Chebyshev Polynomials in the Physics of the One-Dimensional Finite-Size Ising Model: An Alternative View and Some New Results

Abstract

For studying the finite-size behavior of the Ising model under different boundary conditions, we propose an alternative to the standard transfer matrix technique approach based on Abelès theorem and Chebyshev polynomials. Using it, one can easily reproduce the known results for periodic boundary conditions concerning the Lee–Yang zeros, the exact position-space renormalization-group transformation, etc., and can extend them by deriving new results for antiperiodic boundary conditions. Note that in the latter case, one has a nontrivial order parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with the opposite bond in the system. It is interesting to note that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly.