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

4.2. SUPERFLUIDITY AWAY FROM HALF FILLING 37phase is energetically favored. The grand potential of the CDW phase is given as:{ω CDW = −U s 2 + 1 }− ∑ ∫ ∞dεν d (ε) √ 4t42 σ ε2 + U 2 s 2 , (4.6)σ 0where 0 ≤ s ≤ 1 2is the staggered density parameter (see Table 4.1). The self-consistencyequation for s is given as:1 = |U|2∑σ∫ ∞0dεν d (ε)and the self-consistency equation for the superfluid phase writes:1 = |U|∫ ∞0dεν d (ε)1√4t2 σ ε 2 + U 2 s 2 (4.7)1√(t↑ + t ↓ ) 2 ε 2 + U 2 ∆ 2 . (4.8)If we switch on the asymmetry around the symmetric case by replacing:t σ → t + σλ , λ ≪ t , (4.9)the right hand sides of the self-consistency equations (4.7) and (4.8) do not depend on λin first order, so that neither the staggered density parameter s nor the superfluid orderparameter ∆ change their values, implying s = ∆+O(λ 2 ). The difference between the grandpotentials of the phases at s = ∆ has a definite sign:ω Sup − ω CDW > 0 , (s = ∆, T = 0) , (4.10)so that at small hopping asymmetry the CDW phase is preferred at half filling. But from theliterature [29] it is well known that, in the symmetric hopping model at repulsive interaction,the staggered magnetization in x- or y-direction is energetically lower in a magnetic field.This implies (after a special particle-hole transformation) the thermodynamic stability ofthe superfluid phase in the attractive model away from half filling (see Table 4.1). In thisthesis we analyze the asymmetric hopping model away from half filling, hence we assumethat superfluidity occurs. Furthermore, in recent developments both experimentalists andtheoreticians investigate only superfluid phases, while CDW phases seem to be experimentallyirrelevant. (See for example: [9], [30], [11], [31], [32], [10] and [33]). This leads us to restrictconsideration to superfluid phases in this thesis, while possible CDW phases are ignored.4.2 Superfluidity away from half fillingWe will now discuss our model (2.1) at B = 0. We derived the Hartree-Fock self-consistencyequations in chapter 2. As we have seen in section 2.4.3 we cannot expect the system to benon-magnetized away from half filling even if B = 0.4.2.1 The quasiparticle energiesThe quasiparticle energies (B = 0) are for asymmetric hopping amplitudes defined as:E σ (ε) =E(ε) =−Unmσ − (t σ − t −σ )ε + sign ( E(ε) ) √E22 (ε) + (U∆) 2 , (4.11)Un2 − µ − (t ↑ + t ↓ )ε .