Measurement of the correlation length on Ising model

Abstract

We present a method for measuring the correlation length of the Ising model. Starting from a ground state, we consider a quantity $K(t,T) \equiv L ({\langle M^{2}(t) \rangle / \langle M(t) \rangle ^{2}-1})^{1/d}$, where M(t), L and d denote the magnetization, the system size, and substrate dimension of the model, respectively. K(t, T) follows a power-law behavior $K(t,T_{c}) \sim t^{1/z}$ at the critical temperature $T_{c}$ and the saturation value of $K(\infty , T)$ shows that $K_{\mathrm{sat}} (T) \sim |T_{c} - T|^{-\nu }$. The critical exponents $\nu = 1.00(1)$ and $z=2.15(1)$ are estimated in a two dimensional square lattice. By calculating solely K(t, T), we could obtain the correlation-length exponent directly. Also, the correlation length and critical exponents of the three dimensional Ising model on a cubic lattice are discussed. We believe that this is an effective method, which can be feasibly applied to various spin models.