Integrable Kondo impurities in one-dimensional extended Hubbard models

Abstract

Three kinds of integrable Kondo problems in one-dimensional extended Hubbard models are studied by means of the boundary graded quantum inverse scattering method. The boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras acting in a (2s ␣ ϩ1)-dimensional impurity Hilbert space. Furthermore, these models are solved using the algebraic Bethe ansatz method, and the Bethe ansatz equations are obtained.

Results on the symmetries of integrable fermionic models on chains

“…We proposed an example of concrete application of the described techniques to two of the 32 models, for the ground state of which we have explicitly given the eigenvalue, the eigenvector and the pair correlation functions in the different region of the phase diagram ( fig.2). We also point out that all the extended Hubbard models investigated above can also be used as 'bulk systems' to which one can add appropriate boundary interaction terms; this would lead to new results in the context of models with Kondo impurities (see for instance [44]). The X-W plane to describe the 2-parameter class of gl(2, 1)-invariant models.…”

Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices

“…6 A proof for 6 A similar observation (although without proof and only for the case M = 2) has been made for related models in [19,20,21].…”

“…It is from this object that we must identify suitable operators (among them, creation-like operators). In view of the form (3.1) of the left Kmatrix, we follow [19,20,21] (see also [10,34,35,36] and references therein) and write T − a1···LY (u) as follows (as an N × N matrix in the auxiliary space) where…”

“…This approach has been adapted to open spin chains with diagonal K-matrices in [34,35,36]. We further adapt this method, along the lines in [19,20,21] for a related model with M = 2, to solve the GL(N)/(GL(M) × GL(N − M)) model for general values of N and M. The identity (4.3), which relies on a certain factorization property of the "reduced" K-matrices into products of R-matrices, plays an essential role in obtaining a solution. Numerical evidence suggests that the solution is complete.…”

Generalizedq-Onsager algebras and dynamicalK-matrices

“…Among the properties of this model, it is important to notice that for real coupling the boundary amplitudes are self-dual [BK], as for the bulk case [BCorDS], contrary to the case with scalar boundary conditions [G, Cor]. There are also examples of dynamical boundary conditions in quantum integrable spin chains, where the dynamical K-matrices are obtained as 'dressing' of scalar (c-number) boundary matrices [ZGLG1,ZGLG2,FraSlav].…”