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Phase Transition Temperature

The specific temperature at which a distinct phase transition occurs within a substance, marking a change in its physical properties.
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The statement of the theorem

Let H\mathcal{H} be the Hamiltonian of the system, and β=1/(kBT)\beta = 1/(k_B T) be the inverse temperature. Define the Helmholtz free energy density f(T,order parameter)f(T, \text{order parameter}) via the partition function Z=Tr(e−βH)Z = \text{Tr}(e^{-\beta \mathcal{H}}). For a continuous (second-order) phase transition, the critical temperature TcT_c is defined by the condition where the coefficient of the quadratic term in the Landau expansion of the free energy density, A(T)A(T), vanishes, while the coefficient of the quartic term, BB, remains positive. Specifically, let the free energy density expansion be f(T,η)=12A(T)η2+14Bη4+…f(T, \eta) = \frac{1}{2} A(T) \eta^2 + \frac{1}{4} B \eta^4 + \dots. The critical temperature TcT_c is determined by the root of A(T)=0A(T) = 0, provided that the susceptibility χ=∂2f/∂η2\chi = \partial^2 f / \partial \eta^2 diverges at this point, satisfying: A(Tc)=0and∂2f∂η2∣T=Tc→∞A(T_c) = 0 \quad \text{and} \quad \left. \frac{\partial^2 f}{\partial \eta^2} \right|_{T=T_c} \to \infty