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

2.4. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS 15directly leads to a chemical µ = U 2a particle-hole transformationpotential at half-filling. It is well-known that performingc † iσ→ exp(iQ · i)c iσ, (2.56)c iσ→ exp(iQ · i)c † iσ,Q = (π,π,... ,π) T ,leads to the desired result, if the lattice is bipartite. Furthermore, the model is symmetricunder a spin exchange transformation:c † iσ → c† i−σ; c iσ→ c i−σ. (2.57)If we take the Hamiltonian (2.1) at B = 0, particle-hole symmetry still leads to µ = U 2 athalf filling. Spin-exchange symmetry is however broken, the transformation (2.57) leads to amodified hopping termH T → − ∑ t −σc † iσ c jσ, (2.58)(ij)σso that particles hop with the hopping-amplitude of the opposite spin species.The Hamiltonian (2.1) at t ↑ = t ↓ ≡ t is neither symmetric under the transformation (2.56)nor under (2.57). But if we combine both transformations at µ = U 2, we get our symmetryback:HS.E. −−→P.H.− ∑ t c † iσ c jσ + U ∑ (1 − n i↑− n i↓+ n i↑n i↓) (2.59)(ij)i− U ∑(1 − n2 iσ) − B ∑ σ(1 − n iσ)iσiσ= − ∑ t c † iσ c jσ + U ∑(ij)i= H ,so that we still have half filling at µ = U 2 .n i↑ n i↓ − µ ∑ iσn iσ + B ∑ iσσn iσIf both B ≠ 0 and t ↑ ≠ t ↓ , these arguments do not work and therefore the calculation ofthe chemical potential at half filling becomes more complicated. These symmetry relationswere shown for the exact model, but they are still valid in the Hartree-Fock approximation,where some of the operators have been replaced by averages.These symmetry relations at half filling can be used to reduce the numerical problem ofsolving the self-consistency equations by one parameter, since at µ = U 2, under the conditionsdiscussed above, (2.47) reduces exactly to n = 1 at all temperatures and needs not to besolved numerically.2.4.3 Non-polarized solutions of first typeWe will distinguish two types of non-polarized solutions (m = 0). The first type of nonpolarizedsolutions, which we will discuss in this section, arises from the choice of the hoppingamplitudes t σ , the chemical potential µ and the magnetic field B occurring in the Zeemanterm. This type of solutions is non-magnetic even at U = 0, in contrast to the second type