Phase Transitions & Critical Phenomena
Phase Transitions & Critical Phenomena
Phase transitions — from water to ice, from paramagnet to ferromagnet — involve collective reorganization of matter. Near a critical point, systems display universal behavior described by critical exponents independent of microscopic details.
Definition
Order parameter \(m\) vanishes above the critical temperature \(T_c\) and grows below: \(m\sim(T_c-T)^\beta\). Other critical exponents govern correlation length \(\xi\sim|T-T_c|^{-\nu}\) and susceptibility \(\chi\sim|T-T_c|^{-\gamma}\).
Key Result
Mean-field theory (Landau): expand free energy \(F = a_0 + a_2 m^2 + a_4 m^4 + \cdots\). Below \(T_c\), \(a_2 < 0\) gives \(m = \sqrt{-a_2/2a_4}\). Predicts \(\beta = 1/2\) — exact only above 4D.
Example 1
2D Ising model (Onsager exact solution): \(T_c = 2J/(k_B\ln(1+\sqrt{2}))\approx 2.269\,J/k_B\). Critical exponents \(\beta = 1/8\), \(\nu = 1\), \(\eta = 1/4\) — different from mean-field.
Example 2
Renormalization group explains universality: different systems share critical exponents if they have the same dimensionality and symmetry. The 3D Ising model (\(\beta\approx 0.326\)) describes liquid-gas, binary fluid, and uniaxial magnet transitions.
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Practice
- What is spontaneous symmetry breaking?
- Why do fluctuations invalidate mean-field theory near the critical point?
- Explain scaling and universality using the renormalization group.
- What is the role of the upper critical dimension?
Show Answer Key
1. A system's ground state has lower symmetry than the Hamiltonian. Example: ferromagnet above $T_c$ has rotational symmetry (paramagnetic), but below $T_c$, spins align along one direction (broken rotational symmetry). The order parameter ($M$ for magnets) distinguishes phases: $M=0$ above $T_c$, $M\neq 0$ below.
2. Mean-field theory assumes each spin feels the average field of all others, ignoring correlations. Near $T_c$, fluctuations have correlation length $\xi \to \infty$, and correlated regions span the entire system. MFT's neglect of these long-range fluctuations gives wrong critical exponents (e.g., MFT: $\beta=1/2$, Ising 3D: $\beta\approx0.326$).
3. RG: coarse-grain (integrate out short-wavelength fluctuations), rescale, and renormalize parameters. Near a critical point, the RG flow approaches a fixed point where the system looks the same at all scales (scale invariance). Critical exponents are determined by the fixed point, not microscopic details — hence universality (systems with same symmetry and dimension share exponents).
4. Above the upper critical dimension $d_c$ (4 for Ising), fluctuations are irrelevant and mean-field exponents are exact (Ginzburg criterion satisfied). Below $d_c$, fluctuations dominate and non-trivial exponents arise. At $d=d_c$, logarithmic corrections appear. For the Ising model: $d_c=4$ (MFT exact for $d\geq4$).