Symmetry, Sliding Windows and Transfer Matrices

In this paper we study 1D k-neighbor Ising model. Variational approach using modified nearestneighbor interaction strength is developed, but the optimization of the coupling constant appears at least as hard as the exact solution in the general case. For the exact solution two formulations of transfer matrix are studied: the block-spin approach yielding matrix T and the ’sliding window’ approach yielding matrix Ts. Equivalence between the two is established with T k s = T holding univesally. Matrix Ts is sparse and possesses apparent symmetries, giving hope for analytical computation of eigenvalues. Special cases are worked out explicitly. Finally, in the appendix we compute the exact partition function for the 2D Ising model on an anisotropic triangular lattice, using graphical techniques as shown in [1].