Download The one-dimensional Hubbard model by Fabian H. L. Essler, Holger Frahm, Frank Göhmann, Andreas PDF

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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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 [377]. 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 [377].

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 [172].

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