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

50 CHAPTER 5. STRONG COUPLING LIMITBy expanding (5.11) we obtain the coefficients:H U (·) = H U (·) (5.17)H t,1 = [S 1 (d),H U (d)] + H t (d) (5.18)H t,2 = [S 1 (d),[S 1 (d),H U (d)]] + [S 1 (d),H t (d)] + [S 2 (d),H U (d)] . (5.19)We must now arrange S 1 and S 2 such that they satisfy:S † 1/2 = −S 1/2 (5.20)in order to fulfill (5.8). We also have to satisfy (5.14) so that the number of double occupanciesis preserved by the first and second order contributions of H t :[H t,1/2 ,H U ] = 0 . (5.21)Therefore it is useful to decompose the original hopping term as follows:where H +/0/− are given by:H t (d) = −λ[H + + H 0 + H − ] , (5.22)H += ∑ ijσt ijσ¯n i−σ(1 − ¯n j−σ)d † iσ d jσ(5.23)H 0= ∑ ijσt ijσ(1 − ¯n j−σ− ¯n i−σ+ 2¯n i−σ¯n j−σ)d † iσ d jσ(5.24)H −= ∑ ijσt ijσ¯n j−σ(1 − ¯n i−σ)d † iσ d jσ. (5.25)Here H + (H − ) is the part of the hopping term that increases (decreases) the number of doubleoccupancies and H 0 is the part that leaves the number of double occupancies unchanged. Thisis expressed more formally by the commutation relation:In addition they fulfill:[H U (d),H p ] = UpH p p ∈ {+,0, −} . (5.26)H † 0 = H 0 ; H † + = H − ; H † − = H + . (5.27)5.2.2 Choice of the transformation operatorsWe will now choose the transformation operators S 1/2 according to (5.14) in second order ofλ and according to (5.20). Equation (5.14) is obviously satisfied in first order of λ by thechoice:S 1 (d) = 1 U (H + − H − ) . (5.28)With this choice we obtain the first order correction to H U :H t,1 = H 0 . (5.29)This means that first order perturbation theory at strong coupling can be performed byneglecting all terms in (5.6) which change the number of double occupancies. In first order