Finite-size scaling of correlation functions in finite systems

• DOI: 10.1007/s11433-018-9266-x
• Publication Date: 2018-10-13T07:18:13Z
• Authors: Xin Zhang; GaoKe Hu; YongWen Zhang; XiaoTeng Li; XiaoSong Chen
• Keywords: Exponent; Critical point (mathematics); Widom scaling; Percolation (cognitive psychology)

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 $\eta$.