_{k}$, where b$\text{}_{k}$=a$\text{}_{k+}$a$\text{}_{k-}$ and a$\text{}_{kσ}$,a*$\text{}_{k'σ'}$ satisfy Fermi anticommutation relations, is investigated. The trial ground state |G〉 is similar in form to the BCS ground state, with b*$\text{}_{k}$ b*$\text{}_{-k}$ replacing a*$\text{}_{k+}$ a*$\text{}_{-k-}$, but because the excitation energies are not simply additive, the trial density matrix \rho_0 differs from the BCS one. The expectation values 〈G|H|G〉 and Tr(Hρ$\text{}_{0}$) are minimized, revealing the presence of a second-order phase transition, with T$\text{}_{c}$ > T$\text{}_{c(BCS)}$ for appropriately adjusted γ$\text{}_{k}$. It is shown that the minimization procedure applied leads to an expression for the free energy density of H, which is asymptotically exact in the infinite-volume limit. Comparison with experimental data on high-temperature superconductors is made and for a particular choice of \gamma_k qualitative agreement is found with the temperature dependence of the order parameter of the BSCCO superconductor.