Abstract
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
Highlights
The importance of 2D integrable models [1,2,3,4,5] in modern physics is hard to overestimate
The lack of solutions of tetrahedron equations giving rise to models with finite degrees of freedom at the sites, which one expects in any realistic experimentally relevant situation, rises an immediate question: are there criteria sufficient for integrability of 3D models, that have finite degrees of freedom? This is the precise question we address in this letter
Since the model has as much integrals of motion as degrees of freedom, one expects existence of appropriate integrability equations, satisfied by the R-matrix of Kitaev model. Solutions of this integrability equations will lead to the construction of the new type of 3D integrable models, which are essentially different from the Kitaev model
Summary
The importance of 2D integrable models [1,2,3,4,5] in modern physics is hard to overestimate. The basic constituent of 2D integrable systems is the commutativity of the evolution operators, the transfer matrices of the models of different spectral parameters. In a Bethe Ansatz formulation of 3D models their 2D transfer matrices of the quantum states on a plane [8,14,17] can be constructed via three particle R-matrix [9,14,21], which, as an operator, acts on a tensorial cube of linear space V , i.e. R : V ⊗ V ⊗ V → V ⊗ V ⊗ V [11].
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