Part IV: Advanced Topics | Chapter 1

The Renormalization Group

Kadanoff block spins, RG flow equations, fixed points, relevant and irrelevant operators, and real-space RG

Historical Context

The renormalization group (RG) is arguably the deepest conceptual advance in theoretical physics of the 20th century. Leo Kadanoff introduced the block spin concept in 1966, providing the intuitive picture of coarse-graining. Kenneth Wilson formalized this into the renormalization group in 1971, developing both the conceptual framework and computational tools (the epsilon expansion). Wilson received the 1982 Nobel Prize for this work.

The RG explains why universality exists: under successive coarse-graining, different microscopic systems flow to the same fixed point, which determines the critical exponents. The framework extends far beyond statistical mechanics -- it underlies our understanding of quantum field theory, the running of coupling constants in particle physics, and the effective theory approach in condensed matter.

1. Kadanoff Block Spin Transformation

Derivation 1: The Coarse-Graining Idea

Consider a lattice of Ising spins with lattice spacing \(a\). Kadanoff proposed grouping spins into blocks of size \(b^d\) (where \(b > 1\)) and replacing each block by a single effective spin:

\[s_I' = \text{sign}\left(\sum_{i \in \text{block } I} s_i\right)\]

The key assumption is that the coarse-grained system can be described by the same type of Hamiltonian with renormalized couplings:

\[H'[\{s'\}] = -K'\sum_{\langle IJ\rangle}s'_I s'_J - h'\sum_I s'_I + \text{const}\]

The RG transformation maps coupling constants: \((K, h) \to (K', h') = \mathcal{R}_b(K, h)\). Each transformation:

Reduces the number of degrees of freedom by \(b^d\)
Increases the effective lattice spacing: \(a' = ba\)
Decreases the correlation length in lattice units: \(\xi' = \xi/b\)

The last point is crucial: if \(\xi\) is measured in units of the lattice spacing, each RG step brings the system effectively further from criticality (unless \(\xi = \infty\)).

2. Fixed Points and RG Flow

Derivation 2: Fixed Point Condition and Linearization

A fixed point \(\mathbf{K}^*\) satisfies \(\mathcal{R}_b(\mathbf{K}^*) = \mathbf{K}^*\). Since \(\xi' = \xi/b\) at a fixed point either \(\xi^* = 0\) (trivial fixed point: high-T or low-T) or \(\xi^* = \infty\) (critical fixed point).

Near a fixed point, linearize the RG transformation:

\[K_i' - K_i^* = \sum_j M_{ij}(K_j - K_j^*), \qquad M_{ij} = \frac{\partial \mathcal{R}_{b,i}}{\partial K_j}\bigg|_{\mathbf{K}^*}\]

Diagonalizing the matrix \(\mathbf{M}\) gives eigenvalues \(\lambda_i = b^{y_i}\)and corresponding scaling fields \(u_i\). Under RG:

\[\boxed{u_i' = b^{y_i} u_i}\]

The exponents \(y_i\) classify the operators:

Relevant (\(y_i > 0\)): \(u_i\) grows under RG, drives system away from critical point
Irrelevant (\(y_i < 0\)): \(u_i\) shrinks, washed out at large scales -- explains universality
Marginal (\(y_i = 0\)): requires higher-order analysis

Why Universality?

Different microscopic systems correspond to different initial points in coupling space. Under RG flow, irrelevant operators decay away, and the long-distance behavior is controlled entirely by the fixed point and the relevant operators. Systems that flow to the same fixed point share the same critical exponents -- this is universality. The number of relevant operators equals the number of independent thermodynamic variables needed to tune to the critical point (typically 2: temperature and field).

3. Critical Exponents from the RG

Derivation 3: Relating RG Eigenvalues to Exponents

The two relevant scaling fields for a magnetic system are the thermal field\(u_t \propto t = (T - T_c)/T_c\) and the magnetic field \(u_h \propto h\). Under RG:

\[u_t' = b^{y_t}u_t, \qquad u_h' = b^{y_h}u_h\]

The free energy density transforms as \(f_s(u_t, u_h) = b^{-d}f_s(b^{y_t}u_t, b^{y_h}u_h)\). Choosing \(b = |u_t|^{-1/y_t}\):

\[f_s(u_t, u_h) = |u_t|^{d/y_t}\,\Phi_{\pm}\left(\frac{u_h}{|u_t|^{y_h/y_t}}\right)\]

Comparing with the scaling form \(f_s \sim |t|^{2-\alpha}\):

\[\boxed{2 - \alpha = \frac{d}{y_t}, \qquad \beta = \frac{d - y_h}{y_t}, \qquad \gamma = \frac{2y_h - d}{y_t}, \qquad \nu = \frac{1}{y_t}}\]

All six critical exponents are determined by just \(y_t\) and \(y_h\). The scaling and hyperscaling relations follow automatically.

4. Example: 1D Ising Model Decimation

Derivation 4: Exact RG in 1D

For the 1D Ising model, we can perform an exact RG by summing over every other spin (decimation, \(b = 2\)). The partition function for three consecutive spins:

\[\sum_{s_2 = \pm 1} e^{K(s_1 s_2 + s_2 s_3)} = 2\cosh[K(s_1 + s_3)]\]

We want this to equal \(e^{K's_1s_3 + C}\). Setting \(s_1 = s_3 = 1\)and \(s_1 = -s_3 = 1\):

\[2\cosh(2K) = e^{K'+C}, \qquad 2 = e^{-K'+C}\]

Solving:

\[\boxed{K' = \frac{1}{2}\ln\cosh(2K)}\]

The fixed points are \(K^* = 0\) (infinite temperature, trivial) and\(K^* = \infty\) (zero temperature). The linearized RG near \(K = \infty\)confirms \(T_c = 0\): there is no finite-temperature fixed point in 1D.

5. Approximate RG for the 2D Ising Model

Derivation 5: Majority Rule RG

For the 2D triangular lattice, group spins into triangles (\(b = \sqrt{3}\)). The block spin is the majority vote of three spins. After tracing over internal degrees of freedom (approximately), the RG recursion relation is:

\[K' = 2K^3 + \frac{3}{2}K^4 + \cdots \qquad \text{(Niemeijer-van Leeuwen)}\]

The nontrivial fixed point occurs at \(K^* \approx 0.336\), giving\(k_BT_c/J \approx 2.98\) (exact value for the triangular lattice: 3.641). The thermal eigenvalue gives \(y_t \approx 0.88\), yielding\(\nu \approx 1.14\) (exact: \(\nu = 1\)). While approximate, this demonstrates the RG machinery in action.

6. Applications

Key Applications

Quantum field theory: The RG explains the running of coupling constants (e.g., the fine structure constant varies with energy scale). Asymptotic freedom in QCD was discovered using RG methods.

Kondo problem: Wilson's numerical RG solved the Kondo problem of magnetic impurities in metals, demonstrating the power of the method beyond perturbation theory.

Polymer physics: The RG explains why long polymer chains are self-similar and computes the universal exponent \(\nu \approx 0.588\) relating size to length.

Turbulence: RG methods have been applied to the Navier-Stokes equations to study the energy cascade in turbulent flows.

7. Computational Exploration

This simulation demonstrates RG flow, fixed points, the 1D Ising decimation, and how irrelevant operators decay under coarse-graining.

Renormalization Group: RG Flow, Fixed Points, Decimation, and Scaling Fields

Python

script.py166 lines

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

# ============================================================
# Renormalization Group: Flow, Fixed Points, Decimation
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Renormalization Group', 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: 1D Ising RG flow ---
ax1 = axes[0, 0]
K_range = np.linspace(0, 3, 200)
K_prime = 0.5 * np.log(np.cosh(2 * K_range))

ax1.plot(K_range, K_prime, color='#f87171', linewidth=2.5, label="K' = (1/2) ln cosh(2K)")
ax1.plot(K_range, K_range, 'w--', linewidth=1, alpha=0.5, label="K' = K (fixed points)")

# Show flow arrows
for K0 in [0.3, 0.6, 1.0, 1.5, 2.0, 2.5]:
    K1 = 0.5 * np.log(np.cosh(2 * K0))
    # Arrow from (K0, K0) towards (K0, K1) then (K1, K1)
    ax1.annotate('', xy=(K1, K1), xytext=(K0, K0),
                 arrowprops=dict(arrowstyle='->', color='#60a5fa', lw=1.5))

ax1.plot(0, 0, 'o', color='#34d399', markersize=10, zorder=5, label='Stable FP (K*=0, T=inf)')
ax1.set_xlabel('K = J/(k_B T)')
ax1.set_ylabel("K' (renormalized)")
ax1.set_title('1D Ising: RG Decimation')
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Schematic 2D RG flow diagram ---
ax2 = axes[0, 1]

# Create a schematic flow diagram in (K, h) space
K_grid = np.linspace(0, 1.5, 20)
h_grid = np.linspace(-1, 1, 20)
K_mesh, h_mesh = np.meshgrid(K_grid, h_grid)

# Approximate flow: K flows toward 0 or inf depending on Kc, h flows away from 0
Kc = 0.44  # approximate critical coupling for 2D square lattice
dK = np.where(K_mesh < Kc, -0.15 * K_mesh, 0.15 * (K_mesh - Kc))
dh = 0.3 * h_mesh  # h is always relevant

ax2.quiver(K_mesh, h_mesh, dK, dh, color='#60a5fa', alpha=0.5, scale=5)
ax2.plot(Kc, 0, '*', color='#fbbf24', markersize=15, zorder=5, label='Critical FP')
ax2.plot(0, 0, 'o', color='#34d399', markersize=10, zorder=5, label='High-T FP')

# Critical surface (line in 2D)
ax2.axvline(x=Kc, color='#f87171', linestyle='--', alpha=0.5, label='Critical surface')
ax2.set_xlabel('K = J/(k_B T)')
ax2.set_ylabel('h / (k_B T)')
ax2.set_title('Schematic RG Flow (2D Ising)')
ax2.set_xlim(0, 1.5)
ax2.set_ylim(-1, 1)
ax2.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Scaling fields under RG iteration ---
ax3 = axes[1, 0]

# Simulate RG iterations for relevant and irrelevant operators
n_steps = 15
b = 2.0  # scale factor

# Relevant: y_t = 1/nu ~ 1.587 for 3D Ising (nu=0.630)
y_t = 1.0 / 0.630
# Irrelevant: y_irr ~ -0.8 for 3D Ising
y_irr = -0.83

steps = np.arange(n_steps + 1)

for u0, color, label in [(0.01, '#f87171', 'u_t(0) = 0.01'),
                          (0.05, '#ef4444', 'u_t(0) = 0.05')]:
    u_rel = u0 * b**(y_t * steps)
    ax3.semilogy(steps, u_rel, 'o-', color=color, linewidth=2, markersize=4, label=label + ' (relevant)')

for u0, color in [(1.0, '#60a5fa'), (0.5, '#818cf8')]:
    u_irr = u0 * b**(y_irr * steps)
    ax3.semilogy(steps, u_irr, 's--', color=color, linewidth=2, markersize=4,
                 label='u_irr(0) = ' + str(u0) + ' (irrelevant)')

ax3.axhline(y=1.0, color='#fbbf24', linestyle=':', alpha=0.5)
ax3.set_xlabel('RG iteration n')
ax3.set_ylabel('|u_n|')
ax3.set_title('Scaling Fields Under RG (b=2)')
ax3.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Correlation length under RG ---
ax4 = axes[1, 1]

# xi decreases by factor b each RG step
# At criticality xi = inf, off-critical xi is finite

b = 2.0
for xi0, color, label in [(1000, '#f87171', 'xi_0 = 1000 (near T_c)'),
                           (100, '#60a5fa', 'xi_0 = 100'),
                           (10, '#34d399', 'xi_0 = 10'),
                           (3, '#a78bfa', 'xi_0 = 3 (far from T_c)')]:
    xi_vals = xi0 / b**steps
    ax4.semilogy(steps, xi_vals, 'o-', color=color, linewidth=2, markersize=4, label=label)

ax4.axhline(y=1.0, color='#fbbf24', linestyle='--', alpha=0.7, label='xi = a (lattice scale)')
ax4.set_xlabel('RG iteration n')
ax4.set_ylabel('xi / a (lattice units)')
ax4.set_title('Correlation Length Under RG')
ax4.legend(fontsize=7, 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("RENORMALIZATION GROUP: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- 1D Ising Decimation RG ---")
print("  K' = (1/2) ln cosh(2K)")
print("  Fixed points: K* = 0 (stable, T = inf), K* = inf (unstable, T = 0)")
for K0 in [0.1, 0.5, 1.0, 2.0, 3.0]:
    K_val = K0
    trajectory = [K_val]
    for _ in range(10):
        K_val = 0.5 * np.log(np.cosh(2 * K_val))
        trajectory.append(K_val)
    print("  K0 = " + str(K0) + " -> after 10 steps: K = " + str(round(trajectory[-1], 6)))

print("\n--- RG Eigenvalues and Critical Exponents ---")
models = [
    ("2D Ising", 1.0, 1.875, 1.0, 0.125, 1.75),
    ("3D Ising", 1.587, 2.482, 0.630, 0.326, 1.237),
    ("3D XY", 1.489, 2.482, 0.672, 0.348, 1.317),
    ("3D Heisenberg", 1.407, 2.482, 0.711, 0.366, 1.396),
    ("Mean-field (d=4)", 2.0, 3.0, 0.5, 0.5, 1.0),
]
header = "  " + "Model".ljust(18) + "y_t".ljust(8) + "y_h".ljust(8) + "nu".ljust(8) + "beta".ljust(8) + "gamma"
print(header)
for name, yt, yh, nu, beta, gamma in models:
    row = "  " + name.ljust(18) + str(yt).ljust(8) + str(yh).ljust(8) + str(nu).ljust(8) + str(beta).ljust(8) + str(gamma)
    print(row)

print("\n--- Operator Classification ---")
print("  Relevant (y > 0): temperature, magnetic field")
print("  Irrelevant (y < 0): lattice anisotropy, higher-order couplings")
print("  Marginal (y = 0): phi^4 coupling in d=4")
print("  Number of relevant operators = codimension of critical surface")

print("\n--- Real-Space RG Approximations ---")
print("  1D decimation: exact (K' = (1/2)ln cosh(2K))")
print("  2D triangular majority rule: Tc/J ~ 2.98 (exact: 3.641)")
print("  2D square block (b=2): Tc/J ~ 2.55 (exact: 2.269)")
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

Scaling field: \(u_i' = b^{y_i}u_i\)
Correlation length: \(\nu = 1/y_t\)
Free energy: \(f_s = |t|^{d/y_t}\Phi(h/|t|^{y_h/y_t})\)
1D Ising: \(K' = \frac{1}{2}\ln\cosh(2K)\)

Physical Insights

RG = systematic coarse-graining of degrees of freedom
Universality: irrelevant operators wash out
Fixed points determine critical behavior
Only 2 relevant operators for typical magnetic transitions

← Critical Phenomena Universality →