Hubbard Model for Asymmetric Ultracold Fermionic ... - KOMET 337

2.4. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS 17grand potential per site follows as:ω = Ω N= − 1 N ln Z gk{= −U |∆| 2 + n2 (1 − m 2 )}+ 1 ∑{}E4 N k− sign(E k)√Ek 2 + U2 |∆| 2k− 1 ∑ln ( 1 + exp(−β E kσ ) )NβTD−−→limkσ{−U |∆| 2 + n2 (1 − m 2 )}∫4∞+ dεν d (ε){E(ε) − sign ( E(ε) )√ }E 2 (ε) + U 2 |∆| 2− 1 β−∞∑σ∫ ∞−∞(dεν d (ε) ln 1 + exp ( − β E σ (ε) )) . (2.62)At T = 0 (β → ∞) the last term in (2.62) reduces to:− 1 β∑σ∫ ∞−∞(dεν d (ε) ln 1 + exp ( − β E σ (ε) )) → ∑ σ∫ ∞−∞dεν d (ε) E σ (ε)Θ ( − E σ (ε) ) , (2.63)where Θ(x) is the unit step function. With (2.62) we are now able to decide numericallywhich solution has lowest grand potential.