Periodic Ising correlations

Abstract

We consider the finite two-dimensional Ising model on a lattice with periodic boundary conditions. Kaufman determined the spectrum of the transfer matrix on the finite, periodic lattice, and her derivation was a simplification of Onsager's famous result on solving the two-dimensional Ising model. We derive and rework Kaufman's results by applying representation theory, which give us a more direct approach to compute the spectrum of the transfer matrix. We determine formulas for the spin correlation function that depend on the matrix elements of the induced rotation associated with the spin operator. The representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map. We wrap the lattice around the cylinder taking the semi-infinite volume limit. We control the scaling limit of the multi-spin Ising correlations on the cylinder as the temperature approaches the critical temperature from below in terms of a Bugrij-Lisovyy conjecture for the spin matrix elements on the finite, periodic lattice. Finally, we compute the matrix representation of the spin operator for temperatures below the critical temperature in the infinitevolume limit in the pure state defined by plus boundary conditions.