Finite-size scaling of correlation functions in finite systems

Abstract

We propose the finite-size scaling of correlation functions in finite systems near their critical points. At a distance ${\boldsymbol~r}$ in a $d$-dimensional finite system of size $L$, the correlation function can be written as the product of $|{\boldsymbol~r}|^{-(d-2+\eta)}$ and a finite-size scaling function of the variables ${\boldsymbol~r}/L$ and $tL^{1/\nu}$, where $t=(T-T_\text{c})/T_\text{c}$, $\eta$ is the critical exponent of correlation function, and $\nu$ is the critical exponent of correlation length. The correlation function only has a sigificant directional dependence when $|{\boldsymbol~r}|$ is compariable to $L$. We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations. We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponent $\eta$.