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

46 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGT C,∆ , T C,s (a.u.)0.50.40.30.20.10.2 0.4 0.6 0.8 1t ↓Figure 4.9: Critical temperatures T C,∆ for the superfluid order parameter and T C,s for thestaggered density order parameter in dependence of t ↓ at fixed t ↑ + t ↓ = 2 and U = −3.In Figures 4.8 and 4.9 we tune the hopping parameters between the Falicov-Kimball limit andthe isotropic Hubbard limit at fixed U and t ↑ +t ↓ . As we can see, while the critical temperaturefor the superfluid phase T C,∆ increases with the symmetry of the hopping amplitudes, forthe CDW phase the critical temperature T C,s is higher for asymmetric hopping amplitudes.Obviously superfluidity favors a spin-symmetric Hamiltonian, while the CDW phase favorsasymmetry. The critical temperatures are quasi-identical if the asymmetry is small: t ↑ ≈ t ↓ .This is an additional argument for neglegting the CDW phase away from half filling, sincethe difference of the critical temperatures is a measure for the energetical difference of thecompeting phases, and we know that the CDW phase becomes thermodynamically unstableaway from half filling in the symmetric model t ↑ = t ↓ [29], where the critical temperatures areidentical. Interestingly the critical temperature T C,s does not have its maximum at t ↓ = 0:T C,s (a.u.)0.860.850.840.830.820.810.80.2 0.4 0.6 0.8 1t ↓Figure 4.10: Critical temperature T C,s for the staggered density order parameter in dependenceof t ↓ at fixed t ↑ + t ↓ = 2 and U = −4.5. We have chosen a large value of |U| in orderto improve the visibility of the maximum of T C,s .The critical temperature of the superfluid phase at half filling can be seen as an upperbound for the critical temperature away from half filling, since at half filling superfluidity ismaximal. In the isotropic Hubbard model limit it can also be seen as an upper bound forthe critical temperature of a system with a magnetic field, since a magnetic field also tendsto suppress superfluidity.