We investigate the statistical mechanical properties of spin models on the square lattice, using the technique of transfer matrices. With the aid of exact diagonalisation we obtain results which indicate that excitations of the transfer matrix can be thought of as physical objects which we term topological excitations, which are domain walls in the case of the Ising model at low temperature. Inspired by these results, we extend the usefulness of Baker-Campbell-Hausdorff formula by finding and proving a new representation. This mathematical result is the major contribution of this thesis to the wider literature. Applying perturbation theory to it allows us to perturbatively find the eigenvalues of a square lattice transfer matrix in a way reminiscent of a high-temperature expansion. We then do so for the Ising model, comparing our results to known formulae, and extend the calculation to Potts models
