The Ising Model and Phase Transitions, correlation length

The correlation length for a finite lattice size can then be shown to be proportional to $$\begin{equation*} \xi(T) \propto L\sim \left|T_C-T\right|^{-\nu}. \end{equation*}$$ and if we set $T=T_C$ one can obtain the following relations for the magnetization, energy and susceptibility for $T \le T_C$ $$\begin{equation*} \langle {\cal M}(T) \rangle \sim \left(T-T_C\right)^{\beta} \propto L^{-\beta/\nu}, \end{equation*}$$ $$\begin{equation*} C_V(T) \sim \left|T_C-T\right|^{-\gamma} \propto L^{\alpha/\nu}, \end{equation*}$$ and $$\begin{equation*} \chi(T) \sim \left|T_C-T\right|^{-\alpha} \propto L^{\gamma/\nu}. \end{equation*}$$

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