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Physics Letters A

New families of commuting transfer matrices in q-state vertex models

Jacques H. H. Perk, Cherie L. Schultz

Published Aug 1, 1981• 261 citations
Publisher
DOI
10.1016/0375-9601(81)90994-4

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It is found that a new two-dimensional (2D) lattice model of two coupled asymmetric six-vertex models provides a one-parameter family of transfer matrices commuting with the Hamiltonian of the 1D Hubbard model with a chemical potential term, for a special choice of the coupling constants. This new statistical model can be regarded as a four-state model in two dimensions, since the six-vertex model can be viewed as a two-state one. It is conjectured that this "covering" model satisfies the Yang-Baxter equations, i.e., it possesses a commuting family of row-to-row transfer matrices. Moreover, an alternative generating functional for an infinite number of conservation laws in the 1D Hubbard model is written down. This may be useful in understanding the completeness of the set of local conserved currents.
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.