Correlations in Ising chains with non-integrable interactions

TL;DR

This paper investigates two-spin correlations in non-integrable long-range Ising chains, revealing a universal singular structure in scaled correlations at all temperatures through theoretical analysis and Monte Carlo simulations.

Contribution

It introduces a scaling form for correlations in non-integrable Ising chains and demonstrates the universal singular behavior at all temperatures, including criticality.

Findings

Correlations decay with distance as r^{-1-sigma} for -1 < sigma < 0.

Magnified correlations exhibit a universal scaling form F(r/L).

Singular structure in F(x) persists at all temperatures, including the critical point.

Abstract

Two-spin correlations generated by interactions which decay with distance r as r^{-1-sigma} with -1 <sigma <0 are calculated for periodic Ising chains of length L. Mean-field theory indicates that the correlations, C(r,L), diminish in the thermodynamic limit L -> \infty, but they contain a singular structure for r/L -> 0 which can be observed by introducing magnified correlations, LC(r,L)-\sum_r C(r,L). The magnified correlations are shown to have a scaling form F(r/L) and the singular structure of F(x) for x->0 is found to be the same at all temperatures including the critical point. These conclusions are supported by the results of Monte Carlo simulations for systems with sigma =-0.50 and -0.25 both at the critical temperature T=Tc and at T=2Tc.