Decay of Order in Classical Many-Body Systems. I. Introduction and Formal Theory

Abstract

In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contra. st this prediction with the exactly known results for the two-dimensional' Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Frobenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.Additionally, one believes that even at low temperatures and high densities, away from T, , C~w ill be finite and (1.4) will hold. Thus, for example, neutron scattering cross sections at small angles, away from the critical point of a ferromagnetic 946

Variational method and duality in the 2D square Potts model

“…We note also that, for our L(σ, σ ′ ), the Perron-Frobenius (P.F.) theorem is valid [19]. Furthermore we have that L(σ, σ ′ ) is positive definite, that is, for any vector ξ(σ), we obtain…”

“…Since, as H goes from zero to +∞, A D (H, K * ) goes from +∞ to −K * , we deduce from (43), (19), (28) that…”

Spatial Variations of Spontaneous Magnetization in a Layered Ising Model

“…On the basis of the picture developed in the derivation of (6.44) the magnetization responds to the differing coupling constants within the inhomogeneity, i. e. a variation of the local energy density. Accordingly the spinenergy density correlation 27 on the exponential decay of pair correlations hold as well for G^g • So the inconsistencies with the classical Heisenberg model mentioned above persist even when G.^e is taken into consideration. On the other hand, the Monte Carlo results of 23 yield the magnetization only in the 12 to 15 layers next to the surface, which might be too poor to make a reliable statement on asymptotic decay.…”

Numerical transfer-matrix study of surface-tension anisotropy in Ising models on square and cubic lattices

“…A transfer matrix T can be used to find the free energy per unit length in the longitudinal direction. The largest eigenvalue of T , Λ 0 , is related to the partition function Z in the limit of infinite longitudinal size (K → ∞) by [40,41,42,43,29] lim K→∞ Z 1/K = lim…”