By Fabian H. L. Essler, Holger Frahm, Frank Göhmann, Andreas Klümper, Vladimir E. Korepin
This publication provides a self-contained account of the precise answer of the Hubbard version in a single size. the outline of solids at a microscopic point is advanced, regarding the interplay of a big variety of its materials. it truly is most unlikely to resolve the corresponding many-body difficulties, even if perception could be received from research of simplified versions. a massive instance is the Hubbard version, which describes interacting electrons in slender strength bands and which has been utilized to a various diversity of difficulties.
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Additional info for The one-dimensional Hubbard model
38) (2) , since the three-site terms are neglected here. 33) we obtain. L (2) Ht−J = P0 −t † † (c j,a c j+1,a + c j+1,a c j,a ) + j=1 4t 2 U L j=1 S αj S αj+1 − n j n j+1 4 P0 . 39) (2) In this form Ht−J is most frequently encountered in the literature. 39) is not solvable by Bethe ansatz . For the special values 4t/U = 2 and 4t/U = 0 of the coupling constant, however, Bethe ansatz 1 A common short-hand notation in the literature is J = 4t 2 /U , whence the name. 44 Appendices to Chapter 2 solutions exist .
L, which implies that L G(α)TP0 = † t jk (1 − α n j,↑ n j,↓ )c j,a ck,a P0 . 29) j,k=1 Here the right hand side is linear in α. A The strong coupling limit 41 and thus L Ht−J = P0 † t jk c j,a ck,a − j,k=1 L 1 U † j,k,k ,l=1 † t jk tk l c j,a ck,a n k ,↑ n k ,↓ ck ,b cl,b P0 . 31) The second sum in this expression can be further simpliﬁed. Let us concentrate for a moment † on the sum over k . 31) vanish. The term with † k = j also vanishes. 28). 31) reduces to a triple sum over j, k, and l.
We deﬁne the operators of the components of the total spin as Sα = L 2 † c j,a (σ α )ab c j,b , 1 2 α = x, y, z . 66) j=1 a,b=1 In the second summation we identiﬁed 1 with ↑ and 2 with ↓. The matrices σ α are the Pauli matrices σx = 0 1 1 0 , σy = 0 −i i 0 , σz = 1 0 0 −1 . 68) where εαβγ is the totally antisymmetric tensor. 69) and commute with the Hubbard Hamiltonian, [H, S α ] = 0 , α = x, y, z . 70). 2) in the calculation of various commutators. As a convenient means to deal with all of these commutators simultaneously we introduce so-called current operators .