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

= ∑ ijlσ= − ∑ ijlσ= ∑ ijlσ= ∑ ijlσ= − ∑ ijlσ= − ∑ ijlσ5.2. GENERAL PROCEDURE 51(resetting the bookkeeping parameter λ to 1) we get a Hubbard model with“reduced”hopping.In second order in λ we may choose S 2 (d). Equations (5.14) and (5.20) are fullfilled by:The second order correction to the Hamiltonian is given by:S 2 (d) = 1U 2[H + + H − ,H 0 ] . (5.30)H t,2 = 1 U [H +,H − ] . (5.31)We are now able to calculate (5.31) in terms of d † and d in order to analyze the structure ofsecond order perturbation theory at strong coupling.5.2.3 Structure of the second order termNow we focus attention on Hubbard models of the type (5.6) with the constraint:t iiσ = 0 ; ∀i,σ . (5.32)That means, that each site has the same local energy, which is chosen equal to zero. We thenobtain an expression for H t,2 in terms of d and d † . Therefore it is necessary to calculate thecommutation relations appearing in (5.31):The second order contribution can be written as:[d † iσ d jσ ,d† kη d lη ] , (5.33)[d † iσ d jσ , ¯n l−η (1 − ¯n k−η)] . (5.34)H t,2 = 1 U6∑H t2,i (5.35)i=1with the components:H t2,1t ijσt liσ(1 − ¯n i−σ)¯n j−σ¯n l−σd † lσ d jσ(5.36)H t2,2t ijσt jlσ(1 − ¯n i−σ)¯n j−σ(1 − ¯n l−σ)d † iσ d lσ(5.37)H t2,3t ijσt il−σ(1 − ¯n j−σ)¯n lσd † iσ d jσ d† i−σ d l−σ(5.38)H t2,4t ijσt lj−σ¯n i−σd † iσ d jσ (1 − ¯n lσ )d† l−σ d jσ(5.39)H t2,5t ij−σt jlσ¯n l−σ¯n iσd † jσ d lσ d† i−σ d j−σ(5.40)H t2,6t ij−σt liσ(1 − ¯n l−σ)d † lσ d iσ (1 − ¯n jσ )d† i−σ d j−σ. (5.41)At this point we recognize that the second order contributions can be described as pairwisehopping of fermions. The terms H t2,1 and H t2,2 describe double hopping of one fermionto a different or to the same site and the other terms describe pairwise hopping of twofermions with different pseudospin indices. None of the terms changes the number of doubleoccupancies in the d-language.