Фізика фазових переходів

🔄 Types of Phase Transitions

1st Order

Latent heat ΔH, supercooling, superheating, metastable states. Interface: gas-liquid, solid-liquid.

2nd Order

Continuous transitions. Ferromagnetic (T = T_c), superconducting. No latent heat. Critical point.

Critical Point

T = T_c: 1st order → 2nd order. Water: T_c ≈ 374°C, P_c ≈ 22 MPa. Mesoscopic.

Classification

Ehrenfest classification: ∂ⁿG/∂Tⁿ discontinuity. Modern: symmetry breaking, topological.

🔬 Critical Phenomena

Critical exponents

β (order parameter): η ~ (T_c - T)^β, β ≈ 1/3 for 3D Ising. α (specific heat), γ (susceptibility), δ, ν (correlation length), η.

Correlation length

ξ ~ |T - T_c|^(-ν), ν ≈ 2/3 for 3D. Scale invariance. Universal behavior.

Scaling laws

α + 2β + γ = 2. γ = ν(2 - η). Widom scaling, Kadanoff RG.

Universality

Critical exponents: one for universality class. Ising, XY, Heisenberg. d-dimension dependent.

🧲 Landau Theory

Free energy

F(η,T) = a(T)η² + b(T)η⁴ + ... a(T) = a₀(T - T_c). Minima: ∂F/∂η = 0.

Mean-field

Fluctuations ignored. β = 1/2 MF vs 1/3 exact 3D. Valid at high dimensions.

Ginzburg criterion

Mean-field valid if fluctuations are small. ΔF_fluct < ΔF_bulk. Dimension-dependent.

Symmetry breaking

Spontaneous symmetry breaking. Ferromagnetic: η ≠ 0 in the ground state. Goldstone modes.

🔗 Percolation

Critical probability

p_c: bond percolation 2D ≈ 0.5, site ≈ 0.59. p < p_c: disconnected, p > p_c: spanning.

Cluster distribution

n_s ~ s^(-τ) for cluster size s. τ ~ 2.05 in 2D. Dependence on p.

Fractal dimension

D_f ≈ 1.9 in 2D percolation. Scaling: R_s ~ s^(1/D_f). Self-similar.

Applications

Composite materials, porous media, epidemics, forest fires. Resilience, connectivity.

🌊 Topological Phase Transitions

Topological order

Quantum Hall effect: ν = σxy/(e²/h). Integer (ν = 1,2,3...), fractional (ν = 1/3,2/3...).

Topological insulators

Gap bulk, gapless edge. Chern number ≠ 0. 2D (quantum Hall), 3D (Bi₂Se₃).

Edge states

Chiral edge modes. Protected topology. Quantized conductance. Robustness to disorder.

Applications

Quantum computing, spintronics, metrology. Topological qubits, error correction.

🔬 Quantum phase transitions

T = 0 transitions

Quantum fluctuations (not thermal). Mott transition, quantum Hall. Critical point control.

Infinite range

ξ → ∞ at the critical point. Low-T finite-T crossover. D-dimensional quantum → (D+1)-dimensional classical.

RG flow

Renormalization Group: flow of coupling constants. Fixed points: critical, stable.

Examples

Superconductor-insulator transition, Mott, quantum spin liquids, cuprates.

📊 Graphs and Diagrams

Critical exponents

Scaling laws:

Universal exponents: α + 2β + γ = 2. Ising, XY, Heisenberg classes.

Percolation

Cluster percolation:

Fractal dimension D_f ≈ 1.9^2D. Self-similarity at criticality.

η ~ (T_c - T)^β: β ≈ 1/3 (3D Ising) ξ ~ |T - T_c|^(-ν): ν ≈ 2/3 C ~ |T - T_c|^(-α): α ≈ 0.11 χ ~ |T - T_c|^(-γ): γ ≈ 4/3

🧪 Practical Examples

Example 1: Ferromagnetic transition

Ising model: T_c, β ≈ 1/3 (3D). Magnetic susceptibility χ ~ |T - T_c|^(-γ). Critical point.

Example 2: Quantum Hall effect

ν = 1: integer Hall. ν = 1/3: fractional (FQHE). Topological, robust. Edge states.

Example 3: Percolation 2D

p_c ≈ 0.59 site. Spanning cluster. Fractal D_f ≈ 1.9. Universal scaling.

Example 4: Superconductor transition

YBCO: T_c ~ 90 K. Order parameter: gap Δ. BCS mean-field. Fluctuations.

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Phase transitions: change of macroscopic states