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These are self-study notes on Renormalization Group (RG) theory, covering critical phenomena, phase transitions, scaling laws, and universality classes. Notes are organized as articles with accompanying Python code for deeper understanding.

Course Overview¶

The renormalization group was born from puzzles over divergent integrals and critical phenomena, yet within half a century it has evolved into a general language for understanding nature. From fundamental particle interactions to starlings swirling in the sky, from layer-by-layer weights in neural networks to fluctuations in Earth's climate, RG thinking pervades the study of multi-scale complex systems. Macroscopic laws are not simple superpositions of microscopic rules, but emergent products of scale transformations. Through RG, we gain insight into why vastly different systems share the same behavioral principles and how to extract the factors governing the whole from a welter of details.

The common challenge of these multi-scale systems is this: when the correlation length tends to diverge, the traditional "separation of scales" assumption completely breaks down—local fluctuations no longer cancel out, but are amplified layer by layer through long-range correlations, coupling directly to macroscopic observables. Facing strong correlations among \(10^{23}\) degrees of freedom, the Renormalization Group adopts a "divide and conquer" strategy: progressively reducing short-range degrees of freedom, tracking how system parameters "flow" with observation scale, and ultimately converging to fixed points that govern macroscopic behavior.

It is precisely this flow that explains the origin of universality—why systems with vastly different microscopic mechanisms share the same critical exponents. Kenneth Wilson received the 1982 Nobel Prize in Physics for developing this theory. This tutorial series starts from the physical picture of critical phenomena, establishes statistical mechanics foundations, introduces the quantitative language of scaling laws and critical exponents, and ultimately delves into field-theoretic RG and modern frontier applications.

Completed Notes¶

Part 1: Motivation, Statistical Physics, and Critical Phenomena¶

Planned Topics¶

Part 1: Motivation, Statistical Physics, and Critical Phenomena (In Progress)

Landau Theory and Ginzburg Criterion—Success and Failure of Mean Field
The World of the Ising Model—From 1D Exact Solution to 2D Critical Point
Block Spins and Coarse-Graining—Kadanoff's Intuitive RG Picture

Part 2: Real-Space RG and Numerical Methods

Real-Space RG Recursion Relations—Relevant Dimensions and Phase Diagrams
Finite-Size Scaling and Data Collapse
Monte Carlo Simulation of the Ising Model—From Metropolis to Cluster Algorithms
Monte Carlo RG—Constructing RG Transformations from Numerical Data
Ginzburg-Landau Functional and Continuum Field Description
From Lattice Ising to φ⁴ Field Theory

Part 3: Field-Theoretic RG and Quantum Field Theory

Path Integrals and Gaussian Fields—From Harmonic Oscillator to Free Scalar Field
φ⁴ Theory and Feynman Diagrams—Where Do Divergences Come From?
Renormalization Techniques—Regularization, Counterterms, and Physical Parameters
Callan-Symanzik Equation and β Function
Wilson's Perspective on RG—"Integrating Out High-Momentum Modes"
ε Expansion and Wilson-Fisher Fixed Point
RG in Quantum Field Theory—QED, QCD, and Asymptotic Freedom
Non-Perturbative RG—Functional RG and Large-N Techniques

Part 4: Tensor Networks, DMRG, Complex Networks, and Non-Equilibrium RG

Density Matrix Renormalization Group (DMRG) and Matrix Product States (MPS)
Tensor Networks and Multi-Scale Entanglement Renormalization (MERA, TNR)
RG on Networks and Graphs—From Laplacian RG to Network RG
RG in Non-Equilibrium Systems and Active Matter
Quantum Gravity, Asymptotic Safety, and Holographic RG

Part 5: RG × Machine Learning & Interdisciplinary Frontiers

RG, Information Theory, and Parameter Compression—From Fisher Information to "Emergent Theory"
Deep Learning and Variational RG—RBM, Neural RG, and Mutual Information RG
Learning RG with Deep Learning—FRG+NN, Neural Tensor Network, etc.
New Directions for RG in Complex Systems and Interdisciplinary Research
Frontier Paper Reviews (Throughout the Tutorial)

Code Demonstrations¶

Lecture 2: Why Do We Need the Renormalization Group

Real-Space Renormalization Group Flow

Kadanoff block spin coarse-graining: 128×128 → 64×64 → 32×32, microscopic fluctuations fade away as macroscopic order emerges

Lecture 3: Statistical Mechanics Review—Partition Function, Free Energy, and Fluctuations

Energy-Entropy Trade-off	Free Energy Landscape	Feature Importance

TERP Explainable AI: Using thermodynamic free energy principles to find optimal explanations

Lecture 5: Phase Transitions and Critical Exponents—Scaling Laws and Universality Classes (Part II)

3D Percolation Animation	Finite-Size Scaling Analysis

Left: 3D Site Percolation cluster evolution; Right: Finite-size scaling analysis

License¶

This project is licensed under CC BY-NC-ND 4.0.

Citation¶

If you use or reference these notes, we welcome a citation. BibTeX format:

@misc{liu2024renormalization,
  author       = {Liu, Zhihang},
  title        = {Renormalization Group},
  year         = {2025},
  url          = {https://github.com/Liu-Zhihang/Renormalization-Group},
  note         = {Self-study notes on critical phenomena, phase transitions, scaling laws and universality classes}
}