Finite- Size Scaling of Correlation Function

TL;DR

This paper introduces a finite-size scaling approach for correlation functions near critical points, validated by Monte Carlo simulations of Ising and percolation models, enabling precise determination of critical parameters.

Contribution

It presents a novel finite-size scaling form for correlation functions that accounts for directional dependence and system size effects near criticality.

Findings

Finite-size scaling form accurately describes correlation functions.

Monte Carlo simulations confirm the scaling near critical points.

Method allows precise estimation of critical exponents and points.

Abstract

We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance ${\bf r}$ in the finite system with size $L$, the correlation function can be written as the product of $|{\bf r}|^{-(d-2+\eta)}$ and its finite-size scaling function of variables ${\bf r}/L$ and $tL^{1/\nu}$, where $t=(T-T_c)/T_c$. The directional dependence of correlation function is nonnegligible only when $|{\bf r}|$ becomes compariable with $L$. This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent $\eta$.