Abstract
In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times1616×16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2)𝔰𝔲(2)⊕𝔰𝔲(2) symmetry, which include the one-dimensional Hubbard model and the SS-matrix of the {AdS}_5 \times {S}^5AdS5×S5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.
Highlights
The Yang-Baxter equation (YBE) appears in many different areas of physics [2,3,4,5]
The Heisenberg spin chain and the Hubbard model [7] are just some of the famous integrable models and were important for our understanding of low-dimensional statistical and condensed matter systems and, over the last few years, exceptional progress has been made in understanding the AdS/CFT correspondence [8,9,10] due to the discovery of integrable structures [11]
By exploiting identifications we found that we could bring all of the corresponding R-matrices to a form closely resembling the XXX spin chain which we remind the reader is of the form
Summary
The Yang-Baxter equation (YBE) appears in many different areas of physics [2,3,4,5]. It signals the presence of integrability which implies the existence of higher conservation laws. A priori there is no reason for the two constructed operators 2 and 3 to commute with each other, but if we impose this it will place a number of constraints on the entries of the density H12 in the form of a system of ODEs. We solve the set of constraints and show that the resulting Hamiltonian defines an integrable system, meaning it can be obtained from a solution of the YBE. In particular the conserved charges r (θ ) become independent of θ and so the set of ODEs arising from the condition [ 2(θ ), 3(θ )] = 0 reduces to a set of coupled cubic polynomial equations This simplification was exploited in the papers [1, 40] in order to find a plethora of new integrable systems with a range of interesting physical properties. In Appendix B we provide some details on this notebook
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