Questions: Renormalization Group and Scaling Analysis

Question 1

Iron near its Curie point and carbon dioxide near its liquid-gas critical point have nearly identical critical exponents despite completely different microscopic physics. The RG explains this universality by showing that:

• A. Both systems have the same microscopic Hamiltonian when written in reduced units
• B. Both systems flow to the same fixed point under RG transformations, and at the fixed point, all irrelevant microscopic differences have washed out
• C. Critical exponents are always rational numbers, so coincidences among different systems are mathematically inevitable
• D. Both systems are described by the same equation of state, which is determined by thermodynamics alone

Answer: (B)

Under repeated RG coarse-graining, coupling constants flow in parameter space. Systems in the same universality class are governed by the same symmetry group, dimensionality, and interaction range, and these properties dictate which fixed point the flow approaches. At the fixed point, all microscopic differences — lattice details, interaction strengths — are 'irrelevant' in the technical sense: they shrink to zero under RG. The universal critical behavior is read off from the fixed point, which is the same for both systems.

Question 2

In RG analysis, what distinguishes a 'relevant' coupling from an 'irrelevant' one near a fixed point?

• A. Relevant couplings appear in the microscopic Hamiltonian; irrelevant couplings are generated during coarse-graining
• B. Relevant perturbations grow under RG transformations (driving the system away from the fixed point); irrelevant perturbations shrink (washing out at long distances)
• C. Relevant couplings are experimentally measurable; irrelevant couplings are mathematical artifacts
• D. Relevant couplings describe short-range interactions; irrelevant couplings describe long-range interactions

Answer: (B)

Near a fixed point, the RG transformation linearizes: perturbations in coupling space either grow (relevant directions) or shrink (irrelevant directions) under successive coarse-graining. Relevant perturbations drive the system away from the fixed point — temperature distance from criticality (t = (T-Tc)/Tc) and external field are the canonical relevant operators. Irrelevant perturbations shrink with each RG step, their effect on long-distance physics vanishing. Critical exponents are determined by the eigenvalues of the linearized RG at the fixed point.

Question 3

The universality class of a system at its critical point is determined by the detailed form of its microscopic Hamiltonian — different lattice models with different interaction strengths will generically belong to different universality classes.

• T. True
• F. False

Answer: False

This is precisely what the RG overturns. Universality class is determined by the symmetry of the order parameter, the spatial dimensionality, and the range of interactions — not by microscopic details of the Hamiltonian. An Ising model on a square lattice and on a triangular lattice, or with different coupling strengths, belong to the same universality class because all microscopically different perturbations around the fixed point are irrelevant — they wash out under coarse-graining, leaving only universal long-wavelength physics.

Question 4

A fixed point of the RG transformation represents a scale-invariant system — one that looks statistically the same at all length scales — which corresponds to the condition at a critical point where the correlation length diverges.

• T. True
• F. False

Answer: True

By definition, a fixed point is a Hamiltonian unchanged by the RG coarse-graining transformation. If the system looks the same after averaging over short-distance degrees of freedom, it has no preferred length scale — it is scale-invariant. This is exactly the condition at a critical point: with correlation length ξ → ∞, there is no characteristic scale. Fluctuations are correlated at all scales, and the statistical properties look the same whether observed at a few lattice spacings or at macroscopic scales.

Question 5

Explain what 'relevant' and 'irrelevant' operators mean in the RG framework, and why this distinction explains why very different microscopic systems can share the same critical exponents.

Model answer: Near an RG fixed point, perturbations in coupling-constant space are classified by how they transform under coarse-graining: relevant operators grow (driving the system away from the fixed point), irrelevant operators shrink (vanishing in the long-wavelength limit). Critical exponents are determined by the eigenvalues of the linearized RG at the fixed point — they depend only on the relevant directions. Two systems with different microscopic Hamiltonians that differ only in irrelevant operators will flow to the same fixed point, with the irrelevant differences washing away, leaving identical critical behavior.

This converts a puzzling empirical coincidence (why do iron and water have the same critical exponents?) into a theorem (any two systems with the same symmetry, dimension, and interaction range have the same fixed point, hence the same exponents). Microscopic diversity is expected, and the RG predicts exactly which differences survive to long distances (relevant operators, like temperature distance from criticality) and which do not (irrelevant operators, like lattice structure and coupling ratios).