Part III: Phase Transitions | Chapter 3

Mean-Field Theory

Weiss molecular field, Bragg-Williams approximation, Curie-Weiss law, self-consistency equations, and Landau expansion

Historical Context

Pierre-Ernest Weiss introduced the molecular field hypothesis in 1907 to explain ferromagnetism. He proposed that each magnetic moment experiences an effective field proportional to the average magnetization -- this self-consistent approach was the first mean-field theory. Bragg and Williams independently developed an equivalent variational approach in 1934 for order-disorder transitions in alloys.

Mean-field theory remains the starting point for understanding phase transitions. While it gives incorrect critical exponents in low dimensions, it becomes exact in high dimensions (\(d \ge 4\) for short-range interactions) and on fully connected lattices. The theory also provides the foundation for density functional theory in liquids and the BCS theory of superconductivity.

1. The Weiss Molecular Field

Derivation 1: Self-Consistency Equation

Consider the Ising model \(H = -J\sum_{\langle ij\rangle}s_is_j - h\sum_i s_i\). The Weiss approximation replaces the interaction of spin \(s_i\) with its neighbors by an effective field. Write \(s_j = m + (s_j - m)\) where \(m = \langle s_j\rangle\):

\[s_is_j = s_i[m + (s_j - m)] = ms_i + s_i(s_j - m)\]

Neglecting the fluctuation term \(s_i(s_j - m)\), the Hamiltonian becomes:

\[H_{\text{MF}} = -\sum_i(qJm + h)s_i + \frac{NqJm^2}{2}\]

where \(q\) is the coordination number (number of nearest neighbors). Each spin sees an effective field \(h_{\text{eff}} = qJm + h\). The self-consistency equation is:

\[\boxed{m = \tanh\left(\frac{qJm + h}{k_BT}\right)}\]

At \(h = 0\), this has the trivial solution \(m = 0\) for all \(T\). A nontrivial solution exists when the slope of \(\tanh(qJm/k_BT)\) at \(m = 0\)exceeds 1:

\[\frac{qJ}{k_BT} > 1 \implies T < T_c = \frac{qJ}{k_B}\]

2. The Curie-Weiss Law

Derivation 2: Susceptibility Above T_c

Above \(T_c\), for small \(m\) and small \(h\), linearize the self-consistency equation. Since \(\tanh(x) \approx x\) for small \(x\):

\[m \approx \frac{qJm + h}{k_BT} = \frac{T_c}{T}m + \frac{h}{k_BT}\]

Solving for \(m\):

\[m = \frac{h}{k_B(T - T_c)}\]

The susceptibility is:

\[\boxed{\chi = \frac{\partial m}{\partial h} = \frac{1}{k_B(T - T_c)} = \frac{C}{T - T_c}}\]

This is the Curie-Weiss law with Curie constant \(C = 1/k_B\). It diverges as\(T \to T_c^+\) with exponent \(\gamma = 1\). The law accurately describes paramagnetic susceptibility well above \(T_c\) but fails in the critical region.

3. Bragg-Williams Approximation

Derivation 3: Variational Free Energy

The Bragg-Williams approach constructs the free energy directly. For \(N\) spins with magnetization \(m = (N_+ - N_-)/N\), where \(N_{\pm} = N(1 \pm m)/2\):

Energy: The number of parallel nearest-neighbor pairs is approximately\(\frac{Nq}{2}\cdot\frac{(1+m)^2 + (1-m)^2}{4} = \frac{Nq}{4}(1 + m^2)\), so:

\[E = -\frac{NqJ}{2}m^2 - Nhm\]

Entropy: From the number of ways to choose \(N_+\) up-spins from \(N\):

\[S = k_B\ln\binom{N}{N_+} \approx -\frac{Nk_B}{2}\left[(1+m)\ln\frac{1+m}{2} + (1-m)\ln\frac{1-m}{2}\right]\]

The free energy \(F = E - TS\):

\[\frac{F}{N} = -\frac{qJ}{2}m^2 - hm + \frac{k_BT}{2}\left[(1+m)\ln\frac{1+m}{2} + (1-m)\ln\frac{1-m}{2}\right]\]

Minimizing \(\partial F/\partial m = 0\):

\[-qJm - h + \frac{k_BT}{2}\ln\frac{1+m}{1-m} = 0 \implies m = \tanh\left(\frac{qJm + h}{k_BT}\right)\]

This recovers the Weiss self-consistency equation, confirming the equivalence of the two approaches.

4. Connection to Landau Theory

Derivation 4: Expanding Near T_c

Expand the Bragg-Williams free energy for small \(m\). Using\(\ln(1 \pm m) \approx \pm m - m^2/2 \pm m^3/3 - m^4/4 + \cdots\):

\[\frac{F}{N} = F_0 + \frac{1}{2}\left(k_BT - qJ\right)m^2 + \frac{k_BT}{12}m^4 + \frac{k_BT}{30}m^6 + \cdots - hm\]

Identifying \(a = \frac{1}{2}(k_BT - qJ) = \frac{1}{2}k_B(T - T_c)\) and\(b = k_BT/12 > 0\), this is exactly the Landau form:

\[\frac{F}{N} = F_0 + a\,m^2 + b\,m^4 - hm\]

From this, all mean-field critical exponents follow: \(\beta = 1/2\),\(\gamma = 1\), \(\delta = 3\), and \(\alpha = 0\)(discontinuity in \(C_V\)).

5. Accuracy of Mean-Field T_c

Derivation 5: Comparison Across Lattices

The mean-field critical temperature \(k_BT_c^{\text{MF}} = qJ\) overestimates the true \(T_c\) because it neglects fluctuations. The accuracy improves with increasing coordination number:

Mean-Field vs Exact T_c

1D chain (\(q = 2\)): MF gives \(T_c = 2J/k_B\), exact \(T_c = 0\) (qualitatively wrong)

2D square (\(q = 4\)): MF gives \(T_c = 4J/k_B\), exact \(T_c \approx 2.269J/k_B\) (76% too high)

2D triangular (\(q = 6\)): MF gives \(T_c = 6J/k_B\), exact \(T_c \approx 3.641J/k_B\) (65% too high)

3D simple cubic (\(q = 6\)): MF gives \(T_c = 6J/k_B\), numerical \(T_c \approx 4.511J/k_B\) (33% too high)

3D BCC (\(q = 8\)): MF gives \(T_c = 8J/k_B\), numerical \(T_c \approx 6.355J/k_B\) (26% too high)

3D FCC (\(q = 12\)): MF gives \(T_c = 12J/k_B\), numerical \(T_c \approx 9.795J/k_B\) (22% too high)

When Is Mean-Field Exact?

Mean-field theory becomes exact in several limits: (1) Infinite-range interactions (each spin interacts equally with all others). (2) Infinite dimensions or infinite coordination number. (3) Above the upper critical dimension \(d_c = 4\) for short-range interactions. The Ginzburg criterion quantifies the size of the critical region where fluctuations dominate: \(|t| < t_G\) where\(t_G \propto (a_0^d)^{2/(4-d)}\) and \(a_0\) is the lattice spacing.

6. Applications

Key Applications

BCS superconductivity: The BCS gap equation is a mean-field self-consistency equation for the superconducting order parameter, and gives\(T_c\) with remarkable accuracy because the effective dimensionality of the pairing interaction is infinite-range in momentum space.

Hartree-Fock in quantum chemistry: Electrons experience a mean field from all other electrons, leading to self-consistent orbital equations.

Neural networks: The Hopfield model of associative memory is exactly a fully connected Ising model, and mean-field theory provides its exact solution.

Epidemiology: SIR models with homogeneous mixing are mean-field theories of disease spread on networks.

7. Computational Exploration

This simulation visualizes the Weiss self-consistency equation, Bragg-Williams free energy, magnetization curves, and the comparison of mean-field with exact results.

Mean-Field Theory: Self-Consistency, Free Energy, Magnetization, and Lattice Comparison

Python

script.py181 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
def brentq(f, a, b, args=(), xtol=1e-12, maxiter=200):
    fa, fb = f(a, *args), f(b, *args)
    for _ in range(maxiter):
        c = (a + b) / 2.0; fc = f(c, *args)
        if abs(fc) < xtol or (b - a) / 2.0 < xtol: return c
        if fa * fc < 0: b, fb = c, fc
        else: a, fa = c, fc
    return (a + b) / 2.0

# ============================================================
# Mean-Field Theory: Self-Consistency and Free Energy
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Mean-Field Theory of the Ising Model', 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: Graphical solution of self-consistency equation ---
ax1 = axes[0, 0]
m_range = np.linspace(-1, 1, 300)

for T_ratio, color, label in [(0.5, '#ef4444', 'T/Tc = 0.5'),
                                (0.8, '#f87171', 'T/Tc = 0.8'),
                                (1.0, '#fbbf24', 'T/Tc = 1.0'),
                                (1.5, '#60a5fa', 'T/Tc = 1.5'),
                                (3.0, '#818cf8', 'T/Tc = 3.0')]:
    tanh_vals = np.tanh(m_range / T_ratio)
    ax1.plot(m_range, tanh_vals, color=color, linewidth=2, label=label)

ax1.plot(m_range, m_range, 'w--', linewidth=1, alpha=0.5, label='y = m')
ax1.set_xlabel('m')
ax1.set_ylabel('tanh(m / (T/Tc))')
ax1.set_title('Self-Consistency: m = tanh(qJm/kBT)')
ax1.set_xlim(-1.1, 1.1)
ax1.set_ylim(-1.1, 1.1)
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Bragg-Williams free energy ---
ax2 = axes[0, 1]
m_range2 = np.linspace(-0.99, 0.99, 500)

for T_ratio, color, label in [(0.5, '#ef4444', 'T/Tc = 0.5'),
                                (0.8, '#f87171', 'T/Tc = 0.8'),
                                (1.0, '#fbbf24', 'T/Tc = 1.0'),
                                (1.5, '#60a5fa', 'T/Tc = 1.5')]:
    # F/N = -qJ/2 * m^2 + kBT/2 * [(1+m)ln((1+m)/2) + (1-m)ln((1-m)/2)]
    # Set qJ = 1 (so kBTc = 1), kBT = T_ratio
    energy_part = -0.5 * m_range2**2
    entropy_part = T_ratio / 2.0 * ((1 + m_range2) * np.log((1 + m_range2) / 2) +
                                      (1 - m_range2) * np.log((1 - m_range2) / 2))
    F = energy_part + entropy_part
    F = F - np.min(F)  # shift for visibility
    ax2.plot(m_range2, F, color=color, linewidth=2, label=label)

ax2.set_xlabel('Magnetization m')
ax2.set_ylabel('F/N (shifted)')
ax2.set_title('Bragg-Williams Free Energy')
ax2.set_ylim(-0.05, 0.5)
ax2.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Magnetization vs T ---
ax3 = axes[1, 0]

# Solve self-consistency numerically
T_mf = np.linspace(0.01, 1.5, 200)
m_mf = []
for t in T_mf:
    if t >= 1.0:
        m_mf.append(0.0)
    else:
        try:
            sol = brentq(lambda m: m - np.tanh(m / t), 0.001, 0.9999)
            m_mf.append(sol)
        except:
            m_mf.append(0.0)

m_mf = np.array(m_mf)

# Exact 2D Ising (Yang)
Tc_exact = 2.0 / np.log(1 + np.sqrt(2))
T_yang = np.linspace(0.01, Tc_exact - 0.01, 200)
K_yang = 1.0 / T_yang
M_yang = (1 - np.sinh(2 * K_yang)**(-4))**(1.0/8.0)

ax3.plot(T_mf, m_mf, color='#f87171', linewidth=2.5, label='Mean-field')
ax3.plot(T_yang / Tc_exact, M_yang, color='#60a5fa', linewidth=2, linestyle='--', label='Exact 2D Ising')
ax3.axvline(x=1.0, color='#fbbf24', linestyle=':', alpha=0.5)
ax3.set_xlabel('T / T_c')
ax3.set_ylabel('|m|')
ax3.set_title('Magnetization: MF vs Exact')
ax3.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Tc comparison across lattices ---
ax4 = axes[1, 1]

lattices = [
    ('1D chain', 2, 0, '#ef4444'),
    ('2D square', 4, 2.269, '#f87171'),
    ('2D triangular', 6, 3.641, '#fbbf24'),
    ('3D SC', 6, 4.511, '#34d399'),
    ('3D BCC', 8, 6.355, '#60a5fa'),
    ('3D FCC', 12, 9.795, '#a78bfa'),
]

names = [l[0] for l in lattices]
Tc_mf_vals = [l[1] for l in lattices]
Tc_exact_vals = [l[2] for l in lattices]

x_pos = np.arange(len(lattices))
width = 0.35

bars1 = ax4.bar(x_pos - width/2, Tc_mf_vals, width, color='#f87171', alpha=0.7, label='Mean-field (qJ/kB)')
bars2 = ax4.bar(x_pos + width/2, Tc_exact_vals, width, color='#60a5fa', alpha=0.7, label='Exact/numerical')

ax4.set_xticks(x_pos)
ax4.set_xticklabels(names, fontsize=7, rotation=30, ha='right', color='white')
ax4.set_ylabel('k_B T_c / J')
ax4.set_title('Mean-Field vs Exact T_c')
ax4.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

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("MEAN-FIELD THEORY: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Self-Consistency Solutions m(T) ---")
for t in [0.1, 0.3, 0.5, 0.7, 0.9, 0.95, 0.99, 1.0, 1.5]:
    if t >= 1.0:
        m_val = 0.0
    else:
        try:
            m_val = brentq(lambda m: m - np.tanh(m / t), 0.001, 0.9999)
        except:
            m_val = 0.0
    print("  T/Tc = " + str(t) + ": m = " + str(round(m_val, 6)))

print("\n--- Curie-Weiss Susceptibility chi = C/(T-Tc) ---")
for t in [1.01, 1.1, 1.5, 2.0, 3.0, 5.0]:
    chi = 1.0 / (t - 1.0)
    print("  T/Tc = " + str(t) + ": chi * kB * Tc = " + str(round(chi, 4)))

print("\n--- Mean-Field vs Exact Critical Temperatures ---")
header = "  " + "Lattice".ljust(16) + "q".ljust(6) + "MF".ljust(8) + "Exact".ljust(8) + "Error"
print(header)
for name, q, tc_ex, _ in lattices:
    if tc_ex > 0:
        err = str(round(100 * (q - tc_ex) / tc_ex, 1)) + "%"
    else:
        err = "inf"
    row = "  " + name.ljust(16) + str(q).ljust(6) + str(q).ljust(8) + str(tc_ex).ljust(8) + err
    print(row)

print("\n--- Mean-Field Critical Exponents ---")
exponents = [
    ("beta (order parameter)", "1/2"),
    ("gamma (susceptibility)", "1"),
    ("delta (critical isotherm)", "3"),
    ("alpha (specific heat)", "0 (jump)"),
]
for name, val in exponents:
    print("  " + name + " = " + val)
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

Weiss: \(m = \tanh[(qJm + h)/(k_BT)]\)
Curie-Weiss: \(\chi = C/(T - T_c)\)
Mean-field \(T_c = qJ/k_B\)
Landau from Bragg-Williams: \(a = \frac{1}{2}k_B(T - T_c)\)

Physical Insights

MF replaces fluctuating neighbors by an average field
Exact for infinite range or \(d \ge 4\)
Overestimates \(T_c\); improves with higher coordination
Bragg-Williams and Weiss are equivalent variational approaches

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