Part III: Phase Transitions | Chapter 4

Critical Phenomena

Critical exponents, scaling relations, universality classes, correlation functions, and the Ginzburg criterion

Historical Context

The study of critical phenomena underwent a revolution in the 1960s and 1970s. Experiments by Guggenheim (1945) showed that the liquid-gas coexistence curve follows a universal power law with exponent $\beta \approx 1/3$, different from the mean-field value of 1/2. Ben Widom proposed scaling relations in 1965, Leo Kadanoff introduced block spin ideas in 1966, and Kenneth Wilson developed the renormalization group in 1971, earning the 1982 Nobel Prize.

The discovery of universality -- that systems as different as magnets, fluids, and binary alloys share the same critical exponents -- was one of the most profound insights in statistical physics. It revealed that near a critical point, the only relevant features are symmetry, dimensionality, and the range of interactions.

1. The Six Critical Exponents

Derivation 1: Definitions and Physical Meaning

Near the critical point, thermodynamic quantities exhibit power-law singularities. Define the reduced temperature $t = (T - T_c)/T_c$:

Critical Exponent Definitions

$\alpha$: Specific heat: $C \sim |t|^{-\alpha}$ (both sides of $T_c$)

$\beta$: Order parameter: $m \sim (-t)^{\beta}$ ($T < T_c$, $h = 0$)

$\gamma$: Susceptibility: $\chi \sim |t|^{-\gamma}$ (both sides)

$\delta$: Critical isotherm: $m \sim h^{1/\delta}$ (at $T = T_c$)

$\nu$: Correlation length: $\xi \sim |t|^{-\nu}$

$\eta$: Correlation function at $T_c$: $G(r) \sim r^{-(d-2+\eta)}$

The correlation function deserves special attention. Away from $T_c$, it decays exponentially: $G(r) = \langle s_0 s_r\rangle - \langle s\rangle^2 \sim r^{-(d-2+\eta)}e^{-r/\xi}$. At $T_c$, the correlation length diverges ($\xi \to \infty$) and the decay becomes purely algebraic -- the system is scale-invariant, with correlations (and fluctuations) at all length scales.

2. Scaling Relations

Derivation 2: Widom and Hyperscaling Relations

The six exponents are not independent. Widom's scaling hypothesis states that the singular part of the free energy is a generalized homogeneous function:

\[f_s(t, h) = |t|^{2-\alpha}\,\Phi_{\pm}\left(\frac{h}{|t|^{\beta\delta}}\right)\]

This immediately yields four scaling relations:

\[\boxed{\alpha + 2\beta + \gamma = 2 \qquad (\text{Rushbrooke})}\]

\[\boxed{\gamma = \beta(\delta - 1) \qquad (\text{Widom})}\]

The hyperscaling relations connect thermodynamic exponents to the spatial dimension $d$through the correlation length:

\[\boxed{2 - \alpha = d\nu \qquad (\text{Josephson})}\]

\[\boxed{\gamma = (2 - \eta)\nu \qquad (\text{Fisher})}\]

These four relations reduce six exponents to only two independent ones. Hyperscaling holds below the upper critical dimension $d_c = 4$ but is violated above it (where mean-field exponents apply).

Let us verify for the 2D Ising model: $\alpha = 0, \beta = 1/8, \gamma = 7/4, \delta = 15, \nu = 1, \eta = 1/4$.

Rushbrooke: $0 + 2(1/8) + 7/4 = 2$ (check)
Widom: $7/4 = (1/8)(15 - 1) = 7/4$ (check)
Josephson: $2 - 0 = 2 \cdot 1 = 2$ (check)
Fisher: $7/4 = (2 - 1/4) \cdot 1 = 7/4$ (check)

3. Universality Classes

Derivation 3: What Determines the Universality Class

A universality class is determined by three features:

Spatial dimensionality $d$
Order parameter dimensionality $n$ (Ising: $n=1$; XY: $n=2$; Heisenberg: $n=3$)
Range of interactions (short-range vs long-range)

Microscopic details (lattice structure, coupling strength, chemical composition) are irrelevant -- they affect $T_c$ but not the critical exponents. This explains why the liquid-gas critical point of water, the Curie point of iron, and the demixing point of a binary alloy all share the 3D Ising exponents.

Common Universality Classes (d = 3)

Ising (n=1): $\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$ -- liquid-gas, uniaxial magnets, binary alloys

XY (n=2): $\beta \approx 0.348$, $\gamma \approx 1.317$, $\nu \approx 0.672$ -- superfluid He-4, superconductors, planar magnets

Heisenberg (n=3): $\beta \approx 0.366$, $\gamma \approx 1.396$, $\nu \approx 0.711$ -- isotropic magnets (EuO, Ni)

Mean-field (d ≥ 4): $\beta = 1/2$, $\gamma = 1$, $\nu = 1/2$ -- all models above upper critical dimension

4. The Ginzburg Criterion

Derivation 4: When Do Fluctuations Matter?

Mean-field theory fails when fluctuations become as large as the mean. Consider a correlation volume $\xi^d$. The fluctuation in the order parameter within this volume is:

\[\langle(\delta\phi)^2\rangle_{\xi^d} \sim \frac{k_BT_c}{\xi^d \cdot |a|} \sim \frac{1}{\xi^d |t|}\]

The mean-field order parameter squared is $\phi_0^2 \sim |t|$. Fluctuations dominate when:

\[\frac{\langle(\delta\phi)^2\rangle}{\phi_0^2} \sim \frac{1}{\xi^d |t|^2} \sim \frac{|t|^{d\nu - 2}}{1} \gtrsim 1\]

Using the mean-field value $\nu = 1/2$:

\[\boxed{|t| \lesssim t_G \sim \left(\frac{k_BT_c}{E_0}\right)^{2/(4-d)}}\]

where $E_0$ is a microscopic energy scale. For $d > 4$,$t_G \to 0$: mean-field theory is self-consistent. For $d < 4$, there is always a critical region where fluctuations dominate. This explains why$d_c = 4$ is the upper critical dimension.

5. Scaling of the Correlation Function

Derivation 5: Ornstein-Zernike Form and Beyond

In Fourier space, the mean-field (Ornstein-Zernike) correlation function is:

\[\hat{G}(q) = \frac{k_BT\chi}{1 + q^2\xi^2}\]

In real space (for $d = 3$): $G(r) \sim e^{-r/\xi}/r$. At the critical point, $\xi \to \infty$ and:

\[\hat{G}(q) \sim \frac{1}{q^{2-\eta}}, \qquad G(r) \sim \frac{1}{r^{d-2+\eta}}\]

The anomalous dimension $\eta$ characterizes the deviation from the mean-field ($\eta = 0$) form. For the 3D Ising model, $\eta \approx 0.036$: a small but measurable correction verified by neutron scattering experiments on magnets.

The scaling form of the full correlation function is:

\[G(r, t) = \frac{1}{r^{d-2+\eta}}\,g\left(\frac{r}{\xi}\right)\]

where $g(x)$ is a universal scaling function with $g(0) = \text{const}$and $g(x) \sim e^{-x}$ for $x \gg 1$.

6. Applications

Key Applications

Data collapse: Plotting experimental data as $m|t|^{-\beta}$vs $h|t|^{-\beta\delta}$ should collapse data from different temperatures onto a single universal curve -- a stringent test of scaling.

Polymer physics: Self-avoiding walks in $d$ dimensions belong to the $n = 0$ universality class (de Gennes). The end-to-end distance scales as $R \sim N^{\nu}$ with $\nu \approx 0.588$ in 3D.

Percolation: The percolation threshold is a geometric phase transition with its own set of critical exponents, forming a distinct universality class.

QCD phase transition: The deconfinement transition in quantum chromodynamics at high temperature is expected to be in the 3D Ising universality class for 2+1 quark flavors.

7. Computational Exploration

This simulation visualizes critical exponents, scaling relations, universality class comparisons, and the correlation function near criticality.

Critical Phenomena: Exponents, Scaling Relations, Universality Classes, and Correlations

Python

script.py174 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Critical Phenomena: Exponents, Scaling, and Universality
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Critical Phenomena and Universality', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#1a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('#fca5a5')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Order parameter with different beta values ---
ax1 = axes[0, 0]
t_neg = np.linspace(0.001, 1.0, 200)  # |t| = (Tc - T)/Tc

universality_classes = [
    ("Mean-field", 0.5, '#f87171'),
    ("3D Ising", 0.326, '#60a5fa'),
    ("3D XY", 0.348, '#34d399'),
    ("3D Heisenberg", 0.366, '#a78bfa'),
    ("2D Ising", 0.125, '#fbbf24'),
]

for name, beta, color in universality_classes:
    m = t_neg**beta
    ax1.plot(t_neg, m, color=color, linewidth=2, label=name + ' (beta=' + str(beta) + ')')

ax1.set_xlabel('|t| = (T_c - T) / T_c')
ax1.set_ylabel('m / m_0')
ax1.set_title('Order Parameter: m ~ |t|^beta')
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Susceptibility with different gamma values ---
ax2 = axes[0, 1]
t_pos = np.linspace(0.01, 1.0, 200)  # t = (T - Tc)/Tc

classes_gamma = [
    ("Mean-field", 1.0, '#f87171'),
    ("3D Ising", 1.237, '#60a5fa'),
    ("3D XY", 1.317, '#34d399'),
    ("3D Heisenberg", 1.396, '#a78bfa'),
    ("2D Ising", 1.75, '#fbbf24'),
]

for name, gamma, color in classes_gamma:
    chi = t_pos**(-gamma)
    ax2.loglog(t_pos, chi, color=color, linewidth=2, label=name + ' (gamma=' + str(gamma) + ')')

ax2.set_xlabel('t = (T - T_c) / T_c')
ax2.set_ylabel('chi / chi_0')
ax2.set_title('Susceptibility: chi ~ |t|^(-gamma)')
ax2.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Correlation function G(r) ---
ax3 = axes[1, 0]
r = np.linspace(0.5, 30, 300)

# At T = Tc: G ~ 1/r^(d-2+eta)
d = 3
for eta, color, label in [(0.0, '#f87171', 'Mean-field (eta=0)'),
                           (0.036, '#60a5fa', '3D Ising (eta=0.036)'),
                           (0.25, '#fbbf24', '2D Ising (eta=0.25)')]:
    G_crit = r**(-(d - 2 + eta))
    ax3.loglog(r, G_crit / G_crit[0], color=color, linewidth=2, label=label)

# Away from Tc: exponential decay
for xi, ls in [(5.0, '--'), (10.0, ':')]:
    G_off = np.exp(-r / xi) / r
    G_off = G_off / G_off[0]
    ax3.loglog(r, G_off, color='white', linewidth=1, alpha=0.4, linestyle=ls,
               label='Off-critical xi=' + str(int(xi)))

ax3.set_xlabel('Distance r')
ax3.set_ylabel('G(r) / G(r_0)')
ax3.set_title('Correlation Function (d=3)')
ax3.set_ylim(1e-5, 2)
ax3.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Scaling relations verification ---
ax4 = axes[1, 1]

# For each universality class, compute and verify scaling relations
classes_full = {
    "Mean-field": {"alpha": 0, "beta": 0.5, "gamma": 1.0, "delta": 3.0, "nu": 0.5, "eta": 0.0, "d": 4},
    "2D Ising": {"alpha": 0, "beta": 0.125, "gamma": 1.75, "delta": 15.0, "nu": 1.0, "eta": 0.25, "d": 2},
    "3D Ising": {"alpha": 0.110, "beta": 0.326, "gamma": 1.237, "delta": 4.789, "nu": 0.630, "eta": 0.036, "d": 3},
    "3D XY": {"alpha": -0.015, "beta": 0.348, "gamma": 1.317, "delta": 4.780, "nu": 0.672, "eta": 0.038, "d": 3},
    "3D Heisen.": {"alpha": -0.133, "beta": 0.366, "gamma": 1.396, "delta": 4.814, "nu": 0.711, "eta": 0.037, "d": 3},
}

# Plot bars showing how well each scaling relation is satisfied
relations = {
    "Rushbrooke": lambda e: e["alpha"] + 2*e["beta"] + e["gamma"],  # should = 2
    "Widom": lambda e: e["gamma"] / (e["beta"] * (e["delta"] - 1)),  # should = 1
    "Josephson": lambda e: (2 - e["alpha"]) / (e["d"] * e["nu"]),  # should = 1
    "Fisher": lambda e: e["gamma"] / ((2 - e["eta"]) * e["nu"]),  # should = 1
}

x_pos = np.arange(len(relations))
width = 0.15
offsets = np.linspace(-2*width, 2*width, 5)
colors_bar = ['#f87171', '#fbbf24', '#60a5fa', '#34d399', '#a78bfa']

for i, (cls_name, exps) in enumerate(classes_full.items()):
    vals = []
    for rel_name, rel_func in relations.items():
        if rel_name == "Rushbrooke":
            vals.append(rel_func(exps) / 2.0)  # normalize to 1
        else:
            vals.append(rel_func(exps))
    ax4.bar(x_pos + offsets[i], vals, width, color=colors_bar[i], alpha=0.7, label=cls_name)

ax4.axhline(y=1.0, color='#fbbf24', linestyle='--', alpha=0.7, linewidth=2)
ax4.set_xticks(x_pos)
ax4.set_xticklabels(list(relations.keys()), fontsize=9, color='white')
ax4.set_ylabel('Relation value (should = 1)')
ax4.set_title('Scaling Relations Verification')
ax4.set_ylim(0.9, 1.1)
ax4.legend(fontsize=6, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white', ncol=2)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

# Numerical results
print("\n" + "=" * 60)
print("CRITICAL PHENOMENA: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Critical Exponents by Universality Class ---")
header = "  " + "Class".ljust(16) + "alpha".ljust(8) + "beta".ljust(8) + "gamma".ljust(8) + "delta".ljust(8) + "nu".ljust(8) + "eta"
print(header)
for cls_name, exps in classes_full.items():
    row = "  " + cls_name.ljust(16)
    row += str(exps["alpha"]).ljust(8)
    row += str(exps["beta"]).ljust(8)
    row += str(exps["gamma"]).ljust(8)
    row += str(exps["delta"]).ljust(8)
    row += str(exps["nu"]).ljust(8)
    row += str(exps["eta"])
    print(row)

print("\n--- Scaling Relations Verification ---")
for cls_name, exps in classes_full.items():
    print("  " + cls_name + ":")
    rush = exps["alpha"] + 2*exps["beta"] + exps["gamma"]
    widom = exps["gamma"] - exps["beta"]*(exps["delta"] - 1)
    joseph = 2 - exps["alpha"] - exps["d"]*exps["nu"]
    fisher = exps["gamma"] - (2 - exps["eta"])*exps["nu"]
    print("    Rushbrooke: alpha + 2*beta + gamma = " + str(round(rush, 4)) + " (should = 2)")
    print("    Widom: gamma - beta*(delta-1) = " + str(round(widom, 4)) + " (should = 0)")
    print("    Josephson: 2-alpha - d*nu = " + str(round(joseph, 4)) + " (should = 0)")
    print("    Fisher: gamma - (2-eta)*nu = " + str(round(fisher, 4)) + " (should = 0)")

print("\n--- Ginzburg Criterion ---")
print("  Upper critical dimension: d_c = 4")
print("  For d < 4: critical region |t| < t_G where mean-field fails")
print("  Conventional superconductors: t_G ~ 10^-14 (MF excellent)")
print("  High-Tc superconductors: t_G ~ 10^-2 (fluctuations important)")
print("  Liquid-gas: t_G ~ 1 (MF poor near Tc)")
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Summary and Key Results

Core Formulas

Rushbrooke: $\alpha + 2\beta + \gamma = 2$
Widom: $\gamma = \beta(\delta - 1)$
Josephson: $2 - \alpha = d\nu$
Fisher: $\gamma = (2 - \eta)\nu$

Physical Insights

Only 2 independent exponents (out of 6)
Universality: exponents depend on $d$, $n$, range only
At $T_c$: scale invariance, algebraic correlations
Ginzburg: MF fails for $|t| < t_G$ when $d < 4$

Practice Problems

Problem 1: Critical Exponents from Landau TheoryThe Landau free energy is $F = F_0 + a(T-T_c)\phi^2 + b\phi^4 - h\phi$ with $a, b > 0$. Derive the critical exponents $\beta$, $\gamma$, and $\delta$ from mean-field theory.

Solution:

1. Minimize $F$ with respect to $\phi$ at $h = 0$: $\partial F/\partial\phi = 2a t\,\phi + 4b\phi^3 = 0$ where $t = (T-T_c)/T_c$:

$$\phi(2at + 4b\phi^2) = 0$$

2. For $t < 0$ (below $T_c$), the non-trivial solution is:

$$\phi^2 = \frac{-at}{2b} = \frac{a|t|}{2b} \implies \phi \propto |t|^{1/2}$$

$$\boxed{\beta = 1/2}$$

3. For the susceptibility, take $\partial^2 F/\partial\phi^2 = 2at + 12b\phi^2$. Above $T_c$ ($\phi = 0$): $\chi^{-1} = 2at$, so $\chi \propto t^{-1}$. Below $T_c$: $\chi^{-1} = 2at + 12b(-at/2b) = -4at = 4a|t|$, so $\chi \propto |t|^{-1}$:

$$\boxed{\gamma = 1}$$

4. At $T = T_c$ ($t = 0$), minimize with $h \neq 0$: $4b\phi^3 = h$, giving $\phi \propto h^{1/3}$:

$$\boxed{\delta = 3}$$

5. Verify the Widom scaling relation: $\gamma = \beta(\delta - 1) = \frac{1}{2}(3 - 1) = 1$. Confirmed. These mean-field exponents ($\beta = 1/2$, $\gamma = 1$, $\delta = 3$) are exact above the upper critical dimension $d_c = 4$ but receive corrections for $d < 4$ due to fluctuations.

Problem 2: Scaling Relation VerificationThe 3D Ising model has exponents $\alpha \approx 0.110$, $\beta \approx 0.326$, $\gamma \approx 1.237$, $\delta \approx 4.789$, $\nu \approx 0.630$, $\eta \approx 0.036$. Verify the four scaling relations.

Solution:

1. Rushbrooke's identity ($\alpha + 2\beta + \gamma = 2$):

$$0.110 + 2(0.326) + 1.237 = 0.110 + 0.652 + 1.237 = 1.999 \approx 2 \quad \checkmark$$

2. Widom's identity ($\gamma = \beta(\delta - 1)$):

$$\beta(\delta - 1) = 0.326(4.789 - 1) = 0.326 \times 3.789 = 1.235 \approx 1.237 \quad \checkmark$$

3. Josephson's (hyperscaling) relation ($2 - \alpha = d\nu$, with $d = 3$):

$$2 - 0.110 = 1.890, \quad d\nu = 3 \times 0.630 = 1.890 \quad \checkmark$$

4. Fisher's identity ($\gamma = (2 - \eta)\nu$):

$$(2 - 0.036)(0.630) = 1.964 \times 0.630 = 1.237 \quad \checkmark$$

5. All four relations are satisfied:

$$\boxed{\alpha + 2\beta + \gamma = 2, \quad \gamma = \beta(\delta-1), \quad 2-\alpha = d\nu, \quad \gamma = (2-\eta)\nu}$$

These are not independent: only 2 of the 6 exponents are independent. The scaling relations follow from the assumption that the free energy near $T_c$ is a generalized homogeneous function. Hyperscaling ($d\nu = 2 - \alpha$) involves the spatial dimension $d$ explicitly and breaks down above $d = 4$.

Problem 3: Ginzburg CriterionUsing the Ginzburg criterion, estimate the reduced temperature $t_G$ below which mean-field theory fails for a 3D system with correlation length $\xi_0 = 3$ angstroms and lattice spacing $a = 3$ angstroms. Compare with a superconductor where $\xi_0 = 1000$ angstroms.

Solution:

1. The Ginzburg criterion states that mean-field theory breaks down when fluctuations in the order parameter within a correlation volume become comparable to the mean-field value. This gives:

$$t_G \sim \left(\frac{a}{\xi_0}\right)^{2d/(4-d)}$$

2. For $d = 3$:

$$t_G \sim \left(\frac{a}{\xi_0}\right)^6$$

3. For the magnetic system ($\xi_0 = a = 3$ angstroms):

$$\boxed{t_G \sim \left(\frac{3}{3}\right)^6 = 1}$$

This means mean-field theory fails over essentially the entire critical region — fluctuations dominate at all temperatures near $T_c$.

4. For the superconductor ($\xi_0 = 1000$ angstroms, $a = 3$ angstroms):

$$\boxed{t_G \sim \left(\frac{3}{1000}\right)^6 = (3 \times 10^{-3})^6 \approx 7.3 \times 10^{-16}}$$

5. For the superconductor, $t_G$ is astronomically small: mean-field (BCS/Ginzburg-Landau) theory is essentially exact in conventional superconductors. This is because the large Cooper pair size ($\xi_0 \gg a$) means many pairs overlap, suppressing fluctuations. In contrast, high-$T_c$ superconductors have smaller $\xi_0 \sim 10$ angstroms and larger $t_G$, making fluctuation effects observable.

Problem 4: Correlation Length DivergenceNear the critical point, the correlation length diverges as $\xi = \xi_0|t|^{-\nu}$. For the 3D Ising model ($\nu = 0.630$, $\xi_0 = 2$ angstroms), calculate $\xi$ at $t = 10^{-2}$, $10^{-4}$, and $10^{-6}$. At what $t$ does $\xi$ reach 1 $\mu$m?

Solution:

1. Apply the correlation length formula $\xi = \xi_0|t|^{-\nu}$ at each temperature:

2. At $t = 10^{-2}$:

$$\xi = 2\;\text{\AA} \times (10^{-2})^{-0.630} = 2 \times 10^{1.26} = 2 \times 18.2 = 36.4\;\text{\AA} = 3.64\;\text{nm}$$

3. At $t = 10^{-4}$:

$$\xi = 2 \times (10^{-4})^{-0.630} = 2 \times 10^{2.52} = 2 \times 331 = 662\;\text{\AA} = 66.2\;\text{nm}$$

4. At $t = 10^{-6}$:

$$\xi = 2 \times (10^{-6})^{-0.630} = 2 \times 10^{3.78} = 2 \times 6026 = 12{,}050\;\text{\AA} \approx 1.2\;\mu\text{m}$$

5. For $\xi = 1\;\mu\text{m} = 10^4$ angstroms, solve $10^4 = 2 \times |t|^{-0.630}$:

$$|t|^{-0.630} = 5000 \implies |t| = 5000^{-1/0.630} = 5000^{-1.587}$$

$$\log|t| = -1.587\log 5000 = -1.587 \times 3.699 = -5.87$$

$$\boxed{|t| \approx 1.3 \times 10^{-6} \implies |T - T_c| \approx 1.3 \times 10^{-6}\,T_c}$$

For iron ($T_c = 1043$ K), this means $|T - T_c| \approx 1.4$ mK to reach micron-scale correlations. The power-law divergence with non-classical exponent $\nu = 0.630 > 0.5$ (mean-field) reflects the role of fluctuations in 3D.

Problem 5: Universality and the Specific Heat ExponentA fluid near its critical point has measured exponents $\beta = 0.325$ and $\gamma = 1.24$. Using scaling relations, predict $\alpha$, $\delta$, and $\nu$. Which universality class does this system belong to?

Solution:

1. From the Rushbrooke relation, find $\alpha$:

$$\alpha = 2 - 2\beta - \gamma = 2 - 2(0.325) - 1.24 = 2 - 0.650 - 1.24$$

$$\boxed{\alpha = 0.110}$$

2. From the Widom relation, find $\delta$:

$$\delta = 1 + \frac{\gamma}{\beta} = 1 + \frac{1.24}{0.325} = 1 + 3.815$$

$$\boxed{\delta = 4.82}$$

3. From the Josephson relation ($d = 3$), find $\nu$:

$$\nu = \frac{2 - \alpha}{d} = \frac{2 - 0.110}{3} = \frac{1.890}{3}$$

$$\boxed{\nu = 0.630}$$

4. From Fisher's relation, find $\eta$:

$$\eta = 2 - \frac{\gamma}{\nu} = 2 - \frac{1.24}{0.630} = 2 - 1.968 = 0.032$$

5. The complete set of exponents matches the 3D Ising universality class ($n = 1$ scalar order parameter, short-range interactions):

$$\boxed{\alpha = 0.110,\;\beta = 0.325,\;\gamma = 1.24,\;\delta = 4.82,\;\nu = 0.630,\;\eta = 0.032}$$

The liquid-gas critical point belongs to the Ising universality class because both have a scalar order parameter ($n = 1$: magnetization or density difference), $\mathbb{Z}_2$ symmetry, and live in $d = 3$. This is universality in action: the critical exponents depend only on $(d, n)$ and symmetry, not on microscopic details.

← Mean-Field Theory Renormalization Group →