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Order Parameter

A quantity that characterizes the order in a system undergoing a phase transition, often exhibiting a discontinuous change at the transition point.
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The statement of the theorem

Let H\mathcal{H} be the Hamiltonian of a system defined on a lattice Λ\Lambda with local interactions JijJ_{ij}. Consider the partition function Z=Tr(e−βH)Z = \text{Tr}\left(e^{-\beta \mathcal{H}}\right), where β=1/kBT\beta = 1/k_B T. Define the order parameter Φ\Phi as the expectation value of a symmetry-breaking field operator ϕ(r)\phi(\mathbf{r}): Φ=⟨ϕ(r)⟩=1ZTr(ϕ(r)e−βH)\Phi = \langle \phi(\mathbf{r}) \rangle = \frac{1}{Z} \text{Tr}\left(\phi(\mathbf{r}) e^{-\beta \mathcal{H}}\right). In the context of mean-field theory (e.g., Ising model), Φ\Phi minimizes the Landau free energy functional F(Φ)F(\Phi): F(Φ)=a(T−Tc)Φ2+bΦ4+cΦ6+12∑r(∇Φ(r))2F(\Phi) = a(T - T_c)\Phi^2 + b\Phi^4 + c\Phi^6 + \frac{1}{2} \sum_{\mathbf{r}} \left( \nabla \Phi(\mathbf{r}) \right)^2 The equilibrium value Φ0\Phi_0 is determined by the minimization condition ∂F∂Φ∣Φ=Φ0=0\frac{\partial F}{\partial \Phi} \bigg|_{\Phi=\Phi_0} = 0. For a second-order transition, Φ0∝Tc−T\Phi_0 \propto \sqrt{T_c - T} as T→Tc+T \to T_c^+.