Questions: Renormalization Group: Introduction

Question 1

The Ising model and certain liquid-gas systems show nearly identical critical exponents despite having completely different microscopic interactions. What does renormalization group theory offer as an explanation?

• A. Both systems have the same number of degrees of freedom per lattice site, so their thermodynamics must be equivalent
• B. Both systems share the same symmetry and dimensionality, so their RG flows converge to the same fixed point, producing identical critical behavior
• C. Critical exponents are universal constants of nature shared by all phase transitions
• D. Mean-field theory gives the correct exponents for both systems, so microscopic details do not matter

Answer: (B)

The RG explanation of universality is precise: systems with the same symmetry (here, Z₂) and dimensionality lie in the same basin of attraction of the same RG fixed point. Repeated coarse-graining drives both toward that fixed point, at which the critical exponents are determined by the eigenvalues of the linearized RG transformation. The microscopic details — lattice type, interaction range, whether the system is magnetic or a fluid — are irrelevant perturbations that flow to zero under coarse-graining. Option C is wrong because different universality classes have *different* exponents. Option D is wrong because mean-field theory gives incorrect exponents in low dimensions.

Question 2

Under the RG, 'irrelevant' perturbations near a fixed point shrink under coarse-graining. What is the physical significance of this for universality?

• A. Irrelevant perturbations are mathematical artifacts with no physical meaning and can always be ignored
• B. Irrelevant perturbations correspond to the microscopic details that different systems in the same universality class can differ in — they vanish under coarse-graining, allowing different systems to converge to the same fixed point
• C. Only irrelevant perturbations can be measured experimentally; relevant ones are too large-scale to detect
• D. Irrelevant perturbations are stable under coarse-graining, which is why they define the universality class

Answer: (B)

The relevant/irrelevant distinction is the mechanism by which universality class membership is determined. Irrelevant perturbations are exactly the microscopic details — lattice constant, specific interaction form, short-range cutoff — that vanish under repeated coarse-graining. This is why systems with different microscopic details converge to the same fixed point: their differences are irrelevant. Relevant perturbations (like temperature distance from criticality) grow under coarse-graining and drive the system away from the fixed point; they control the approach to and departure from critical behavior. Option D inverts the definition: relevant perturbations grow, not shrink.

Question 3

A fixed point of the RG flow corresponds to a theory that looks the same at all length scales — which is why the correlation length diverges at the critical point.

• T. True
• F. False

Answer: True

A fixed point is a coupling configuration that maps to itself under coarse-graining: blocking spins and rescaling produces the same effective theory. If the theory is unchanged by rescaling, then no characteristic length scale is introduced — meaning the system has no preferred length scale. A system with no characteristic length scale must have correlation length ξ = ∞. The divergence of ξ at criticality is not a coincidence — it is the direct consequence of the critical point being an RG fixed point. Scale invariance and diverging correlation length are two faces of the same fact.

Question 4

Mean-field theory fails to predict critical exponents correctly in low dimensions because it corresponds to the correct fixed point for those dimensions.

• T. True
• F. False

Answer: False

Mean-field theory fails *because* it corresponds to the fixed point of a hypothetical infinite-dimensional system, not the actual fixed point of 2D or 3D systems. In lower dimensions, fluctuations are stronger and the relevant directions of the RG flow push the system toward a *different* fixed point — one with different eigenvalues and therefore different critical exponents. Mean-field theory ignores fluctuations, which amounts to assuming you are in infinite dimensions where fluctuation contributions become negligible. In 2D or 3D, this captures the wrong fixed point and predicts the wrong universality class.

Question 5

Explain how the renormalization group provides a quantitative explanation for why microscopically different systems can display identical critical exponents.

Model answer: The RG coarse-grains the system by integrating out short-distance degrees of freedom and rescaling, generating a flow in the space of coupling constants. If two systems share the same symmetry and dimensionality, their coupling constants lie in the same basin of attraction of the same RG fixed point. Under repeated coarse-graining, the microscopic differences — lattice structure, interaction form — correspond to irrelevant perturbations that flow to zero. Both systems converge to the same fixed point, at which the critical exponents are determined by the eigenvalues of the linearized RG transformation. Since both reach the same fixed point, they exhibit the same critical exponents despite their different microscopic origins.

This is the deep conceptual achievement of RG theory: universality is not a coincidence but a consequence of the coarse-graining flow. The fixed point is the universal object; individual systems are different starting points that flow toward it. The critical exponents are properties of the fixed point, not of the individual systems — which is why they are universal across the entire universality class. It also explains why mean-field theory fails: it corresponds to the wrong fixed point (infinite-dimensional) and therefore predicts the wrong exponents for systems that live in lower dimensions.