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arXiv:1008.0352 · Journal of Mathematical Physics

Spin matrix for the scaled periodic Ising model

John A. Palmer, Grethe Hystad

Published Dec 1, 2010• 7 citations
arXivHTMLPDF

Abstract

We compute the matrix elements for the spin operator in the two dimensional Ising model with respect to the eigenvectors for the transfer matrix in the continuum limit with periodic boundary conditions. This matrix can be used to calculate the correlations for the Ising model on the cylinder or the torus.
DOI
10.1063/1.3515312

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The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.
Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors — matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.