4.3 Critical exponents

Definition of critical exponents

Critical exponents characterize how physical quantities behave as a system approaches a continuous (second-order) phase transition. They capture the power-law divergences and vanishings that occur near the critical point, and they turn out to depend only on broad features like symmetry and dimensionality rather than microscopic details. This makes them central to the idea of universality in phase transitions.

Significance in phase transitions

Near a critical point, thermodynamic quantities don't change smoothly. Instead, they follow power laws: some quantities diverge (blow up), while others vanish. Critical exponents describe exactly how fast these power laws grow or shrink. The remarkable fact is that very different physical systems (a ferromagnet and a liquid-gas system, for example) can share the same set of critical exponents. This lets you classify phase transitions into universality classes based on the symmetry of the order parameter, the spatial dimensionality, and the range of interactions.

Mathematical representation

All critical exponents are defined in terms of the reduced temperature:

$t = \frac{T - T_c}{T_c}$

where $T_c$ is the critical temperature. This dimensionless quantity measures how far you are from the transition. As $t \to 0$, a physical quantity $X$ obeys a power law of the form:

$X \propto |t|^{\lambda}$

where $\lambda$ is the relevant critical exponent. The standard exponents are denoted by Greek letters: $\alpha$, $\beta$, $\gamma$, $\delta$, $\nu$, and $\eta$. Depending on which quantity you're looking at, the exponent can be positive (the quantity vanishes at $T_c$), negative (it diverges), or zero (logarithmic behavior).

Types of critical exponents

Each critical exponent tracks a different physical quantity as the system approaches criticality. Together, they give a complete picture of how a material behaves at the transition.

Order parameter exponent

The exponent $\beta$ describes how the order parameter $m$ vanishes as you approach $T_c$ from below:

$m \propto |t|^{\beta} \quad \text{for } T < T_c$

What counts as the order parameter depends on the system. In a ferromagnet, it's the spontaneous magnetization. In a liquid-gas transition, it's the density difference between liquid and gas phases. Since $\beta > 0$, the order parameter smoothly goes to zero at the critical point.

Correlation length exponent

The exponent $\nu$ governs the correlation length $\xi$, which measures the typical size of correlated fluctuations:

$\xi \propto |t|^{-\nu}$

Because the exponent appears with a minus sign, $\xi$ diverges as $T \to T_c$. Physically, this means fluctuations become correlated over arbitrarily large distances at the critical point. This diverging correlation length is what drives phenomena like critical opalescence in fluids, where density fluctuations scatter light at all wavelengths.

Susceptibility exponent

The exponent $\gamma$ describes the divergence of the susceptibility $\chi$:

$\chi \propto |t|^{-\gamma}$

Susceptibility measures how strongly the system responds to a small external field. In a ferromagnet, it's the magnetic susceptibility; in a fluid, it's the compressibility. The divergence of $\chi$ reflects the fact that near criticality, even a tiny perturbation can produce a large response because fluctuations in the order parameter become enormous.

Specific heat exponent

The exponent $\alpha$ characterizes the specific heat $C$ near the critical point:

$C \propto |t|^{-\alpha}$

The behavior here is more subtle than for other exponents:

• $\alpha > 0$: specific heat diverges (true power-law singularity)
• $\alpha = 0$: logarithmic divergence (as in the 2D Ising model)
• $\alpha < 0$: specific heat remains finite but has a cusp

This exponent reflects the nature of energy fluctuations in the system near the transition.

Critical isotherm exponent

There's one more standard exponent worth knowing. The exponent $\delta$ describes the relationship between the order parameter and the conjugate field right at $T_c$ (i.e., at $t = 0$):

$m \propto |h|^{1/\delta} \quad \text{at } T = T_c$

where $h$ is the external field. This exponent captures how nonlinear the response is exactly at criticality.

Scaling relations

The six critical exponents ($\alpha, \beta, \gamma, \delta, \nu, \eta$) are not all independent. Scaling relations connect them, so that knowing just two exponents (typically $\nu$ and $\eta$, or any two independent ones) is enough to determine the rest. These relations arise from the scaling hypothesis and thermodynamic consistency requirements.

Widom scaling

The Widom relation connects $\beta$, $\gamma$, and $\delta$:

$\gamma = \beta(\delta - 1)$

It follows from the assumption that the free energy near the critical point is a generalized homogeneous function. This relation holds across a wide range of systems, from ferromagnets to fluids.

Rushbrooke inequality

The Rushbrooke relation connects the specific heat, order parameter, and susceptibility exponents:

$\alpha + 2\beta + \gamma \geq 2$

This is derived from thermodynamic stability requirements. For systems that obey hyperscaling (which includes most physical systems below the upper critical dimension), the inequality becomes an exact equality:

$\alpha + 2\beta + \gamma = 2$

Fisher equality

The Fisher relation ties the correlation function exponents to the thermodynamic ones:

$\gamma = \nu(2 - \eta)$

Here $\eta$ is the anomalous dimension, which describes how the correlation function at criticality deviates from the naive (Ornstein-Zernike) form. At $T_c$, the equal-time correlation function decays as:

$G(r) \propto \frac{1}{r^{d-2+\eta}}$

where $d$ is the spatial dimension. Fisher's relation connects this spatial decay directly to the thermodynamic susceptibility.

Josephson (hyperscaling) relation

One additional scaling relation worth knowing is the Josephson hyperscaling relation:

$d\nu = 2 - \alpha$

where $d$ is the spatial dimension. Unlike the other scaling relations, this one explicitly involves dimensionality. It holds below the upper critical dimension $d_c$ (which is 4 for standard $\phi^4$ theory). Above $d_c$, mean field exponents take over and hyperscaling breaks down.

Universality classes

Systems with completely different microscopic physics can share the same critical exponents if they have the same spatial dimensionality, order parameter symmetry, and range of interactions. Each such group is called a universality class.

Ising model

• Symmetry: Discrete $Z_2$ (up/down)
• Physical systems: Uniaxial ferromagnets, binary alloys, liquid-gas transitions
• 3D critical exponents: $\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$, $\alpha \approx 0.110$
• The 2D Ising model was solved exactly by Onsager, giving $\beta = 1/8$, $\gamma = 7/4$, $\nu = 1$, and $\alpha = 0$ (logarithmic). These exact results serve as a crucial benchmark for all theoretical methods.

XY model

• Symmetry: Continuous $O(2)$ (planar rotational symmetry)
• Physical systems: Superfluid helium (the lambda transition), thin-film superconductors, easy-plane magnets
• 3D critical exponents: $\beta \approx 0.348$, $\gamma \approx 1.316$, $\nu \approx 0.672$
• In 2D, the XY model does not have a conventional phase transition. Instead, it exhibits the Kosterlitz-Thouless transition, a topological transition driven by vortex-antivortex unbinding, with no standard power-law critical exponents.

Heisenberg model

• Symmetry: Continuous $O(3)$ (full rotational symmetry in spin space)
• Physical systems: Isotropic ferromagnets, antiferromagnets, some liquid crystals
• 3D critical exponents: $\beta \approx 0.366$, $\gamma \approx 1.395$, $\nu \approx 0.707$
• No exact solution exists. Results come from renormalization group calculations, high-temperature series expansions, and Monte Carlo simulations.

Notice the trend: as the symmetry of the order parameter increases (from $Z_2$ to $O(2)$ to $O(3)$), $\beta$ and $\nu$ increase while $\gamma$ also increases. More continuous symmetry means fluctuations are "easier" (more directions to fluctuate in), which systematically shifts the exponents.

Experimental determination

Measuring critical exponents requires getting very close to $T_c$ while maintaining precise control over temperature and other variables. Several practical challenges arise: sample impurities can shift or smear the transition, finite sample sizes cut off the divergence of the correlation length, and critical slowing down (relaxation times diverging near $T_c$) makes equilibration difficult.

Scattering techniques

Scattering experiments probe spatial correlations directly. The key idea is that the scattering cross-section is proportional to the structure factor $S(q)$, which is the Fourier transform of the correlation function.

• Neutron scattering is the primary tool for magnetic systems, since neutrons couple to spin. It can measure both the correlation length (from the width of the scattering peak, giving $\nu$) and the anomalous dimension $\eta$ (from the peak shape at $T_c$).
• X-ray scattering probes electron density correlations and is used for structural transitions.
• Light scattering is effective for fluid systems near the liquid-gas critical point, where density fluctuations scatter visible light (critical opalescence).

Thermodynamic measurements

Bulk measurements target the thermodynamic exponents directly:

• Calorimetry measures specific heat to extract $\alpha$. High-resolution AC calorimetry is often needed because the singularity can be weak (logarithmic for $\alpha = 0$).
• Magnetometry (SQUID, vibrating sample) measures magnetization vs. temperature to get $\beta$, and susceptibility vs. temperature to get $\gamma$.
• PVT measurements in fluids determine the coexistence curve ($\beta$) and compressibility ($\gamma$).

Extracting clean power-law behavior requires fitting data over a range of reduced temperatures, typically $10^{-4} < |t| < 10^{-1}$. Too far from $T_c$ and corrections to scaling matter; too close and finite-size or impurity effects dominate.

Renormalization group theory

The renormalization group (RG) provides the theoretical foundation for universality and scaling. It explains why systems with different microscopic physics share the same critical exponents, and it gives a systematic method for calculating those exponents.

Wilson's approach

Kenneth Wilson's key insight was to treat the problem by progressively integrating out short-wavelength fluctuations. The procedure works in three steps:

1. Coarse-grain: Integrate out degrees of freedom at length scales shorter than some cutoff.
2. Rescale: Shrink the system back to its original size so you can compare with the original Hamiltonian.
3. Renormalize: Adjust (renormalize) the coupling constants so the long-wavelength physics is preserved.

Repeating this process generates a flow in the space of coupling constants. Near the critical point, the system is scale-invariant, meaning it looks statistically the same at all length scales. This is the physical origin of power-law behavior.

Fixed points and critical behavior

The RG flow has fixed points where the coupling constants don't change under further coarse-graining. These fixed points correspond to universality classes.

• Relevant operators grow under the RG flow and drive the system away from the fixed point. The reduced temperature $t$ and external field $h$ are the standard relevant operators.
• Irrelevant operators shrink under the RG flow. They affect corrections to scaling but don't change the leading critical exponents.

Critical exponents are determined by the eigenvalues of the linearized RG transformation around the fixed point. If $y_t$ and $y_h$ are the RG eigenvalues associated with temperature and field, then:

$\nu = \frac{1}{y_t}, \quad \beta = \frac{d - y_h}{y_t}, \quad \gamma = \frac{2y_h - d}{y_t}$

All scaling relations follow automatically from these two eigenvalues, which is why only two critical exponents are independent.

Mean field theory vs exact results

Mean field theory gives a useful first approximation to critical behavior, but it systematically gets the exponents wrong because it ignores fluctuations. Comparing mean field predictions with exact or numerical results highlights just how important fluctuations are.

Limitations of mean field theory

Mean field theory replaces the fluctuating local environment with a uniform average field. This yields the classical (mean field) critical exponents:

$\beta = \frac{1}{2}, \quad \gamma = 1, \quad \nu = \frac{1}{2}, \quad \alpha = 0 \text{ (jump)}, \quad \delta = 3, \quad \eta = 0$

These are exact above the upper critical dimension $d_c = 4$, where fluctuations become negligible relative to the mean. Below $d_c$, mean field theory fails quantitatively. For example, in the 3D Ising model, the true $\beta \approx 0.326$ vs. the mean field value of $0.5$. Mean field theory also fails qualitatively in low dimensions: it predicts a phase transition in the 1D Ising model, which doesn't actually have one.

Beyond mean field approximations

Several methods systematically improve on mean field theory:

• Epsilon expansion: Wilson and Fisher showed that critical exponents can be computed as a power series in $\epsilon = 4 - d$, where $d$ is the spatial dimension. Setting $\epsilon = 1$ gives approximate 3D exponents. For example, to first order, $\nu = 1/2 + \epsilon/12 + \ldots$
• Exact solutions: The 2D Ising model (Onsager) provides exact exponents that serve as a benchmark. No exact solution exists for the 3D case.
• Monte Carlo simulations: Generate statistical samples of configurations and use finite-size scaling to extract exponents. Modern cluster algorithms (Wolff, Swendsen-Wang) overcome critical slowing down.
• Non-perturbative methods: The functional renormalization group and conformal bootstrap have pushed the precision of 3D Ising exponents to six significant figures in recent years.

Critical phenomena in real systems

The universality framework has been confirmed across a wide range of physical systems. Here are the major examples.

Liquid-gas transitions

The liquid-gas critical point is the classic example. Near the critical temperature and pressure, the density difference between liquid and gas phases vanishes as a power law with $\beta \approx 0.326$. The system belongs to the 3D Ising universality class because the order parameter (density difference) has a scalar, discrete symmetry. Critical opalescence, where the fluid becomes milky white, is a direct visual signature of the diverging correlation length.

Ferromagnetic transitions

Below the Curie temperature, a ferromagnet develops spontaneous magnetization. The universality class depends on the spin symmetry:

• Uniaxial magnets (strong crystal-field anisotropy forcing spins along one axis): 3D Ising class
• Easy-plane magnets (spins confined to a plane): 3D XY class
• Isotropic magnets (no preferred direction): 3D Heisenberg class

Neutron scattering is particularly powerful here because it directly probes the spin-spin correlation function, giving access to $\nu$ and $\eta$ in addition to the thermodynamic exponents.

Superconducting transitions

The superconducting order parameter is a complex scalar (the pair condensate wave function), which has $O(2)$ symmetry. The zero-field transition in a bulk superconductor therefore belongs to the 3D XY universality class. In type-II superconductors in a magnetic field, the situation is more complex: the vortex lattice melting transition and the upper critical field transition involve additional physics (gauge fluctuations, vortex pinning) that can modify or obscure the critical behavior.

Finite-size effects

Real experiments and numerical simulations always deal with finite systems. When the correlation length $\xi$ grows to be comparable to the system size $L$, the system can no longer exhibit true singular behavior, and the sharp phase transition gets rounded and shifted.

Scaling in finite systems

Finite-size scaling theory handles this by introducing the ratio $L/\xi$ as an additional scaling variable. A thermodynamic quantity $Q$ near the critical point takes the form:

$Q(t, L) = L^{x/\nu} \tilde{Q}(L^{1/\nu} t)$

where $x$ is the appropriate critical exponent for $Q$ and $\tilde{Q}$ is a universal scaling function. By measuring $Q$ for several system sizes $L$ and collapsing the data onto a single curve, you can extract both $T_c$ and the critical exponents.

Numerical simulations

Monte Carlo simulations are the workhorse for computing critical exponents numerically. The standard procedure is:

1. Simulate the model at several temperatures near the expected $T_c$ for multiple system sizes $L$.
2. Compute observables (order parameter, susceptibility, Binder cumulant, etc.) as functions of temperature.
3. Use finite-size scaling to collapse data from different $L$ values onto universal curves.
4. Extract $T_c$ and critical exponents from the best collapse or from the size-dependence of peak heights and positions.

Cluster algorithms (Wolff, Swendsen-Wang) are essential near criticality because they flip correlated clusters of spins in a single move, dramatically reducing critical slowing down compared to single-spin-flip algorithms like Metropolis.

Critical dynamics

So far, all the exponents discussed are static: they describe equilibrium properties. But near a critical point, the dynamics also become anomalous. Relaxation times diverge, and transport coefficients develop singular behavior.

Dynamic critical exponent

The dynamic critical exponent $z$ relates the characteristic relaxation time $\tau$ to the correlation length:

$\tau \propto \xi^z \propto |t|^{-\nu z}$

The value of $z$ depends not just on the universality class but also on the dynamics: whether the order parameter is conserved or not, and whether it couples to other slow modes (like energy density or momentum). For example, the 3D Ising model with non-conserved order parameter (Model A in the Hohenberg-Halperin classification) has $z \approx 2.0$, while the conserved case (Model B) has $z \approx 4 - \eta$.

Time-dependent correlation functions

Near criticality, the time-dependent correlation function obeys a scaling form:

$C(r, t_{\text{time}}) = \frac{1}{r^{d-2+\eta}} \tilde{C}\!\left(\frac{r}{\xi}, \frac{t_{\text{time}}}{\tau}\right)$

These are measured experimentally through:

• Dynamic light scattering (photon correlation spectroscopy) for fluids
• Neutron spin echo spectroscopy for magnetic systems

Both techniques probe the time decay of fluctuations at a given wavevector, giving direct access to the relaxation rate and hence $z$.