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

54 CHAPTER 5. STRONG COUPLING LIMITand the Pauli matrices:( ) 0 1σ 1 =1 0; σ 2 =( 0 i−i 0); σ 3 =( 1 00 −1), (5.56)so that we can write the single-site spin-operators ( = 1) as:S i,ν = 1 2 d† i · σ ν · d i; ν ∈ {1,2,3} , (5.57)where ν denotes the spatial component of S i . With the help of these tools we can rewrite oursecond order contributions as:H t2,2 = (t 2 ↑ + t2 ↓{S )∑ i,3 S j,3 − 1 }4 (d† i · d i )(d† j · d j ) (5.58)(ij)H t2,6∑}= 2t ↑ t ↓{S i,1 S j,1 + S i,2 S j,2(ij)(5.59)by using that spin-flip terms may be expressed by linear combinations of S i,1 and S i,2 :where S i,± are the spin-raising/lowering operators.S i,± = S i,1 ± iS i,2 , (5.60)The term (d † i · d i )(d† j · d j) can be neglegted, since at one fermion per site it is a constantand hence physically inactive. In hypercubic lattices it reduces to:∑(d † i · d i )(d† j · d j) = 2dN , (5.61)(ij)where d is the dimensionality and N is the total number of lattice sites. Finally we are ableto write the effective Hamiltonian as:H = H U + 1 U{H t2,2 + H t2,6}= 4t ↑t ↓U∑〈ij〉S i · S j + 2(t ↑ − t ↓ ) 2U∑S i3 S j3 , (5.62)since H U gives no contribution because of the absence of doubly occupied states. Note thatin this notation each bond contributes only once in contrast to the“(ij)”-notation. As we see,we have derived a antiferromagnetic H XXZ model with the constraint J Z ≥ J X . The model isable to reproduce the correct isotropic Hubbard model limit (t ↑ = t ↓ ) and the Falicov-Kimballlimit (t ↑ = 0, t ↓ > 0 or t ↑ > 0, t ↓ = 0). In the first case we obtain an isotropic Heisenbergmodel:H Hub → 4t2 ∑S i · S j (5.63)Uand in the second case we obtain an Ising model:H Fal → 2t2U〈ij〉〈ij〉∑S i,3 S j,3 . (5.64)In one dimension these effective models correspond to the results presented in [35].〈ij〉