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arXiv:1904.12005 · Journal of Physics A Mathematical and Theoretical

Classifying integrable spin-1/2 chains with nearest neighbour interactions

Marius de Leeuw, Anton Pribytok, Paul Ryan

Published Oct 30, 2019• 50 citations
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Abstract

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.

DOI
10.1088/1751-8121/ab529f

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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.
We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the AdS_{2} and AdS_{3} integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.
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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.
This work is based on the author's PhD thesis. The main result of the thesis is the use of the boost operator to develop a systematic method to construct new integrable spin chains with nearest-neighbour interaction and characterized by an R-matrix of non-difference form. This method has the advantage of being more feasible than directly solving the Yang-Baxter equation. We applied this approach to various contexts, in particular, in the realm of open quantum systems, we achieved the first classification of integrable Lindbladians. These operators describe the dynamics of physical systems in contact with a Markovian environment. Within this classification, we discovered a novel deformation of the Hubbard model spanning three sites of the spin chain. Additionally, we applied our method to classify models with su(2)⊕su(2)\mathfrak{su}(2)\oplus \mathfrak{su}(2)su(2)⊕su(2) symmetry and we recovered the matrix part of the S-matrix of AdS5×S5AdS_5 \times S^5AdS5​×S5 derived by requiring centrally extended su(2∣2)\mathfrak{su}(2|2)su(2∣2) symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type and we showed that the models of this class satisfy the free fermion condition. This enables us to express the transfer matrix associated to some of the models in a diagonal form, simplifying the computation of the eigenvalues and eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231, 2011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.
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