Integrability of three dimensional models: cubic equations

TL;DR

This paper generalizes integrability concepts from 2D to 3D models using algebraic Bethe Ansatz, establishing conditions for transfer matrix commutativity via cubic equations, and includes the Kitaev model as an example.

Contribution

It introduces a framework for 3D integrable models based on cubic equations and extends the algebraic Bethe Ansatz to higher dimensions.

Findings

Derived sufficient conditions for transfer matrix commutativity in 3D models.

Established the role of R-matrices in 3D integrability.

Showed that the Kitaev model fits within this integrable framework.

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.