Yang-Baxter Algebras and Integrable Models in Field Theory and Statistical Mechanics

Abstract

The investigation of two-dimensional classical and quantum theories has shown in recent years that the Yang-Baxter equations and the Yang-Baxter algebras are the basic concepts of integrability. In statistical models and field theories the commutativity of transfer matrices t(λ) at different values of the spectral parameter follows directly from the Yang-Baxter algebra. The expansion in powers of λ of log t(λ) (or t(λ)) provides an infinite number of commuting operators including the Hamiltonian. So, we can say that the theory is integrable since there are as many commuting operators as degrees of freedom (infinity). More precisely, one associates in many theories a local transition matrix L_n(λ) and the monodromy operator $T\left( \lambda \right) = {\overleftarrow \Pi _n}{L_n}\left( \lambda \right)$, the trace of which is the transfer matrix t(λ). In an integrable theory T(λ) verifies the Yang-Baxter algebra.