Phase Transitions and Critical Phenomena

When Matter Changes Its Face

Water turns into ice at 0°C — a first-order phase transition. Iron loses magnetism at 770°C (the Curie point) — a second-order phase transition. At first glance, these are ordinary phenomena. In reality: phase transitions are one of the richest areas of theoretical physics, linking symmetry, statistics, and critical behavior.

Especially intriguing is the critical point: as one approaches it, fluctuations increase at all scales, and the system becomes scale-invariant. Critical exponents turn out to be universal — the same for physically completely different systems!

First and Second Order Phase Transitions

First Order: Jump of the first derivatives of the free energy (volume, magnetization). Latent heat $L = T\Delta S$. Coexistence of phases. Examples: melting of ice ($\Delta S = L/T \approx 22$ J/(mol·K)); boiling water; transition paramagnet → ferromagnet in an external field.

Second Order (Continuous): Second derivatives of $F$ (heat capacity) diverge or have a discontinuity. No latent heat, no phase coexistence. The order parameter grows continuously from zero at $T_c$. Examples: ferromagnet (magnetization $M \to 0$ as $T \to T_c$); superconductivity; $\lambda$-transition of liquid helium.

The Ising Model: The Simplest Nontrivial Model

Hamiltonian: $H = -J \sum_{\langle ij \rangle} s_i s_j - h \sum_i s_i$, $s_i = \pm 1$

Symbols: $J > 0$ — exchange integral (ferromagnetic interaction); the sum $\langle ij \rangle$ — over nearest neighbors; $h$ — external magnetic field; $s_i = \pm 1$ — spin projection onto the axis.

For $J > 0$ and $T < T_c$: ordered phase ($M \neq 0$, ferromagnet). For $T > T_c$: disordered ($M = 0$, paramagnet).

1D Ising: No phase transition at $T > 0$ (thermal fluctuations destroy order).

2D Ising: Onsager (1944) obtained an exact solution. $T_c = \frac{2J}{k_B \ln(1 + \sqrt{2})} \approx 2.27, J/k_B$. Magnetization: $M \sim (T_c - T)^{1/8}$ as $T \to T_c^-$.

Critical Phenomena and Exponents

Close to $T_c$, power laws are observed with critical exponents:

$M \sim (T_c - T)^{\beta}$ (magnetization), $\beta$(Ising 2D) $= 1/8$ $C_V \sim |T - T_c|^{-\alpha}$ (heat capacity diverges!) $\chi \sim |T - T_c|^{-\gamma}$ (magnetic susceptibility) $\xi \sim |T - T_c|^{-\nu}$ (correlation length $\to \infty$!)

Correlation length $\xi \to \infty$ as $T \to T_c$ means: fluctuations arise at all spatial scales. The system becomes similar to itself under any scaling — scale invariance.

Landau Theory and Mean Field

Landau suggested expanding the free energy in terms of the order parameter $M$ (in the absence of a field):

$ F(M, T) = F_0 + a(T - T_c) M^2 + b M^4 + O(M^6) $

When $a > 0, b > 0$: minimum at $M = 0$ (disordered phase). For $T < T_c$: $a(T - T_c) < 0$, minimum at $M^2 = \frac{a(T_c - T)}{2b}$, $M \sim (T_c - T)^{1/2}$.

Prediction of Landau theory (mean field): $\beta = 1/2$. But for 2D Ising $\beta = 1/8$! Landau theory is incorrect near $T_c$ — it ignores fluctuations.

Ginzburg Criterion: Mean field theory works when fluctuations are small. In spatial dimensions $d > 4$ (upper critical dimension) Landau is accurate. In $d = 3$ fluctuations are important — renormalization group is needed.

Renormalization Group: Explanation of Universality

Wilson (1971, Nobel 1982) proposed an explanation of the universality of critical exponents through the renormalization group (RG):

Near $T_c$: the physics does not depend on details at small scales. When "coarse-graining" (block-spin transformation), the system "flows" to one of the RG's "fixed points". Critical exponents are determined only by the fixed point — depend only on the spatial dimension $d$ and the symmetry of the order parameter $n$. This is universality.

Universality classes: Ising 3D ($n = 1$): $\beta = 0.326$, $\nu = 0.630$. XY model ($n = 2$): $\beta = 0.346$, $\nu = 0.671$. Heisenberg ($n = 3$): $\beta = 0.365$, $\nu = 0.707$.

Real Examples

Mixture of oils and water near the critical point (38°C at a certain composition): it scatters white light — critical opalescence. Fluctuations of scales from nanometers to the wavelength of visible light → strong scattering of all wavelengths.

Superconductivity (second-order transition in zero field): order parameter — macroscopic wave function of Cooper pairs. Fluctuations are responsible for the finite width of the transition.

Phase Transitions in Materials Science and Neuroscience

The theory of second-order phase transitions and critical phenomena covers a huge range of systems. In materials science, the ferroelectric–paraelectric transition in perovskites (BaTiO₃, PZT) is described by the Landau theory and used in the manufacture of piezoelectric sensors and actuators, ultrasonic medical scanners, and memory elements. In superconductors, the critical temperature for the transition from normal phase to superconducting phase is described by the Ginzburg–Landau theory — a direct generalization of Landau theory for phase transitions. Magnetic materials for hard drives use the ferromagnet–paramagnet transition (Curie point) for heat-assisted recording, increasing data density. In neuroscience, critical phenomena describe the state of the brain: the brain operates near the critical point, maximizing the dynamic range and sensitivity to external stimuli. In economics, financial crises are considered phase transitions in systems with feedback, where "critical points" correspond to thresholds of systemic collapse. Universality of critical exponents explains why such different systems demonstrate identical behavior near a transition.

Assignment: (a) Landau theory: $F = a(T - T_c)M^2 + bM^4$, $h = 0$. Find $M(T)$ for $T < T_c$ and $T > T_c$. Show $\beta = 1/2$. (b) For $h \neq 0$ and $T = T_c$: $M \sim h^{1/\delta}$. Find $\delta$ from Landau theory. (c) Why does mean field yield incorrect critical exponents in $d = 3$? What physics does it ignore?