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

48 CHAPTER 5. STRONG COUPLING LIMIT⎛⎜⎝1000⎞⎛⎟⎠ = |1 ↑,1 ↓〉;⎜⎝0100⎞⎛⎟⎠ = |1 ↑,2 ↓〉;The eigenvalues of H can easily be calculated:√E 3/4 = U 2 ±⎜⎝0010⎞⎛⎟⎠ = |2 ↑,1 ↓〉;√2U+ (t ↑ − t ↓ )42 ; E 5/6 = U 2 ±These eigenvalues may now be analyzed at strong coupling.5.1.2 “Band structure” at strong coupling⎜⎝0001⎞⎟⎠ = |2 ↑,2 ↓〉 .(5.4)2U+ (t ↑ + t ↓ )42 . (5.5)In Figures 5.1 and 5.2 we demonstrate the energy levels at strong coupling graphically (parameters:U = ±15; t ↑ = 3; t ↓ = 1.5; half filling):Energy levels a.u.1512.5107.552.50Figure 5.1: E 1−6 atEnergy levels a.u.-2.5-7.5-10-12.5-15Umax{t ≫ 1 σ}0-5Figure 5.2: E 1−6 atUmax{t σ} ≪ −1Obviously the energy levels are close to the values 0 and U. A Taylor expansion ( of E)1−6with respect to the hopping amplitude shows that each level behaves like 0 + O t 2 σUor( )U + O t 2 σU. From now on, at strong coupling, we call levels close to an integer n multipleof U, levels with effective double occupancy n. Note that effective double occupancy differsfrom real double occupancy in the language of the c operators, since real double occupancyis not a good quantum number at t ↑ or t ↓ different from 0 in this language. Levels withn = 0, for example, may have 〈n i↑n i↓〉 ≠ 0. In the next section we will perform a unitarytransformation, which transforms the c operators into d operators, so that effective doubleoccupancy in the c language is mapped onto real double occupancy in the new d language.5.2 General procedureIn this section we treat the general modelH = U ∑ in i↑n i↓− ∑ ijσt ijσc † iσ c jσ; t ijσ≥ 0 , (5.6)which we rewrite as H(c) = H U (c)+H t (c) in the strong coupling limit (U/max{t ijσ } → ±∞).We perform an unitary transformationc † iσ = exp[S(d)]d† iσexp[−S(d)] , (5.7)