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

52 CHAPTER 5. STRONG COUPLING LIMIT5.3 Treatment of pure nearest neighbor hoppingIn this section we treat a simplified model with the special choice of t ijσ :{tσ if i,j nearest neighborst ijσ =0 else .(5.42)This will reduce our second order contribution to:H t2,1 = ∑t 2 σ¯n i+α,−σ (1 − ¯n i,−σ )¯n i+β,−σ d† i+α,σ d i+β,σ(5.43)H t2,2H t2,3H t2,4H t2,5H t2,6iαβσ= − ∑t 2 σ¯n i,−σ(1 − ¯n i+α,−σ)(1 − ¯n i+β,−σ)d † i+α,σ d i+β,σ(5.44)iαβσ∑= t ↑ t ↓ (1 − ¯n i+α,−σ)¯n i+β,σd † i,σ d i+α,σ d† i,−σ d i+β,−σ(5.45)iαβσ= t ↑ t ↓∑iαβσ= −t ↑ t ↓∑iαβσ¯n i+α,−σd † i+α,σ d i,σ (1 − ¯n i+β,σ )d† i+β,−σ d i,−σ(5.46)¯n i+α,−σ¯n i+β,σd † i,σ d i+α,σ d† i+β,−σ d i,−σ(5.47)∑= −t ↑ t ↓ (1 − ¯n i+α,−σ)d † i+α,σ d i,σ (1 − ¯n i+β,σ )d† i,−σ d i+β,−σ. (5.48)iαβσIn this notation α and β are summed over all lattice vectors with a length of one, so thatthey connect the lattice site i to its nearest neighbor. We have used that t σ t −σ = t ↑ t ↓ forboth σ =↑ and for σ =↓.5.4 Low temperature limitWe will show that under the condition (5.42) the model (5.6) can be mapped onto a H XXZmodel for U ≫ max{t σ } and can be mapped onto a hard-core-boson model with nearestneighbor interaction for U ≪ −max{t σ } , by restricting consideration to low temperatures:k B T ≪ |U| . (5.49)With the restriction (5.49) we can make the assumption that only states with the lowestenergy in 0 th order are occupied. This means that for a fixed number of particles in theU ≫ max{t σ } case the number of double occupancies is minimal and in the U ≪ −max{t σ }case it is maximal. We will first treat the repulsive U model and then we will treat theattractive U model. In the special cases we analyze, all contributions either vanish or can bedrastically simplified. We will focus on the repulsive U model at half filling and the attractiveU model with an even number of fermions.5.4.1 Repulsive U model at half fillingWe will restrict consideration to half filling in the repulsive case. In this case, if we minimizethe number of double occupancies, in 0 th order we have a highly degenerate ground state.Each site is singly occupied, there is no doubly occupied or empty site left. The system iscompletely degenerate with respect to the spin configuration. We will show that second orderperturbation theory will lead to an effective H XXZ model.