The transfer matrix model

Takeo Kojima

We diagonalize infinitely many commuting operators T B ( z ). We call these operators T B ( z ) the boundary transfer matrix associated with the quantum group and the elliptic quantum group. The boundary transfer matrix is related to the solvable model with a boundary. When we diagonalize the boundary transfer matrix, we can calculate the correlation functions for the solvable model with a boundary. We review the free field approach to diagonalization of the boundary transfer matrix T B ( z ) associated with U q ( A (2) 2 ) and . We construct the free field realizations of the eigenvectors of the boundary transfer matrix TB ( z ). This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ). It is thought that this diagonalization method can be extended to more general quantum group U q ( g ) and elliptic quantum group U q,p ( g ).

Bruce W. Shore

S. Rao, Somendra M. Bhattacharjee

We introduce the correlation length for the square-tiling model for the glass transition through the transfer-matrix approach developed earlier [S. M. Bhattacharjee and E. Helfand, Phys. Rev. A 36, 3332 (1987)] for this model. We show that this transfer-matrix approach, in the limit when the competing internal energy reduces the model to the hard-square gas problem with nearest- and next-nearest-neighbor exclusion, is equivalent to the conventional row-to-row transfer matrix. Based on this, we propose that the transitions for other values of the coupling constant for the competing term are also nonuniversal, as is known for the hard-square gas problem.

Peter Høyer

M. A. Novotny

A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.