Finite- Size Scaling of Correlation Function

Abstract

We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance $\mathbf{r}$ in the finite system with size $L$, the correlation function can be written as the product of $|\mathbf{r}{|}^{-(d-2+\eta )}$ and its finite-size scaling function of variables $\mathbf{r}/L$ and $t{L}^{1/\nu }$, where $t=(T-T_c)/T_c$. The directional dependence of correlation function is nonnegligible only when $|\mathbf{r}|$ 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 $\eta$.

7 pages, 13 figures