4.5 Renormalization group

Renormalization group (RG) provides a systematic framework for understanding how physical systems behave across different length scales. Near a phase transition, fluctuations occur on all scales simultaneously, and RG gives you the mathematical machinery to handle that. It explains why microscopically different systems can share the same critical behavior, and it connects the short-distance physics you can write down in a Hamiltonian to the long-distance physics you actually measure.

Scale invariance sits at the heart of RG theory. At a critical point, a system looks statistically the same whether you zoom in or out. Correlation functions decay as power laws rather than exponentials, and this self-similarity is what RG exploits. The payoff is universality: systems as different as a ferromagnet and a liquid near its critical point can have identical critical exponents, because RG shows that their differences become irrelevant at long wavelengths.

Fundamentals of Renormalization Group

The core idea behind RG is deceptively simple: systematically remove short-distance degrees of freedom, then rescale the system so it looks like the original. After this procedure, the coupling constants in your effective Hamiltonian will generally change. Tracking how they change as you repeat this process reveals the large-scale physics of the system, including whether it's ordered, disordered, or sitting right at a critical point.

Concept of Scale Invariance

A system is scale-invariant when its statistical properties don't change under a rescaling of length. This doesn't happen at arbitrary temperatures; it's a special property of the critical point.

• Correlation functions at criticality follow power laws: $G(r) \sim r^{-(d-2+\eta)}$, where $d$ is the spatial dimension and $\eta$ is the anomalous dimension.
• Physical observables like the susceptibility and specific heat also diverge as power laws in $|T - T_c|$.
• A familiar example is critical opalescence in fluids: density fluctuations occur on all length scales near the liquid-gas critical point, scattering light of all wavelengths and making the fluid appear milky.
• This self-similarity across scales is exactly what makes RG the natural tool for analyzing critical points.

Universality in Critical Phenomena

One of the most striking predictions of RG is that systems with completely different microscopic physics can share the same critical exponents. These systems belong to the same universality class.

What determines the universality class is not the detailed form of the interactions, but a small set of global properties:

• Spatial dimensionality $d$ of the system
• Symmetry of the order parameter (scalar, vector, etc.)
• Range of interactions (short-range vs. long-range)

This is why a uniaxial ferromagnet (Ising symmetry) and a liquid-gas transition (also a scalar order parameter in $d = 3$) have the same critical exponents. The microscopic details, such as whether you're dealing with atomic spins or fluid molecules, get washed out under repeated coarse-graining.

Wilson's Approach to Renormalization

Kenneth Wilson transformed RG from a set of tricks in perturbation theory into a conceptual framework for multi-scale physics. His key insight was to think of RG as a flow in the space of all possible Hamiltonians.

The procedure works in three steps:

1. Coarse-grain: Integrate out (or average over) degrees of freedom at the shortest length scales, removing fluctuations below some cutoff.
2. Rescale: Shrink the system back to its original size so you can compare the new effective Hamiltonian to the old one.
3. Renormalize: Adjust the fields so the partition function retains the same form.

After each iteration, the coupling constants shift. The equations governing this shift are the RG flow equations, and their structure (fixed points, flow directions) encodes the phase diagram and critical behavior of the system.

Renormalization Group Transformations

There are two main arenas for performing RG: directly in real space, or in momentum (Fourier) space. Each has strengths depending on the problem.

Real-Space Renormalization

Real-space RG works directly on the lattice or physical configuration of the system. The idea is to group nearby degrees of freedom into blocks and define new effective interactions between the blocks.

For a concrete example, consider the 1D Ising model:

1. Group every pair (or triplet) of neighboring spins into a single block spin.
2. Sum over the internal configurations of each block to obtain an effective coupling between block spins.
3. Rescale lengths so the new lattice spacing equals the original.
4. Compare the new Hamiltonian to the old one to extract how couplings have changed.

This approach is intuitive and works well for lattice models like the Ising model and percolation. The trade-off is that the blocking procedure isn't unique, and in dimensions higher than one, it often generates new couplings not present in the original Hamiltonian, forcing you to truncate.

Momentum-Space Renormalization

Momentum-space RG, the approach Wilson used most powerfully, operates in Fourier space. Instead of grouping real-space blocks, you integrate out high-momentum (short-wavelength) modes.

• Start with a field theory defined up to some ultraviolet cutoff $\Lambda$.
• Integrate out modes with momenta in a thin shell: $\Lambda/b < |k| < \Lambda$, where $b > 1$ is the rescaling factor.
• Rescale momenta so the new cutoff is again $\Lambda$.
• Read off the new effective couplings.

This framework connects naturally to perturbative expansions. Wilson and Fisher's famous $\epsilon$-expansion sets $\epsilon = 4 - d$ and treats the deviation from the upper critical dimension $d = 4$ as a small parameter. For the Ising universality class in $d = 3$ ($\epsilon = 1$), this gives approximate but remarkably accurate critical exponents.

Decimation and Blocking Techniques

These are specific implementations of real-space RG:

• Decimation removes a subset of degrees of freedom (e.g., every other spin on a lattice) and sums over them exactly. It works cleanly in 1D but becomes approximate in higher dimensions because new longer-range couplings are generated.
• Blocking (Kadanoff's original idea) groups clusters of spins into single block variables using a majority-rule or similar prescription.

Both methods must be designed carefully to preserve the symmetries of the original Hamiltonian. If your blocking rule breaks a symmetry the system has, the resulting flow will give incorrect results.

Fixed Points and Critical Exponents

Stable vs. Unstable Fixed Points

A fixed point of the RG transformation is a Hamiltonian $H^*$ that maps to itself under coarse-graining. At a fixed point, the system is scale-invariant.

The stability of a fixed point tells you about the physics:

• Stable (attractive) fixed points draw in nearby RG trajectories. These correspond to phases: the high-temperature disordered phase and the low-temperature ordered phase each have their own stable fixed point.
• Unstable fixed points repel trajectories along at least one direction in coupling-constant space. The critical point of a phase transition is an unstable fixed point. The system flows away from it unless you tune the temperature (or another parameter) to exactly $T_c$.

The number of unstable directions at a fixed point equals the number of parameters you need to tune to reach that critical point. A standard second-order transition has one unstable direction (temperature), so you tune one parameter.

Calculation of Critical Exponents

Critical exponents come from linearizing the RG transformation around the unstable fixed point. Here's the procedure:

1. Write the RG transformation near the fixed point: $K' = R(K)$, where $K$ represents the set of coupling constants.
2. Linearize: $\delta K' = M \cdot \delta K$, where $M$ is the Jacobian matrix $\partial R / \partial K$ evaluated at $K^*$.
3. Find the eigenvalues $\lambda_i$ of $M$. Each eigenvalue defines a scaling exponent $y_i$ through $\lambda_i = b^{y_i}$, where $b$ is the spatial rescaling factor.
4. Eigenvalues with $y_i > 0$ are relevant (they grow under RG, corresponding to unstable directions). Those with $y_i < 0$ are irrelevant (they shrink and don't affect critical behavior). Those with $y_i = 0$ are marginal.

The familiar critical exponents ($\nu$, $\eta$, $\alpha$, $\beta$, $\gamma$, $\delta$) are all determined by the relevant eigenvalues. For instance, the correlation length exponent is $\nu = 1/y_T$, where $y_T$ is the thermal eigenvalue. The scaling relations (e.g., $\alpha + 2\beta + \gamma = 2$) follow automatically from the RG structure.

Universality Classes

Since critical exponents depend only on the fixed-point structure, and the fixed point is determined by symmetry, dimensionality, and interaction range, all systems flowing to the same fixed point share the same exponents. Some important universality classes:

Applications in Condensed Matter

Ising Model and Phase Transitions

The Ising model is the simplest system exhibiting a nontrivial phase transition, and it serves as the testing ground for RG methods.

• In $d = 2$, Onsager's exact solution gives $\nu = 1$, $\beta = 1/8$, $\gamma = 7/4$. RG reproduces these and provides the conceptual framework for why they take these values.
• In $d = 3$, no exact solution exists. RG methods ($\epsilon$-expansion, Monte Carlo RG, conformal bootstrap) give the best available estimates: $\nu \approx 0.630$, $\beta \approx 0.326$, $\gamma \approx 1.237$.
• The same universality class describes binary alloys (order-disorder transitions), lattice gas models, and the liquid-gas critical point, all confirmed experimentally.

Kondo Effect and Scaling

The Kondo effect describes the anomalous increase in resistivity at low temperatures when a dilute magnetic impurity sits in a non-magnetic metal. Perturbation theory breaks down because the effective coupling between the impurity spin and conduction electrons grows under RG flow toward low energies.

• At high temperatures, the impurity spin is weakly coupled (the coupling is irrelevant in the RG sense at the free-electron fixed point).
• As temperature decreases, the coupling grows and flows toward a strong-coupling fixed point where the impurity spin is fully screened by conduction electrons, forming a singlet.
• The crossover scale is the Kondo temperature $T_K$, and physical quantities scale as universal functions of $T/T_K$.

Wilson's numerical RG solution of the Kondo problem was one of the early triumphs of the method. The physics is analogous to asymptotic freedom in QCD, where the coupling also grows at low energies.

Quantum Criticality

Quantum phase transitions occur at $T = 0$ and are driven by a non-thermal control parameter (pressure, magnetic field, doping) rather than temperature. Quantum fluctuations play the role that thermal fluctuations play at classical transitions.

• The quantum critical point influences a wide "fan" of the finite-temperature phase diagram, the quantum critical region, where neither classical thermal physics nor simple ground-state quantum mechanics applies.
• RG for quantum critical points maps the $d$-dimensional quantum problem onto a $(d+z)$-dimensional classical problem, where $z$ is the dynamical critical exponent relating time and space scaling: $\omega \sim k^z$.
• This framework applies to quantum magnets, heavy-fermion compounds (like $\text{CeCu}_6$), and is relevant to understanding the phase diagrams of cuprate high-$T_c$ superconductors.

Numerical Renormalization Methods

Monte Carlo Renormalization Group

When analytical RG is intractable (complex lattice geometries, competing interactions), Monte Carlo RG provides a numerical alternative.

1. Generate equilibrium configurations of the full system using Monte Carlo sampling.
2. Apply a blocking transformation to produce coarse-grained configurations.
3. Measure correlation functions at both the original and blocked scales.
4. Extract the RG transformation and its linearization near the fixed point from the relationship between configurations at different scales.
5. Compute critical exponents from the eigenvalues of the linearized transformation.

This method is especially valuable for disordered systems and models with complex phase diagrams where perturbative approaches fail.

Density Matrix Renormalization Group (DMRG)

DMRG, developed by Steven White in 1992, is the most accurate numerical method for 1D quantum many-body systems. Despite its name, it's conceptually different from Wilson's RG: rather than coarse-graining in scale, it optimally truncates the Hilbert space.

• The system is grown iteratively, and at each step the reduced density matrix of a subsystem determines which states to keep.
• States with the largest density-matrix eigenvalues capture the most entanglement and are retained; the rest are discarded.
• For 1D gapped systems, DMRG converges exponentially fast and gives ground-state energies accurate to 10 or more significant figures.
• Applications include spin chains ($S = 1$ Haldane chain), quantum impurity problems, and topological phases.
• Extensions to 2D and time-dependent problems exist (tensor network methods) but are more computationally demanding.

Functional Renormalization Group (FRG)

FRG extends Wilson's ideas to continuous field theories in a non-perturbative way. Instead of tracking a finite set of coupling constants, FRG tracks the flow of the entire effective action $\Gamma_k$ as the momentum cutoff $k$ is lowered.

• The central equation is the Wetterstein equation (also called the exact RG equation): $\partial_k \Gamma_k = \frac{1}{2} \text{Tr}\left[(\Gamma_k^{(2)} + R_k)^{-1} \partial_k R_k\right]$, where $R_k$ is a regulator function and $\Gamma_k^{(2)}$ is the second functional derivative of the effective action.
• This equation is exact but must be approximated in practice (truncations of the effective action).
• FRG bridges perturbative and non-perturbative regimes and has been applied to Fermi liquid instabilities, frustrated magnets, and quantum criticality in correlated electron systems.

Limitations and Extensions

Non-Perturbative Effects

Standard RG often relies on perturbative expansions (in $\epsilon$, in the coupling constant, etc.), which can fail for strongly coupled systems.

• The $\epsilon$-expansion is an asymptotic series, not a convergent one. Resummation techniques (Borel summation, Padé approximants) are needed to extract reliable numbers.
• Topological effects (vortices, instantons, domain walls) are non-perturbative and invisible to any finite order in perturbation theory. The Berezinskii-Kosterlitz-Thouless transition in the 2D XY model is a classic example where topological defects drive the transition.
• Modern non-perturbative approaches include the conformal bootstrap (which gives the most precise critical exponents known for the 3D Ising model) and exact RG/FRG methods.

Conformal Field Theory Connection

At a critical point, scale invariance is often enhanced to the larger conformal symmetry group, which includes rotations, translations, dilations, and special conformal transformations.

• In $d = 2$, conformal symmetry is infinite-dimensional (the Virasoro algebra), and conformal field theory (CFT) provides exact solutions for critical exponents and correlation functions. The 2D Ising model corresponds to a CFT with central charge $c = 1/2$.
• In $d > 2$, conformal symmetry is finite-dimensional but still highly constraining. The conformal bootstrap program uses consistency conditions (unitarity, crossing symmetry) to bound and determine critical exponents without any Lagrangian input.
• CFT also connects to the AdS/CFT correspondence in string theory, providing a bridge between condensed matter and high-energy physics.

Renormalization in Disordered Systems

Quenched disorder (impurities frozen in place) introduces new complications for RG because you must average over disorder realizations.

• The Harris criterion states that disorder is relevant (changes the critical behavior) if the pure-system specific heat exponent satisfies $\alpha > 0$.
• The replica trick handles disorder averaging by introducing $n$ copies of the system and taking $n \to 0$, which can lead to replica symmetry breaking in systems like spin glasses.
• Random-field systems (e.g., the random-field Ising model) have their own universality classes, and dimensional reduction arguments that naively connect them to pure systems in lower dimensions turn out to fail due to non-perturbative effects.
• Anderson localization, where disorder causes quantum wavefunctions to become spatially localized, is also analyzed using RG (the nonlinear sigma model approach).

Experimental Verification

Critical Exponent Measurements

Precise experimental tests of RG predictions require careful measurements very close to $T_c$, where power-law behavior dominates.

• Neutron scattering measures the correlation length $\xi$ and the structure factor $S(q)$ directly, giving access to $\nu$ and $\eta$.
• Specific heat measurements near $T_c$ determine $\alpha$. For the superfluid transition in $^4\text{He}$, space-based experiments (to minimize gravity effects) measured $\alpha = -0.0127 \pm 0.0003$, in excellent agreement with RG predictions for the 3D XY universality class.
• Susceptibility and magnetization measurements give $\gamma$ and $\beta$.
• Challenges include achieving sufficient temperature resolution near $T_c$, eliminating impurity effects, and accounting for corrections to scaling from irrelevant operators.

Scaling Relations in Experiments

RG predicts not just individual exponents but relationships between them, as well as universal scaling functions.

• Data collapse: If you plot a thermodynamic quantity as a function of the appropriate scaled variable (e.g., $\xi / L$ for finite-size scaling, or $t \cdot |h|^{-1/\beta\delta}$ for the equation of state), data from different temperatures or fields should collapse onto a single universal curve. Achieving good collapse is strong evidence for the scaling hypothesis.
• Scaling relations like $\gamma = \nu(2 - \eta)$ and the hyperscaling relation $2 - \alpha = d\nu$ have been verified across many systems.
• These tests apply to superconductors, liquid crystals, polymer solutions, and many other systems.

Universality Across Different Systems

The most dramatic confirmation of RG is that systems with completely different microscopic physics yield the same critical exponents when they share the same universality class.

• The liquid-gas critical point of $\text{CO}_2$ and the Curie point of a uniaxial ferromagnet both belong to the 3D Ising class, and their measured exponents agree within experimental error.
• The superfluid transition of $^4\text{He}$ matches the 3D XY predictions with extraordinary precision.
• These comparisons across different materials and different types of phase transitions provide the most compelling evidence that RG captures something fundamental about how nature organizes itself near critical points.