Renormalization Group

Takeaway

Renormalization group (RG) explains how a system’s description changes as you zoom in or out; fixed points of that zooming flow determine universal behavior like critical exponents near phase transitions and the running of couplings in quantum field theory.

The problem (before → after)

Before: Your model works at one resolution but fails at another. Mean-field theory predicts the wrong critical exponents; a coupling that seemed small at short distances becomes important at large distances. Predictions depend on arbitrary cutoffs.
After: You track how parameters evolve with scale. Instead of a single set of parameters, you have a flow through parameter space. Scale-invariant behavior emerges at fixed points, giving predictions that do not depend on arbitrary measurement scale.

Mental model first

Think of viewing a coastline on a map. As you zoom out, small bays and peninsulas blur into the outline. “Coarse-graining” throws away details below your current pixel size. RG is the rulebook for how the remaining, larger-scale features—and the parameters that describe them—change as you repeatedly zoom out and rescale the map so the coastline fits your screen again.

Just-in-time concepts

Coarse-graining: Average or integrate out fine-scale degrees of freedom; keep the long-wavelength modes.
Rescaling: After coarse-graining by a factor b > 1, stretch lengths by b so the system’s size stays fixed; also rescale fields to restore normalization.
Couplings and flow: Model parameters become scale-dependent, g → g(ℓ). Their evolution is governed by β(g) = d g / d log ℓ.
Fixed point: A parameter set g⋆ with β(g⋆) = 0. Near g⋆, linearize the flow to classify directions as relevant (grow under RG), irrelevant (shrink), or marginal.

First-pass solution

Naive approaches (e.g., mean-field theory) assume fluctuations are small. Near critical points, fluctuations occur on all scales, breaking that assumption. Introducing a microscopic cutoff Λ tames divergences but makes results cutoff-dependent. We need a principled way to change Λ while keeping physics invariant: enter RG.

Start with a Hamiltonian or action with cutoff Λ.
Integrate out high-momentum modes in a thin shell Λ/b < |k| ≤ Λ.
Rescale momenta and fields to restore the cutoff to Λ.
Read off the new couplings. This defines an RG transformation T_b.
Iterate: the sequence g, T_b(g), T_b^2(g), … traces a flow. Fixed points govern large-scale physics. Linearizing around g⋆ gives scaling dimensions and critical exponents.

Consequences:

Universality: Microscopic details are washed out. Systems with different microphysics but the same relevant directions share critical exponents.
Running couplings: In QFT, couplings depend on energy scale μ; e.g., asymptotic freedom in nonabelian gauge theories arises from β(g) < 0 at weak coupling.
Operator classification: Only relevant and marginal operators control long-distance physics; irrelevant ones die away.

Code as a byproduct (toy RG flow)

Below is a minimal numerical RG for a single coupling g with a schematic β(g) = a g − b g^3 that has fixed points at g = 0 and g = ±√(a/b):

def rg_flow(g0: float, a: float, b: float, steps: int, dlogl: float):
    g = g0
    traj = [g]
    for _ in range(steps):
        beta = a * g - b * (g ** 3)
        g = g + beta * dlogl
        traj.append(g)
    return traj

# Example: approach to a nontrivial fixed point
trajectory = rg_flow(g0=0.2, a=1.0, b=2.0, steps=2000, dlogl=1e-3)

This illustrates flows toward (or away from) fixed points and how initial conditions become irrelevant near attractors.

Principles, not prescriptions

Only differences across scales that survive coarse-graining matter.
Fixed points and their relevant directions control macroscopic behavior.
Dimensionless quantities and scaling laws are the right invariants to track.
Regularization introduces a scale; renormalization removes dependence on that arbitrary choice by flowing parameters.

Common pitfalls

Confusing regularization with renormalization: the former introduces a cutoff; the latter shows how to remove its arbitrariness by absorbing it into scale-dependent parameters.
Thinking RG is only for QFT: it equally explains critical phenomena, turbulence spectra, and even learning dynamics in large models.
Treating critical exponents as model-specific: they are universal within a universality class determined by symmetries and dimensionality.

Connections and contrasts

See also: [/blog/spontaneous-symmetry-breaking] (symmetry changes under scale), [/blog/black-hole-information] (scale/entropy ideas in gravity), [/blog/network-effects] (scaling in economics), and [/blog/emergent-complexity-systems] (phenomena across scales).

Quick checks

What is a fixed point and why does it matter? — A parameter set unchanged by RG; it controls scale-invariant behavior and critical exponents.
What makes an operator “relevant”? — Its coefficient grows under RG, so it influences long-distance physics.
Why do different materials share exponents near criticality? — Microscopic details become irrelevant; only symmetry and dimension remain.
How does RG remove cutoff dependence? — By letting parameters run with scale so predictions for observables are invariant.