Analysis of Projections of the Transfer Matrix in 2D Ising Models

摘要

The Ising model, originally proposed to explain properties of ferromagnets,consists of a regular lattice whose vertices are considered to be 'sites' that can be in exactly one of two possible states. Of interest is the partition function, which is the sum of the energy of the lattice over all possible configurations. There are two main approaches to computing the partition function: the combinatorial method uses an expansion whose coefficients are the number of subgraphs satisfying certain criteria; the algebraic approach introduces a transfer matrix whose spectral radius is the partition function per spin. In the semi-infinite 2D model with n rows, the associated transfer matrix Mn is duodiagonal of order 2n. This thesis introduces a special class of subspaces for approximating the dominant eigenvectors of Mn, and analyzes the projections of Mn, and its adjoint onto these subspaces. We shall show that the projections are sparse (with 2 or 4 nonzero entries per column), and are of order 0(n221-1) where 1 is a parameter of the subspaces. Some optimal properties of these subspaces are established.