Phase Transitions

Water boils. Iron loses its magnetism when heated. The early universe underwent dramatic symmetry-breaking transitions that determined the forces and particles we observe today. Phase transitions are among the most striking phenomena in nature: smooth changes in a control parameter (temperature, pressure, magnetic field) produce sudden, qualitative changes in the state of matter.

What makes phase transitions remarkable from a theoretical perspective is that they involve singular behavior of thermodynamic quantities -- discontinuities, divergences, power laws -- emerging from smooth microscopic interactions. Understanding how this happens has required some of the deepest ideas in theoretical physics, ideas that reach far beyond condensed matter into particle physics and cosmology.

Classification of Phase Transitions

First-Order Transitions

In a first-order transition, the first derivative of the free energy is discontinuous. This means a sudden jump in quantities like entropy, volume, or magnetization. The system releases or absorbs a latent heat $L = T \Delta S$.

Examples include water boiling (liquid to gas), ice melting (solid to liquid), and the ferromagnetic transition in an applied field. The two phases can coexist at the transition point -- think of ice floating in water at $0$°C.

Continuous (Second-Order) Transitions

In a continuous transition, the first derivatives of the free energy are continuous, but the second derivatives (heat capacity, susceptibility, compressibility) diverge or are discontinuous. There is no latent heat and no phase coexistence. Instead, the system changes gradually through a critical point.

The paradigmatic example is the ferromagnetic transition: below the Curie temperature $T_c$, a magnet has spontaneous magnetization; above $T_c$, the magnetization vanishes continuously. At $T_c$ itself, the system exhibits remarkable behavior: fluctuations on all length scales, power-law correlations, and universal critical exponents.

Order Parameters

An order parameter is a quantity that is zero in the disordered (high-symmetry) phase and nonzero in the ordered (broken-symmetry) phase. It characterizes what changes at the transition:

System	Order Parameter	Symmetry Broken
Ferromagnet	Magnetization $\mathbf{M}$	Rotational (spin direction)
Liquid-gas	$\rho - \rho_c$	Liquid/gas distinction
Superfluid	Condensate wavefunction $\psi$	U(1) gauge symmetry
Superconductor	Cooper pair condensate	U(1) gauge symmetry
Higgs field	Vacuum expectation value $v$	Electroweak SU(2)×U(1)

The Ising Model

The simplest model of a phase transition is the Ising model: a lattice of spins $s_i = \pm 1$ that interact with their nearest neighbors. The energy is:

$$E = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$$

where $J > 0$ favors aligned spins (ferromagnetism) and $h$ is an external magnetic field. Despite its simplicity, this model captures the essential physics of phase transitions: competition between energy (favoring order) and entropy (favoring disorder).

In one dimension, there is no phase transition at finite temperature -- thermal fluctuations always destroy order. In two dimensions, Onsager solved the model exactly in 1944, finding $k_B T_c = 2J/\ln(1+\sqrt{2}) \approx 2.27 J$. This exact solution was a tour de force of mathematical physics and confirmed that phase transitions can arise from simple microscopic interactions.

Mean Field Theory

Mean field theory replaces the effect of all neighboring spins on a given spin with their average. Each spin feels an effective field:

$$h_{\text{eff}} = h + zJm$$

where $z$ is the number of neighbors and $m = \langle s_i \rangle$ is the average magnetization. Self-consistency requires:

$$m = \tanh\left(\beta(h + zJm)\right)$$

For $h = 0$, this equation has a nonzero solution below $T_c^{\text{MF}} = zJ/k_B$. Near $T_c$, expanding the $\tanh$ gives $m \propto (T_c - T)^{1/2}$, predicting a critical exponent $\beta = 1/2$.

Mean field theory is qualitatively correct but quantitatively wrong near the critical point, because it ignores fluctuations. The true exponent for the 3D Ising model is $\beta \approx 0.326$, not $1/2$. Mean field theory becomes exact in dimensions $d \geq 4$ (the "upper critical dimension").

Critical Exponents and Universality

Near a continuous phase transition, thermodynamic quantities obey power laws characterized by critical exponents. Defining $t = (T - T_c)/T_c$:

The most remarkable discovery is universality: systems with completely different microscopic physics share the same critical exponents if they have the same spatial dimension and the same symmetry of the order parameter. A liquid near its critical point and an Ising ferromagnet belong to the same universality class and have identical exponents.

The 3D Ising universality class has $\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$. These are known to extraordinary precision from both theory (conformal bootstrap methods) and experiment.

Spontaneous Symmetry Breaking

Above $T_c$, the Ising model has a $\mathbb{Z}_2$ symmetry: flipping all spins ($s_i \to -s_i$) leaves the energy unchanged. Below $T_c$, the system chooses either mostly up or mostly down spins. The symmetry of the Hamiltonian is not reflected in the ground state -- the symmetry is spontaneously broken.

Spontaneous symmetry breaking is one of the most important ideas in modern physics. It appears everywhere:

• Ferromagnetism: rotational symmetry is broken by the magnetization direction
• Crystals: translational and rotational symmetry is broken by the lattice
• The Higgs mechanism: the electroweak symmetry SU(2)$\times$U(1) is broken by the Higgs vacuum expectation value, giving mass to the W and Z bosons
• QCD chiral symmetry breaking: creates most of the visible mass in the universe

Landau Theory

Landau's approach, developed in the 1930s, constructs the free energy as a power series in the order parameter $m$, respecting the symmetries of the system. For a system with $m \to -m$ symmetry:

Landau theory is essentially mean field theory expressed in the language of symmetry. It correctly predicts the qualitative structure of phase transitions and the relationship between symmetry breaking and order parameters. It fails quantitatively near the critical point due to fluctuations but remains the starting point for more sophisticated theories.

The Renormalization Group (Preview)

The true understanding of critical phenomena came from Kenneth Wilson's renormalization group (RG) in the 1970s, which earned him the 1982 Nobel Prize. The key insight: at a critical point, the correlation length $\xi$ diverges, meaning fluctuations exist on all length scales simultaneously. No single scale matters -- the system is scale-invariant.

The RG procedure systematically "zooms out," integrating over short-distance fluctuations to produce an effective theory at longer distances. The flow of coupling constants under this rescaling determines:

• Which microscopic details are irrelevant (explaining universality)
• The exact values of critical exponents
• The upper critical dimension above which mean field theory is exact

The renormalization group connects statistical physics to quantum field theory in a profound way. The same mathematical framework describes critical phenomena in magnets, the behavior of elementary particles at different energy scales, and the running of coupling constants in the Standard Model.

Phase Transitions in Fundamental Physics

The language of phase transitions pervades modern fundamental physics:

The electroweak transition: In the early universe, at temperatures around $10^{15}$ K, the Higgs field acquired its vacuum expectation value, breaking electroweak symmetry and giving masses to the W, Z bosons and fermions. Whether this was first-order or a crossover has profound implications for baryogenesis.

The QCD transition: At temperatures around $10^{12}$ K, quarks and gluons confined into hadrons. This confinement/deconfinement transition is studied experimentally at RHIC and the LHC by colliding heavy nuclei.

Cosmic phase transitions: Phase transitions in the early universe may have produced topological defects (cosmic strings, domain walls) and gravitational waves that could be detected by future experiments.

Key Insights

• First-order transitions have latent heat and discontinuous order parameters; continuous transitions have power-law singularities
• Order parameters characterize what changes at a transition; they are zero in the disordered phase and nonzero in the ordered phase
• The Ising model, despite its simplicity, captures the essential physics of phase transitions
• Mean field theory (Landau theory) correctly predicts qualitative behavior but gives wrong exponents near the critical point
• Universality means that critical exponents depend only on dimension and symmetry, not microscopic details
• Spontaneous symmetry breaking -- the ground state has less symmetry than the Hamiltonian -- appears in magnets, superfluids, the Higgs mechanism, and QCD
• The renormalization group explains universality and connects statistical physics to quantum field theory

Looking Ahead

We've seen that entropy, ensembles, quantum statistics, and phase transitions form a coherent framework connecting microscopic physics to macroscopic phenomena. In our final lesson on statistical physics, we'll explore the deepest connection of all:

• Shannon entropy -- the mathematical identity between information and thermodynamic entropy
• Landauer's principle -- why erasing information necessarily produces heat
• Black hole entropy -- the most dramatic example of the information-thermodynamics connection

The bridge between information and physics may be the deepest theme in modern theoretical physics.