Rk 1 [U 2M 4diak Qk 1jg2k C 22I11I222 VI11I22

with g(k) = — U + The spectral functions a(n)(k) have the following expressions

122 2

The determination of GR(k,rn) and C(k,rn) requires the knowledge of the chemical potential and of the two bosonic thermal averages A and p, a fermionic correlator and a bosonic one, respectively. These quantities are self-consistently determined by means of the following set of coupled equations n = 2(1 - Cu - C22)

where Cab = {^a(i)^l(i)) and C"b = {f"(i)fl(i)}. The first equation fixes the chemical potential in terms of all other parameters; the second comes from the definition of A [cf. (4.37)]; the third comes from the constraint (4.28). Once this set of coupled self-consistent equations has been solved, we can calculate all relevant single-particle and thermodynamical properties of the model. The double occupancy per site: D = l/N J2i{n"(i)n#(i)) = -22 — C22; the energy bands En(k); the Fermi surface by means of the equation En(k) = 0; the momentum distribution function n(k) and the density of states N(rn):

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