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R-matrices for integrable SU(2) × U(1)-symmetric spin-orbital chains

P. N. Bibikov

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The Yang-Baxter equation for a SU(2) × U(1)-symmetric S = 1 2 spin-orbital chain was solved using the special computer algorithm developed by the author. The 7 new R-matrices separated on 4 groups are presented. Among the obtained integrable models there are special cases related to 1D ferromagnet TDAE − C60, 1D superconductors AC60 (A=K, Cs, Rb), the quarter filled ladder compound NaV2O5 and the model of correlated electrons on a chain of Berry phase molecules. 1

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R-matrices for integrableSU(2) ×U(1)-symmetric spin-orbital chains

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The Yang–Baxter equation for an SU (2) × U (1)-symmetric spin-orbital chain was solved using the special computer algorithm developed by the author. The seven new R -matrices separated into four groups are presented. Among the obtained integrable models there are special cases related to 1D ferromagnet TDAE-C 60 , 1D superconductors AC 60 (A = K, Cs, Rb), the quarter-filled ladder compound NaV 2 O 5 and the model of correlated electrons on a chain of Berry phase molecules.

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The Yang-Baxter equation for a SU(2) × U(1)-symmetric S = 1 2 spin-orbital chain was solved using the special computer algorithm developed by the author. The 7 new R-matrices separated on 4 groups are presented. Among the obtained integrable models there are special cases related to 1D ferromagnet TDAE − C60, 1D superconductors AC60 (A=K, Cs, Rb), the quarter filled ladder compound NaV2O5 and the model of correlated electrons on a chain of Berry phase molecules. 1
Abstract In this paper we find new integrable one-dimensional lattice models of electrons. We describe all such nearest-neighbour integrable models with s u ( 2 ) × s u ( 2 ) symmetry by classifying solutions of the Yang–Baxter equation following the procedure first introduced in [1]. We find 12 R-matrices of difference form, some of which can be related to known models such as the XXX spin chain and the free Hubbard model, and some are new models. In addition, integrable generalizations of the Hubbard model are found by keeping the kinetic term of the Hamiltonian and adding all terms which preserve fermion number. We find that most of the new models cannot be diagonalized using the standard nested Bethe Ansatz.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract We classify all fundamental integrable spin chains with two-dimensional (2D) local Hilbert space which have regular R -matrices of difference form. This means that the R -matrix underlying the integrable structures is of the form and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, eight of which correspond to well-known eight or lower vertex models. The remaining six models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R -matrices, we find a bijection between solutions of the Yang–Baxter equation (YBE) and the graded YBE which extends our results to the graded 2D case.
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable statistical mechanics and their avatars are also based on YBE. Therefore, new solutions of the YBE could be used to construct new interesting 1D quantum or 2D classical systems with many other far-reaching applications. In this work, we attempt to find (almost) exhaustive set of solutions for the YBE in the lowest dimensions corresponding to a two-qubit case. We develop an algorithm, which can potentially be used for generating new higher-dimensional solutions of the YBE.
Exactly solvable models have played a key role in our understanding of low-dimensional statistical systems and condensed matter physics. Underlying these models is the famous Yang-Baxter equation, and the authors present here a large class of new integrable quantum many-body models following a rational solution to this equation. Moreover, a duality symmetry connecting different integrable models is uncovered. This is illustrated using an interacting topological system, where the topological phase transition is related to the modular subgroup of the duality symmetry.