Solving the star-triangle relation(Development of Soliton Theory)

Lincoln D. Carr, J. Brand

J.-M. Maillard

The study of the two-dimensional models in lattice statistics and the related exactly integrable 1+1 dimensional field theory has developed intensively over the past ten years and this evolution, this developement does not seem to decrease. There are several reasons why one can be interested in such models: the need to leave the framework of perturbation theory with the hope to describe particle spectrum and scattering, the fact that some 1+1 dimensional exactly solved models have physical applications for some important solid state theories (Solitons in magnets [61,11], Peierls-Frohlich model, Anderson localization, Kondo effects [19,37,1]),the importance of exactly solved two-dimensional models of lattice statistics to understand the critical phenomena (universality...). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

George L. Lamb

Yuan Miao, Eric Vernier

The star-triangle relation plays an important role in the realm of exactly solvable models, offering exact results for classical two-dimensional statistical mechanical models. In this article, we construct integrable quantum circuits using the star-triangle relation. Our construction relies on families of mutually commuting two-parameter transfer matrices for statistical mechanical models solved by the star-triangle relation, and differs from previously known constructions based on Yang-Baxter integrable vertex models. At special value of the spectral parameter, the transfer matrices are mapped into integrable quantum circuits, for which infinite families of local conserved charges can be derived. We demonstrate the construction by giving two examples of circuits acting on a chain of Q− state qudits: Q -state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and ZQ circuits, which are novel to our knowledge. In the first example, we present for Q=3 a connection to the Zamolodchikov-Fateev 19-vertex model.

Carlo F. Barenghi, Nick G. Parker

Barry M. McCoy

Abstract This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.