Thermodynamics · Advanced Topics

The Renormalization Group

The renormalization group (RG) is a framework for understanding how physics changes with scale. It explains why phase transitions exhibit universal behavior independent of microscopic details, and connects condensed matter physics to quantum field theory through a profound mathematical analogy.

Prerequisites

Phase transitions (Ch. PT) \cdot Statistical mechanics (Ch. S) \cdot Fourier analysis (Ch. F) \cdot Gr

oup theory (Ch. GT)

Learning Goals

Explain why different systems near a continuous phase transition share identical critical exponents and belong to the same universality class.
Perform a real-space RG decimation on the 1D Ising model and show that K flows to zero at any finite temperature, confirming the absence of a phase transition.
$Write the Landau-Ginzburg-Wilson \phi^{4} functional and derive the one-loop RG flow equations f$ $or r and u near d = 4$ .
Classify RG perturbations as relevant, irrelevant, or marginal, and us $e this classification to derive all critical exponents from two independent eigenvalues y_$
$Verify the scaling relations \alpha + 2\beta + \gamma = 2 and d\nu = 2 - \alpha using the 3D Ising critical exp$ onents.

RG.1 Scale and Universality

Near a continuous phase transition (T → T_c), the correlation length diverges: ξ ∼ |T − T_c|^(−ν) → ∞. At T_c, the system looks the same at all length scales — it is scale invariant. This is why different systems (magnets, liquids, superconductors) show identical critical exponents — the same universality class.

Critical exponents (defined near T_c, h = 0 unless noted):

ξ ∼ |t|^(−ν), C ∼ |t|^(−α), m ∼ |t|^β (t = (T−T_c)/T_c, h = 0), χ ∼ |t|^(−γ), m ∼ h^(1/δ) (T = T_c).

Mean-field values: α = 0, β = 1/2, γ = 1, δ = 3, ν = 1/2. Ising 3D (Wilson-Fisher): β ≈ 0.326, γ ≈ 1.237, ν ≈ 0.630. These differ from mean-field because of fluctuations.

RG.2 Real-Space Renormalization Group

The RG procedure: coarse-grain the system by a factor b, then rescale to restore the original lattice spacing. Under this transformation, the coupling constants of the Hamiltonian flow:

K' = R(K) \qquad (RG transformation: couplings K \to K' after coarse-graining by b)

(RG.1)

Fixed points K* satisfy R(K*) = K* — scale-invariant systems. The nature of the fixed point determines the phase:

K = 0 (high T): high-T fixed point — disordered phase. K = ∞ (low T): low-T fixed point — ordered phase. K = K* (critical): unstable fixed point — the phase transition.

Example RG.1 — 1D Ising Model — Exact RG

$Apply the RG to the 1D Ising model H = -K \sum_{i} s_{i} s_{i+1} by summing out every other spin$ .

Partition function:

Z = \sum_

{s}

e^{K \sum s_{i}s_{i+1}}. Integrate out even spins s_{2}, s_{4}, \cdots (b = 2 decimation

Sum s₂:

For each s_{2}: \sum(s_{2}=\pm1) e^{K s_{1} s_{2} + K s_{2} s_{3}} = 2 cosh(K(s_{1}+s_{3})). This depends on s_{1}s_{3} only

Write as e^(K' s_{1} s_{3} + const

New coupling:

Comparing: 2 cosh(2K) = A e^{K'} (if s_{1}=s_{3}=+1). 2 cosh(0) = A e^{-K'} (if s_{1}=-s_{3}). Dividin

g: tanh(K') = tanh^{2}(K). So K' = arctanh(tanh^{2}(K)) < K

Fixed points:

K' = K only for K = 0 (trivial) or K = \infty (trivial). No finite fixed point \to no phase trans

ition in 1D Ising at finite T! RG confirms the exact result (Peierls argument).

Physical:

K \to 0 under RG: the system always flows to disorder at any finite T. Long-range order requ

ires T = 0. The 2D Ising model has a nontrivial fixed point (K_{c} = ln(1+\sqrt2)/2) — gives the

phase transition.

RG.3 The Wilson-Fisher Fixed Point

Kadanoff and Wilson reformulated RG in momentum space. Start with the Landau free energy functional (the Wilson-Fisher φ⁴ theory):

F[\phi] = \int d^d r [\frac{1}{2}(\nabla\phi)^{2} + \frac{1}{2}r \phi^{2} + u \phi^{4}] \qquad (Landau-Ginzburg-Wilson functional)

(RG.2)

RG in momentum space: integrate out modes with |k| between Λ/b and Λ (shell), then rescale k → bk to restore cutoff Λ. The couplings flow:

dr/dl = 2r + 12u K_{d} \Lambda^d/(r + \Lambda^{2}) \qquad (flow of mass parameter)

(RG.3)

du/dl = (4-d)u - 36u^{2} K_{d} \Lambda^d/(r + \Lambda^{2})^{2} \qquad (flow of coupling)

(RG.4)

where l = ln(b). The crucial observation: the coefficient of u in Eq. RG.4 is (4−d). In d = 4: u is marginal (flows logarithmically). In d < 4: u is relevant — interactions matter. In d > 4: u is irrelevant — mean-field is exact.

The Wilson-Fisher fixed point at d = 4−ε (ε expansion): u* = ε/36 K_d + O(ε²). The critical exponents to first order in ε:

\nu = \frac{1}{2} + \varepsilon/12 + O(\varepsilon^{2}) \qquad \eta = 0 + O(\varepsilon^{2}) \qquad (Ising class, \varepsilon = 4-d)

(RG.5)

At ε = 1 (d = 3): ν ≈ 0.583 (actual 0.630) — good first approximation. Higher orders: ν = 0.630 at 5-loop order, matching experiments and Monte Carlo to 4 significant figures.

RG.4 Scaling and Universality from Fixed Points

At a fixed point K*, linearize the RG transformation: δK' = M × δK where M = dR/dK|(K*). The eigenvalues Λ_i = b^(y_i) classify the perturbations:

Relevant:y_i > 0 (λ_i > 1) — grows under RG, takes system away from fixed point. Controls the distance from T_c and h.

Irrelevant:y_i < 0 (λ_i < 1) — shrinks under RG. Microscopic details that don't affect critical behavior. This is WHY universality exists!

Marginal: y_i = 0 — needs higher-order analysis. Leads to logarithmic corrections (2D Ising at T_c, or QCD asymptotic freedom).

Theorem RG.1 — Scaling Hypothesis and Exponent Relations

At a fixed point with two relevant eigenvalues y_{t} (temperature) and y_{h} (field), the free

energy obeys a homogeneous scaling form:

f(t, h) = b^{-d} f(b^{y_{t}} t, b^{y_{h}} h) for any b

This leads to exact exponent relations — only two exponents are independent: \alpha + 2\beta + \gamma =

2 (Rushbrooke), d\nu = 2 - \alpha (hyperscaling), \gamma = \nu(2 - \eta). All critical exponents of a unive

sality class follow from two numbers (y_{t}, y_{h}

RG.5 Applications Beyond Phase Transitions

The RG idea extends far beyond condensed matter:

Quantum field theory: the beta function β(g) = μ dg/dμ is the RG flow of coupling g with energy scale μ. Asymptotic freedom (QCD) = g → 0 under RG to high energy. The Landau pole in QED = g → ∞ (UV problem).

Chaotic systems: Feigenbaum constants (δ = 4.669..., α = 2.502...) in the period-doubling route to chaos — universal regardless of the map. Explained by a fixed point of the RG acting on maps.

Turbulence: Kolmogorov scaling E(k) ∝ k^(−5/3) can be understood as a fixed point of the RG for the Navier-Stokes equation (though a rigorous derivation remains incomplete).

Disordered systems: random systems have infinite-randomness fixed points where quantum fluctuations are amplified — describes random quantum spin chains, many-body localization.

Definition RG.2 — Common Traps

RG transformations change descriptions, not the underlying system: coarse-graining tracks which couplings matter at large scale.
Relevant does not mean important in everyday language: it means the perturbation grows under repeated rescaling.
Universality ignores irrelevant details: lattice structure and microscopic chemistry often wash out near a fixed point.
Mean-field breaks below the upper critical dimension: fluctuations change exponents in d < 4 for the Ising class.

Exercises — RG.1–RG.5 Renormalization Group

Verify the scaling relations \alpha + 2\beta + \gamma = 2 and d\nu = 2 - \alpha using the 3D Ising exponents \alpha

0.110, \beta = 0.326, \gamma = 1.237, \nu = 0.630. What is the upper critical dimension

(upper critical dimension)

Straightforward

Apply the Migdal-Kadanoff real-space RG to the 2D Ising model with b=2. Find the approxima

te K_{c} and compare to the exact Onsager value K_{c} = ln(1+\sqrt2)/2

Intermediate

3.Describe the Berezinskii-Kosterlitz-Thouless (BKT) transition using the RG. What is the role of vortices and what is the universal jump in the superfluid density?

Intermediate

Derive the Wilson-Fisher fixed point in the \varepsilon = 4-d expansion for the \phi^{4} theory. What are

the flow equations for r and u to one loop, and what is u*?

Challenging

Key Takeaways

$RG: coarse-grain by b, rescale. Couplings flow K \to K'. Fixed points \to phases; unstable fix$ $ed point \to phase transition$ .
Relevant perturbations (y>0): control critical behavior. Irrelevant (y<0): explains universality.
$1D Ising: RG flows to disorder at any T \to no transition. 2D: nontrivial fixed point at K_{c}$ .
$Wilson-Fisher: \phi^{4} theory near d=4. u* = \varepsilon/36, exponents in \varepsilon expansion. Ising 3D at \varepsilon=1$ .
$Exponent relations: \alpha+2\beta+\gamma=2, d\nu=2-\alpha, \gamma=\nu(2-\eta). Only two independent exponents$ .
RG explains QFT running couplings, Feigenbaum universality in chaos, Kolmogorov turbulence scaling.