Corner transfer matrices

Abstract

Abstract For two-dimensional lattice models with interactions only between nearest (and diagonally nearest) neighbour spins, a well-known concept is the row-to-row transfer matrix. Less well-known is the “corner” transfer matrix (CTM). This has some very useful properties. If it is normalized so that its largest eigenvalue is unity, and the eigenvalues are arranged in numerically decreasing order, then each eigenvalue tends to a limit as the lattice becomes large. For those models which have been solved exactly (notably the Ising, eight-vertex and hard hexagon models), this limiting eigenvalue distribution is very simple, being basically that of a direct product of two-by-two matrices. From it the order parameter can easily be obtained. For all models one can write down formally exact matrix relations for the CTM, but the matrices are of infinite size. If one uses a representation in which the CTM is diagonal, and then truncates these relations to finite size, then one obtains a quite accurate approximation. The larger the size the greater the accuracy. I.G. Enting and I have thereby obtained comparatively long series expansions for the Ising model in a field, and for the hard squares model.