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

3.2. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS AT T = 0 23Since δn is small, both E 1 (ε) and E 2 (ε) have the same zeroes as E 1σ (ε) and E 2σ (ε). Hencewe can do a Taylor-expansion in powers of δn, which leads to:∫ ∞|In 1σ − In 2σ | = 1 |U 3 ∆ 2 |dε ν d (ε)4 −∞ } {{ } (E1 2(ε) + ≤ν d (ε U2 ∆ 2 ) 3 2max)[≤|U|4t ν E 1 (ε)d(ε max ) ||δn|| ∞≤|U|2t ν d(ε max ) ||δn|| ∞ ,√E21 (ε) + U 2 ∆ 2 ] d−d|δn ↑ + δn ↓ |} {{ }≤ 2||δn|| ∞(3.13)since E(ε) = U(n ↑+n ↓ )2− µ − 2tε and ν d (ε) = 0 for |ε| ≥ d. Note that this procedure worksonly for d ≥ 3, since in one or two dimensions the interaction-free DOS diverges at thevan-Hove-singularities. As (3.13) is valid for both spin species, we have:so as we need q < 1, we must have||In 2 − In 1 || ∞ ≤ |U|2t ν d(ε max ) ||n 2 − n 1 || ∞} {{ }, (3.14):=q|U|t