Yang-Baxter and the Boost: splitting the difference

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 nondifference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16 × 16 solutions which exhibit su(2) ⊕ su(2) symmetry, which include the one-dimensional Hubbard model and the S-matrix of the AdS 5 × S 5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

Integrable spin chains and cellular automata with medium range interaction

“…Based on this result it is possible to perform a classification of integrable spin chains with 3-site interactions. This is rather analogous to the ideas used in [20,21] and later in [28][29][30][31][32]. The idea is to make an Ansatz for h j,j+1,j+2 and hj,j+1,j+2 , possibly including a number of free parameters, to construct Q 5 using (II.10) and to check the commutation relation [H, Q 5 ] = 0 on spin chains with medium length.…”

“…Conjecture 2 was used in the recent works [28][29][30][31][32] for the classification of integrable nearest neighbor chains in various circumstances. We will not review this classification here, instead we will focus on the generalization of the Reshetikhin condition to medium range spin chains.…”

“…chains, therefore we just mention a few works that performed classifications with certain restrictions [22][23][24][25][26][27]. More recently a systematic method was worked out in [28][29][30][31][32], based on well established ideas [33] but leading to more detailed classifications than in prior works. Despite all of this progress there is a class of models which has so far received relatively little attention: Translationally invariant spin chains where the Hamiltonian has a finite interaction range ≥ 3.…”

“…) (u, 0) = X A,1 (u), (D.4) R A,(31) (u, 0)L A,3 (u) −1 = X A,1 (u). (D.5)The second equation can be written asR (u, 0) = X A,2 L A,1 (u).…”

Introduction to classical and quantum integrability

“…This method was successfully applied to construct models that are interesting both in condensed matter and in high energy physics. For example, the method allowed for a classification of all integrable deformations of AdS 3 and AdS 2 R-matrices [57,87,88]. It also provided a systematic way to construct integrable Lindblad superoperators [60].…”