Families of commuting transfer matrices and integrable models with disorder

Jacques H. H. Perk, Cherie L. Schultz

B. Sriram Shastry

The 1D Hubbard model is shown to be an exactly integrable system. A "covering" model of 2D statistical mechanics which I proposed recently was shown to provide a one-parameter family of transfer matrices, commuting with the Hamiltonian of the Hubbard model. I show in this work that any two transfer matrices of a family commute mutually. At the root of the commutation relation is the ubiquitous Yang-Baxter factorization condition. The form of the $R$ operator is displayed explicitly.

С. М. Сергеев

Francisco Castilho Alcaraz, R. Z. Bariev

Vladimir V. Mangazeev, Yu. G. Stroganov

As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of the corresponding three-dimensional lattice models. We present the modified version of these equations which give the commuting family of more complicated two-layer transfer-matrices. In the static limit we have succeeded in constructing the solution of these equations in terms of elliptic functions.

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.