GL(2,R) in the Finite One-Dimensional Ising Model with Nonuniform Couplings

Using the properties of GL(2,R), the general linear group of invertible 2×2 real matrices, we investigate random fields of spin variables on finite one-dimensional rings with a unit cell of p∈N sites. The interaction parameters are assumed to be periodic with period p. The cases p=1 and p=2 recover the one-dimensional and alternating Ising models, respectively. The couplings between adjacent spins may be ferromagnetic (positive) or antiferromagnetic (negative). Utilising the recurrence relations of the Chebyshev polynomials and a bijection between the number of spins and the polynomial index, we derive explicit formulae well suited to the finite-size analysis of the partition functions, free energy, and specific heat of both models. We show that, for the (p=2) case, the specific heat exhibits a double-Schottky anomaly whenever the characteristic exchange energy scales are sufficiently separated. We prove that this double-peak structure originates from the coexistence of distinct energy scales induced by the periodic modulation of the coupling signs and characterise its dependence on the model parameters. We demonstrate that the universality hypothesis in critical Casimir force theory remarkably holds without requiring small fields or large interaction parameters, suggesting a form of “hyper-universal” behaviour valid for arbitrary model parameters.