6.7 Renormalization group

Fundamentals of Renormalization Group

Renormalization group (RG) theory provides a framework for understanding critical phenomena by analyzing how a system's effective description changes as you zoom out to larger length scales. Near a phase transition, fluctuations exist on all scales, and no single "characteristic length" dominates. The RG handles this by systematically coarse-graining degrees of freedom and tracking how the effective Hamiltonian evolves. This approach reveals why microscopically different systems can share identical critical behavior.

Concept of Scale Invariance

At a critical point, the correlation length $\xi$ diverges, meaning fluctuations span every length scale in the system. The result is that the system looks statistically the same whether you examine it at short or long distances. This is scale invariance.

• Correlation functions decay as power laws rather than exponentials: $G(r) \sim r^{-(d-2+\eta)}$ at criticality, where $\eta$ is the anomalous dimension
• Thermodynamic quantities also follow power laws (e.g., $C \sim |t|^{-\alpha}$, $\chi \sim |t|^{-\gamma}$), which are signatures of an underlying scale-invariant theory
• Scale invariance produces fractal-like spatial structure in the order parameter fluctuations

Universality in Critical Phenomena

One of the most striking results in statistical mechanics is that systems with completely different microscopic physics can share the same critical exponents. A liquid near its critical point and a uniaxial ferromagnet near its Curie temperature have the same exponents, despite being governed by very different interactions.

• Systems are grouped into universality classes determined by just a few features: the spatial dimensionality $d$, the symmetry of the order parameter, and whether interactions are short- or long-ranged
• Microscopic details (lattice structure, exact coupling strengths) are irrelevant in the RG sense: they don't affect the critical exponents
• This is why the RG is so powerful. It explains universality as a consequence of many different Hamiltonians flowing to the same fixed point under coarse-graining

Kadanoff's Blocking Procedure

Before Wilson formalized the RG, Kadanoff proposed a physical picture that captures the essential idea. The procedure works as follows:

1. Divide the lattice into blocks of $b^d$ spins (where $b$ is the rescaling factor and $d$ is the dimension)
2. Replace each block with a single effective spin, defined by some rule (e.g., majority vote or the block's average magnetization)
3. Rescale lengths by $b$ so the new lattice has the same spacing as the original
4. Determine the new effective Hamiltonian that reproduces the same long-distance physics

The key insight: if the system is at the critical point, the effective Hamiltonian after blocking looks the same as the original. Away from criticality, the couplings shift toward one of the stable phases. This picture directly motivates the concept of RG flow and fixed points.

Renormalization Group Transformations

An RG transformation is a map $\mathcal{R}$ that takes a Hamiltonian $H$ (specified by a set of coupling constants) and produces a new Hamiltonian $H' = \mathcal{R}(H)$ describing the same system at a larger scale. The two main strategies differ in where the coarse-graining is performed.

Real-Space Renormalization

This approach works directly on the lattice by grouping degrees of freedom in physical space.

• You define blocks of spins and sum (or "trace") over the internal degrees of freedom within each block, producing effective couplings between blocks
• For the 1D Ising model, this can be carried out exactly and demonstrates how the effective coupling flows to zero (no phase transition in 1D)
• In higher dimensions, approximations are typically needed (e.g., the Migdal-Kadanoff bond-moving approximation)
• The method is intuitive and works well for lattice models, but the truncation of generated couplings introduces uncontrolled errors in most cases

Momentum-Space Renormalization

Instead of grouping spins in real space, this approach integrates out short-wavelength (high-momentum) fluctuations in Fourier space.

1. Start with a field theory defined with a momentum cutoff $\Lambda$
2. Integrate out the "fast" modes with momenta in the shell $\Lambda/b < k < \Lambda$
3. Rescale momenta $k \to bk$ and fields $\phi \to \zeta \phi$ to restore the original cutoff
4. Read off the new coupling constants

This is the basis of Wilson's approach and connects naturally to perturbative methods. It's especially suited to continuum field theories and is the setting for the epsilon expansion.

Decimation Techniques

Decimation is a specific real-space method where you eliminate a subset of degrees of freedom exactly (rather than block-averaging).

• In the 1D Ising model, you can sum over every other spin exactly, producing a new 1D Ising model with renormalized couplings. This is one of the few cases where the RG transformation is exact and closed.
• In 2D and higher, decimation typically generates new couplings not present in the original Hamiltonian (e.g., next-nearest-neighbor or four-spin interactions), so truncation is required
• The technique is valuable for building intuition and for exact solutions of simple models

Fixed Points and Critical Exponents

A fixed point is a Hamiltonian $H^*$ that is unchanged by the RG transformation: $\mathcal{R}(H^*) = H^*$. Fixed points correspond to scale-invariant systems and control the critical behavior of all Hamiltonians that flow toward them.

Stable vs. Unstable Fixed Points

The stability of a fixed point is determined by linearizing the RG transformation around it and examining the eigenvalues.

• Stable (attractive) fixed points: All nearby trajectories flow toward them. These typically correspond to the trivial high-temperature ($T = \infty$, completely disordered) and low-temperature ($T = 0$, fully ordered) limits.
• Unstable fixed points: At least one direction in coupling-constant space is repulsive. These correspond to critical points. A system sitting exactly at the unstable fixed point remains there, but any deviation drives it toward a stable fixed point (i.e., toward one of the phases).
• The critical surface is the set of all Hamiltonians that flow into the unstable fixed point. It separates the basins of attraction of the stable fixed points and defines the phase boundary.

Calculation of Critical Exponents

Critical exponents are extracted from the eigenvalues of the linearized RG transformation at the unstable fixed point.

1. Linearize $\mathcal{R}$ near $H^*$: small deviations $\delta H$ evolve as $\delta H' = M \, \delta H$, where $M$ is the linearization matrix
2. Find the eigenvalues $\lambda_i$ of $M$. Each eigenvalue corresponds to a scaling operator $\mathcal{O}_i$
3. Define the RG eigenvalue $y_i$ by $\lambda_i = b^{y_i}$
4. Relevant operators have $y_i > 0$ (they grow under RG flow and drive the system away from criticality). Irrelevant operators have $y_i < 0$ (they shrink and don't affect critical exponents). Marginal operators have $y_i = 0$ and require further analysis.
5. The thermal exponent $y_t$ and magnetic exponent $y_h$ determine all standard critical exponents through scaling relations, e.g., $\nu = 1/y_t$ and $\beta = (d - y_h)/y_t$

Universality Classes

Because critical exponents depend only on the fixed point (not on where you started in coupling space), all systems flowing to the same fixed point share the same exponents. This is the RG explanation of universality.

• Ising universality class ($Z_2$ symmetry, scalar order parameter): includes the liquid-gas critical point, uniaxial ferromagnets, and binary fluid demixing. In 3D: $\nu \approx 0.630$, $\beta \approx 0.326$, $\gamma \approx 1.237$
• XY universality class ($O(2)$ symmetry): includes the superfluid transition in $^4\text{He}$ (lambda transition) and certain superconducting transitions
• Heisenberg universality class ($O(3)$ symmetry): isotropic ferromagnets
• The determining factors are dimensionality $d$, the number of order parameter components $n$, and the range of interactions

Wilson's Renormalization Group Theory

Kenneth Wilson synthesized Kadanoff's scaling ideas with field-theoretic methods into a rigorous, calculable framework. His approach made it possible to compute critical exponents systematically and earned him the 1982 Nobel Prize in Physics.

Epsilon Expansion

The upper critical dimension for the $O(n)$ $\phi^4$ theory is $d_c = 4$: above this dimension, mean-field theory gives the correct exponents. Wilson and Fisher exploited this by treating $\epsilon = 4 - d$ as a small parameter.

• Critical exponents are expanded as power series in $\epsilon$. For example, for the Ising model ($n=1$): $\eta = \frac{\epsilon^2}{54} + \mathcal{O}(\epsilon^3)$ and $\nu = \frac{1}{2} + \frac{\epsilon}{12} + \mathcal{O}(\epsilon^2)$
• Setting $\epsilon = 1$ gives estimates for 3D systems. Despite being an expansion around 4D, the results are surprisingly accurate (especially when combined with resummation techniques like Padé or Borel methods)
• The method reveals how the Gaussian (free-field) fixed point becomes unstable below 4D and a new, nontrivial Wilson-Fisher fixed point emerges

Perturbative Renormalization Group

This approach applies RG ideas within the framework of Feynman diagram perturbation theory.

• Loop integrals in the $\phi^4$ theory produce divergences that are absorbed by redefining (renormalizing) the coupling constant $u$, the mass (related to $t = (T - T_c)/T_c$), and the field normalization
• The beta function $\beta(u) = \frac{du}{d\ln b}$ governs how the coupling runs with scale. A zero of the beta function is a fixed point
• Anomalous dimensions arise from the field renormalization and give the exponent $\eta$
• The Callan-Symanzik equation provides a differential form of the RG that connects correlation functions at different scales
• This machinery is shared with quantum field theory, creating a deep link between statistical mechanics and particle physics

Renormalization Group Flow

The RG transformation defines a flow in the (potentially infinite-dimensional) space of all possible Hamiltonians.

• Each point in this space represents a set of coupling constants $(t, u, v, \ldots)$. The RG maps one point to another, tracing out a trajectory
• Flow diagrams (typically projected onto 2 or 3 coupling constants) visualize the phase structure. Arrows point in the direction of increasing coarse-graining (larger scales)
• Near a fixed point, the flow is controlled by the relevant and irrelevant directions. Relevant directions point away from the fixed point; irrelevant directions point toward it
• Crossover occurs when the RG flow passes near one fixed point before eventually reaching another, producing intermediate scaling behavior

Applications in Statistical Mechanics

Ising Model and Renormalization

The Ising model is the testing ground for RG methods. In 1D, exact decimation shows the coupling flows to zero under RG, confirming the absence of a phase transition. In 2D, the exact solution (Onsager) provides benchmarks. In 3D, the epsilon expansion and numerical RG give the best available critical exponents.

• Real-space RG on the 2D triangular lattice (using majority-rule blocking) gives approximate but instructive results
• The 3D Ising critical exponents are now known to high precision from conformal bootstrap methods, Monte Carlo, and high-order epsilon expansion, all consistent with each other
• The Ising universality class also describes the liquid-gas critical point, demonstrating the power of universality: you can study a simple spin model to learn about fluid criticality

Lattice Gas Models

A lattice gas assigns occupation variables $n_i = 0, 1$ to lattice sites with nearest-neighbor interactions. Through the mapping $s_i = 2n_i - 1$, the lattice gas Hamiltonian maps exactly onto the Ising model (up to a chemical potential term playing the role of the magnetic field).

• This mapping proves that the liquid-gas critical point is in the Ising universality class
• The lattice gas has a conserved quantity (total particle number), which corresponds to working at fixed magnetization in the Ising language. The RG still applies, but the conserved density introduces the chemical potential as the relevant field conjugate to the order parameter
• Lattice gas models provide a bridge between discrete spin models and continuum fluid descriptions

Polymer Systems

Long flexible polymers in a good solvent behave as self-avoiding random walks (SAWs). The RG connects polymer statistics to critical phenomena through a mapping to the $n \to 0$ limit of the $O(n)$ vector model.

• The end-to-end distance of a polymer of $N$ monomers scales as $R \sim N^{\nu}$, where $\nu$ is the SAW correlation length exponent (the Flory exponent). In 3D, $\nu \approx 0.588$, distinct from the random walk value of $1/2$
• The RG explains why polymer scaling laws are universal: they depend on dimension and the self-avoidance constraint, not on chemical details
• This connection has been fruitful for both polymer physics and field theory

Numerical Renormalization Group Methods

Analytical RG methods rely on perturbative expansions or exactly solvable limits. For strongly interacting systems, numerical approaches extend the RG to regimes where no small parameter exists.

Monte Carlo Renormalization Group

This method combines Monte Carlo simulation with RG blocking transformations.

1. Simulate the system at or near criticality using standard Monte Carlo methods
2. Apply a block-spin transformation to the Monte Carlo configurations
3. Measure correlation functions of both the original and blocked configurations
4. Extract the linearized RG transformation matrix and its eigenvalues, yielding critical exponents

The approach overcomes some finite-size limitations of pure Monte Carlo by using the RG to relate different scales. It has been applied successfully to 3D spin models and lattice gauge theories.

Density Matrix Renormalization Group (DMRG)

DMRG, developed by Steven White in 1992, is the most powerful method for 1D quantum systems. Despite its name, it's conceptually different from the RG methods discussed above. Rather than coarse-graining in the Kadanoff/Wilson sense, DMRG systematically truncates the Hilbert space by keeping the most important states as determined by the density matrix.

• The system is grown iteratively, and at each step the reduced density matrix of a block is diagonalized. States with the largest eigenvalues are retained
• DMRG achieves essentially exact ground-state energies and correlation functions for 1D systems with short-range interactions
• It works because 1D ground states typically have low entanglement (area-law scaling), so a relatively small number of states captures the physics
• Extensions to 2D and to time-dependent problems exist but are more limited due to higher entanglement

Functional Renormalization Group (FRG)

The FRG replaces the perturbative RG with an exact flow equation (the Wettergren equation) for the effective action $\Gamma_k$, where $k$ is a running momentum scale.

• The Wettergren equation is $\partial_k \Gamma_k = \frac{1}{2} \text{Tr}\left[\left(\Gamma_k^{(2)} + R_k\right)^{-1} \partial_k R_k\right]$, where $R_k$ is a regulator function and $\Gamma_k^{(2)}$ is the second functional derivative
• This equation is exact but cannot be solved exactly. In practice, one truncates the effective action (e.g., local potential approximation) and solves the resulting flow equations
• The FRG is non-perturbative and has been applied to strongly correlated fermion systems, frustrated magnets, and quantum gravity
• It provides a unified framework that interpolates between the microscopic action ($k = \Lambda$) and the full effective action ($k = 0$)

Renormalization Beyond Critical Phenomena

Quantum Field Theory Applications

The RG originated in quantum electrodynamics (Stueckelberg and Petermann, Gell-Mann and Low) before Wilson adapted it to statistical mechanics. The ideas flow both ways.

• Running coupling constants: the effective strength of an interaction depends on the energy scale at which you probe it. In QED, the fine structure constant increases at higher energies. In QCD, it decreases (asymptotic freedom, discovered by Gross, Wilczek, and Politzer)
• Effective field theories: the RG justifies treating physics at different scales with different theories. Low-energy nuclear physics doesn't need to resolve quarks, just as critical phenomena don't need to resolve lattice-scale details
• The Wilsonian viewpoint reinterprets renormalizability: a renormalizable theory is simply one controlled by a fixed point with a finite number of relevant operators

Condensed Matter Systems

Beyond classical phase transitions, the RG is central to modern condensed matter physics.

• Quantum phase transitions occur at $T = 0$ as a function of some non-thermal parameter (pressure, doping, magnetic field). The RG applies with an extra "imaginary time" dimension, so a $d$-dimensional quantum system maps to a $(d+1)$-dimensional classical system
• Kondo problem: Wilson's numerical RG was originally developed to solve the Kondo impurity problem, showing how a magnetic impurity in a metal is screened at low temperatures
• Non-Fermi liquid behavior and competing orders in strongly correlated systems (e.g., cuprate superconductors) are analyzed using functional and numerical RG methods

Nonequilibrium Statistical Mechanics

Extending the RG to nonequilibrium systems is an active area of research.

• The Kardar-Parisi-Zhang (KPZ) equation for interface growth exhibits nontrivial scaling exponents that can be studied with dynamic RG
• Reaction-diffusion systems (e.g., $A + A \to \emptyset$) have upper critical dimensions and universality classes analogous to equilibrium systems
• Driven diffusive systems and active matter present new classes of fixed points not found in equilibrium
• Dynamic critical phenomena near equilibrium phase transitions involve both static and dynamic critical exponents, with the dynamic exponent $z$ relating time and length scales as $\tau \sim \xi^z$

Limitations and Extensions

Non-Universal Corrections

The RG predicts universal critical exponents, but real experiments also see non-universal features.

• Corrections to scaling arise from irrelevant operators. Near the critical point, a quantity like the susceptibility behaves as $\chi \sim |t|^{-\gamma}(1 + a|t|^{\Delta} + \cdots)$, where $\Delta > 0$ is the leading correction-to-scaling exponent and $a$ is non-universal
• Non-universal amplitudes (the prefactors of power laws) depend on microscopic details, though certain ratios of amplitudes are universal
• Accounting for these corrections is essential when comparing RG predictions with experimental or numerical data, since measurements are never taken exactly at $t = 0$

Crossover Phenomena

When a system has competing interactions or symmetries, the RG flow may pass near multiple fixed points before reaching its ultimate destination.

• Dimensional crossover: a thin film (quasi-2D) crosses over from 2D to 3D critical behavior as the correlation length grows beyond the film thickness
• Quantum-classical crossover: a quantum system at finite temperature crosses over from quantum critical to classical critical behavior at a temperature scale set by $\hbar \omega \sim k_B T$
• Crossover scaling functions interpolate between the exponents of the two fixed points and are needed to describe the intermediate regime

Finite-Size Scaling

Real systems and simulations are always finite. Finite-size scaling theory, built on the RG, describes how critical behavior is modified when the system size $L$ is comparable to $\xi$.

• A singular quantity $Q$ near criticality takes the form $Q(t, L) = L^{x/\nu} \tilde{Q}(L^{1/\nu} t)$, where $\tilde{Q}$ is a universal scaling function and $x$ is the appropriate critical exponent
• Data from simulations at different $L$ values can be collapsed onto a single curve by plotting $L^{-x/\nu} Q$ vs. $L^{1/\nu} t$. This data collapse is a standard method for extracting critical exponents numerically
• Finite-size scaling connects with conformal field theory in 2D, where the finite-size spectrum of a transfer matrix encodes the operator content of the CFT