6.2 Second-order phase transitions

Definition of phase transitions

Phase transitions mark abrupt changes in a system's macroscopic properties as external parameters (temperature, pressure, magnetic field) are tuned. What makes them fascinating from a statistical mechanics perspective is that they arise from collective behavior: huge numbers of particles acting together produce emergent phenomena that you'd never predict by studying individual interactions alone.

First vs. second order transitions

The distinction between first- and second-order transitions comes down to how thermodynamic potentials behave:

• First-order transitions have discontinuities in the first derivatives of the free energy (entropy, volume, magnetization). They involve latent heat and phase coexistence (think ice melting into water).
• Second-order (continuous) transitions have continuous first derivatives but discontinuities or divergences in second derivatives (specific heat, susceptibility, compressibility). There's no latent heat, and the system changes smoothly through the transition.

The key signature of a second-order transition is critical phenomena: power-law divergences, scaling behavior, and universality emerge as you approach the critical point.

Critical point characteristics

The critical point sits at the terminus of a phase coexistence line, where the distinction between two phases vanishes entirely. Several striking features appear here:

• The susceptibility (response to an external field) diverges
• The correlation length (how far fluctuations extend spatially) diverges
• Universality emerges: systems with completely different microscopic physics show identical critical behavior
• In fluids, critical opalescence occurs because density fluctuations grow to the scale of visible light wavelengths, scattering light strongly

Order parameter

The order parameter is a quantity that captures how much order exists in a system. It's zero in the disordered (high-symmetry) phase and becomes nonzero in the ordered (low-symmetry) phase. Choosing the right order parameter is the first step in any theoretical description of a phase transition, and it plays a central role in Landau theory.

Symmetry breaking

A second-order transition typically involves spontaneous symmetry breaking: the system's equilibrium state has lower symmetry than the underlying Hamiltonian. No external field forces this; the system "chooses" one of several equivalent ground states on its own.

• The ordered phase has degenerate ground states related by the broken symmetry (e.g., a ferromagnet can magnetize "up" or "down")
• When a continuous symmetry is broken, Goldstone modes appear as gapless, low-energy excitations (spin waves in a Heisenberg magnet, phonons in a crystal)

Examples of order parameters

Critical exponents

Near the critical point, physical quantities follow power laws in the reduced temperature $t = (T - T_c)/T_c$ or the external field. The exponents governing these power laws are called critical exponents, and they encode the universal features of the transition.

Definition and significance

The standard critical exponents are defined by how key quantities behave as $t \to 0$:

• $\alpha$: Specific heat diverges as $C \sim |t|^{-\alpha}$
• $\beta$: Order parameter vanishes as $m \sim (-t)^{\beta}$ for $t < 0$
• $\gamma$: Susceptibility diverges as $\chi \sim |t|^{-\gamma}$
• $\delta$: At $t = 0$, the order parameter responds to the field as $m \sim h^{1/\delta}$
• $\nu$: Correlation length diverges as $\xi \sim |t|^{-\nu}$
• $\eta$: At criticality, the correlation function decays as $G(r) \sim r^{-(d-2+\eta)}$

These exponents are not all independent; they obey scaling relations (discussed below), so only two are truly independent.

Universal behavior

The remarkable fact about critical exponents is that they depend only on a few broad features of the system, not on microscopic details:

• Spatial dimensionality $d$
• Symmetry of the order parameter (scalar, vector, etc.)
• Range of interactions (short-range vs. long-range)

Systems sharing these features fall into the same universality class. For example, the liquid-gas critical point and the uniaxial ferromagnet both belong to the 3D Ising universality class, despite being physically very different systems. Other important classes include the XY class (2-component order parameter) and the Heisenberg class (3-component order parameter).

Mean field theory

Mean field theory replaces the complicated many-body interaction problem with a simpler one: each particle feels an average "effective field" from all the others. This is the simplest analytical approach to phase transitions and often serves as a launching point for more accurate treatments.

Landau theory

Landau's approach is phenomenological. You expand the free energy $F$ as a power series in the order parameter $m$, keeping only terms allowed by symmetry:

$F = F_0 + a(T)\,m^2 + b\,m^4 + \cdots$

where $a(T) = a_0(T - T_c)$ changes sign at the critical temperature and $b > 0$ for stability. Minimizing $F$ with respect to $m$ gives:

1. For $T > T_c$: $a > 0$, so the minimum is at $m = 0$ (disordered phase)
2. For $T < T_c$: $a < 0$, so the minimum shifts to $m = \pm\sqrt{-a/2b} \sim |t|^{1/2}$

This yields the classical (mean field) critical exponents: $\alpha = 0$, $\beta = 1/2$, $\gamma = 1$, $\delta = 3$, $\nu = 1/2$, $\eta = 0$. These are qualitatively correct but quantitatively wrong for most physical systems.

Limitations of mean field approach

Mean field theory neglects fluctuations, and this becomes a serious problem near the critical point where fluctuations dominate. Specifically:

• It predicts the wrong critical exponents in dimensions $d < d_c$ (the upper critical dimension, which is $d_c = 4$ for standard $\phi^4$ theory)
• It misses the true universality structure because it treats all systems the same regardless of dimensionality
• It does become exact for $d \geq 4$ or for systems with sufficiently long-range interactions (where each particle interacts with many neighbors, making the "average field" approximation accurate)

The Ginzburg criterion quantifies when mean field theory breaks down by comparing the size of fluctuations to the mean value of the order parameter.

Scaling theory

Scaling theory rests on a powerful hypothesis: near the critical point, the diverging correlation length $\xi$ is the only relevant length scale. All thermodynamic singularities arise from this single diverging quantity, which constrains how different critical exponents relate to each other.

Scaling laws

The assumption that the singular part of the free energy is a generalized homogeneous function of its arguments leads to algebraic relations among exponents. The most important scaling laws are:

• Rushbrooke's law: $\alpha + 2\beta + \gamma = 2$
• Widom's identity: $\gamma = \beta(\delta - 1)$
• Fisher's law: $\gamma = \nu(2 - \eta)$
• Josephson (hyperscaling): $2 - \alpha = d\nu$

These reduce the number of independent exponents to just two (for systems below the upper critical dimension). You can verify that mean field exponents satisfy the first three but violate hyperscaling for $d \neq 4$.

Universality classes

Universality classes group together all systems that share the same critical exponents and scaling functions. The classification depends on:

• Spatial dimension $d$
• Number of components of the order parameter $n$ ($n=1$ for Ising, $n=2$ for XY, $n=3$ for Heisenberg)
• Whether interactions are short-range or long-range

Renormalization group

The renormalization group (RG) is the theoretical framework that explains why universality and scaling occur. It provides a systematic method for calculating critical exponents and understanding how physics changes across length scales.

Basic concepts

The core idea is to progressively coarse-grain a system, integrating out short-wavelength degrees of freedom while tracking how the effective Hamiltonian changes. This defines a flow in the space of coupling constants.

• Fixed points of the RG flow correspond to scale-invariant systems, i.e., critical points
• Near a fixed point, perturbations are classified as relevant (grow under RG, control departures from criticality), irrelevant (shrink, explaining why microscopic details don't matter), or marginal (require higher-order analysis)
• The number of relevant directions at a fixed point equals the number of independent thermodynamic parameters needed to reach criticality

This framework directly explains universality: different microscopic Hamiltonians flow to the same fixed point, so they share the same critical behavior.

Real-space renormalization

Real-space RG applies coarse-graining directly on the lattice. The conceptual foundation is Kadanoff's block-spin transformation:

1. Divide the lattice into blocks of $b^d$ spins
2. Replace each block with a single effective spin (using a rule like majority vote)
3. Rescale lengths by $b$ so the new lattice looks like the original
4. Determine the new effective couplings between block spins

This procedure is particularly intuitive for discrete models. Practical implementations include decimation (summing over a subset of spins exactly) and majority-rule schemes. The approach works well for lattice models but can be hard to make systematic.

Momentum-space renormalization

Momentum-space RG works in Fourier space, integrating out fluctuations with wavevectors in a shell $\Lambda/b < |k| < \Lambda$. Wilson's key insight was to combine this with the $\epsilon$-expansion, setting $\epsilon = 4 - d$ and expanding critical exponents as power series in $\epsilon$:

$\eta = 0 + O(\epsilon^2), \quad \nu = \frac{1}{2} + \frac{n+2}{4(n+8)}\epsilon + \cdots$

This approach is powerful because it's systematic and connects naturally to field-theoretic methods. It also led to the broader concept of effective field theories, which has had enormous impact in both condensed matter and particle physics.

Critical phenomena

Critical phenomena encompass the collective behavior that emerges near a continuous phase transition: power-law correlations, diverging response functions, and scale invariance. These features arise because thermal fluctuations and incipient long-range order compete on all length scales simultaneously.

Fluctuations near critical point

As $T \to T_c$, fluctuations in the order parameter grow both in magnitude and spatial extent. This has several consequences:

• Thermodynamic response functions (specific heat, susceptibility) diverge or show singularities
• Mean field theory breaks down because it assumes fluctuations are small compared to average values
• In fluids, critical opalescence makes the sample appear milky as density fluctuations scatter visible light
• In magnets, critical scattering produces a sharp peak in the neutron scattering cross-section at small wavevectors

Correlation length

The correlation length $\xi$ measures the typical distance over which fluctuations in the order parameter are correlated. It diverges as $\xi \sim |t|^{-\nu}$ when approaching the critical point.

This divergence is central to everything: it's why microscopic details become irrelevant (the system's behavior is dominated by physics at the scale $\xi \gg a$, where $a$ is the lattice spacing), and it's what makes scaling theory work. At the critical point itself, $\xi \to \infty$, and the system becomes scale-invariant with power-law correlations at all distances.

Experimental techniques

Measuring critical exponents and verifying scaling predictions requires extremely precise experiments. The main challenges are controlling temperature to within microkelvins of $T_c$, dealing with long equilibration times, and separating singular critical contributions from smooth backgrounds.

Scattering methods

Neutron and X-ray scattering directly probe spatial correlations by measuring the structure factor $S(\mathbf{q})$, which is the Fourier transform of the real-space correlation function $G(\mathbf{r})$.

• The shape of $S(q)$ near $q = 0$ gives the correlation length $\xi$ (and hence $\nu$)
• The power-law tail of $S(q)$ at criticality gives the anomalous dimension $\eta$
• Techniques include small-angle neutron scattering (SANS) for bulk correlations and X-ray photon correlation spectroscopy (XPCS) for dynamics

Specific heat measurements

Specific heat measurements determine the exponent $\alpha$, but this is often the hardest exponent to pin down because the divergence can be very weak (logarithmic for the 3D Ising model, where $\alpha \approx 0.11$).

• Adiabatic calorimetry and ac calorimetry are standard techniques
• Careful background subtraction is essential since the singular part may be a small fraction of the total specific heat
• Sample purity and homogeneity matter enormously because impurities can shift $T_c$ locally and smear the transition

Examples of second-order transitions

Ferromagnetic transition

In materials like iron, nickel, and EuO, cooling below the Curie temperature $T_c$ causes spontaneous magnetization to appear. The order parameter is the magnetization $M$, which grows continuously from zero as $T$ drops below $T_c$.

For a uniaxial ferromagnet (where spins prefer to align along one axis), the transition belongs to the 3D Ising universality class with exponents $\beta \approx 0.326$, $\gamma \approx 1.237$, and $\nu \approx 0.630$. The magnetic susceptibility diverges on both sides of the transition, and the specific heat shows a weak divergence.

Superfluid transition

Liquid $^4\text{He}$ undergoes a superfluid transition at the lambda point, $T_\lambda = 2.17 \text{ K}$ at atmospheric pressure. The name comes from the shape of the specific heat curve, which resembles the Greek letter $\lambda$.

The order parameter is the complex macroscopic wave function $\Psi$ of the Bose-Einstein condensate, and the relevant symmetry is $U(1)$ (phase rotation), placing this transition in the 3D XY universality class. Below $T_\lambda$, helium exhibits zero viscosity flow and quantized vortices. Space-based experiments (to suppress gravity-driven convection) have measured $\alpha = -0.0127 \pm 0.0003$, providing one of the most precise tests of RG predictions.

Numerical methods

Computational methods are indispensable for studying critical phenomena, especially in systems where exact solutions don't exist (which is most of them). They complement analytical approaches and can achieve high-precision estimates of critical exponents.

Monte Carlo simulations

Monte Carlo methods sample the Boltzmann distribution stochastically, generating a sequence of configurations weighted by their statistical importance. For studying phase transitions:

1. Start with a lattice model (e.g., Ising model on a cubic lattice)
2. Use an update algorithm to propose and accept/reject spin flips
3. Measure thermodynamic quantities (energy, magnetization, susceptibility) as averages over configurations
4. Repeat near $T_c$ for multiple system sizes

The Metropolis algorithm is the simplest update scheme, but it suffers badly from critical slowing down. Cluster algorithms (Wolff, Swendsen-Wang) flip correlated groups of spins at once, dramatically reducing autocorrelation times near $T_c$.

Finite-size scaling

Real simulations use finite lattices of linear size $L$, but critical behavior strictly occurs only in the thermodynamic limit $L \to \infty$. Finite-size scaling bridges this gap by predicting how quantities depend on $L$:

$\chi(t, L) = L^{\gamma/\nu} \tilde{\chi}(tL^{1/\nu})$

where $\tilde{\chi}$ is a universal scaling function. By simulating at several system sizes and plotting $\chi L^{-\gamma/\nu}$ vs. $tL^{1/\nu}$, all data should collapse onto a single curve if you've chosen the right exponents. This data collapse technique is one of the most reliable ways to extract critical exponents from simulations.

Beyond equilibrium

Critical phenomena extend beyond equilibrium thermodynamics. When dynamics are included, new universal behavior emerges that isn't captured by the static exponents alone.

Critical slowing down

As $T \to T_c$, the relaxation time $\tau$ of the system diverges as $\tau \sim \xi^z \sim |t|^{-z\nu}$, where $z$ is the dynamical critical exponent. This happens because the growing correlation length means the system must rearrange over increasingly large regions to equilibrate.

Critical slowing down is a practical headache for both experiments (long equilibration waits) and simulations (high autocorrelation between samples). It can also lead to aging and history-dependent behavior when the system falls out of equilibrium near $T_c$.

Dynamical critical phenomena

The classification of dynamical universality classes, developed by Hohenberg and Halperin, depends on which quantities are conserved and how the order parameter couples to other slow modes:

• Model A: Non-conserved order parameter with purely relaxational dynamics (e.g., Ising magnet with Glauber dynamics)
• Model B: Conserved order parameter with diffusive dynamics (e.g., binary alloy undergoing phase separation)
• Model E/F/G/H: Include coupling to additional conserved densities (energy, momentum)

Each model has its own dynamical exponent $z$ and belongs to a distinct dynamical universality class. This framework connects to the broader study of phase ordering kinetics, pattern formation, and non-equilibrium statistical mechanics.