The Renormalization Group and Condensed Matter Physics

Physics & Astronomy

The Renormalization Group and Condensed Matter Physics

David Nelson and Grace H. Zhang

A graduate-level entr茅e to the application of renormalization group theory to condensed matter physics

Hardcover

Price:: $85.00/拢70.00
ISBN:
Published:: Jul 15, 2025
Copyright:: 2025
Pages:: 312
Size:: 8 x 10 in.
Illus:: 102 color illus. 6 tables.
Main_subject:: Physics & Astronomy

ebook

Price:: $85.00/拢70.00
ISBN:
Published:: Jul 15, 2025
Copyright:: 2025
Pages:: 312
Size:: 8 x 10 in.
Illus:: 102 color illus. 6 tables.
Main_subject:: Physics & Astronomy

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Renormalization group (RG) ideas have had a major impact on condensed matter physics for more than a half century. This book develops the theory and illustrates the broad applicability of the renormalization group to major problems in condensed matter physics. Based on course materials developed and class-tested by the authors at Harvard University, the book will be especially useful for students, as well as researchers in condensed matter physics, soft matter physics, biophysics, and statistical physics. After reviewing Ising models, lattice gases, and critical point phenomena, the book covers quantum critical phenomena; the statistical mechanics of linear polymer chains; fluctuating sheet polymers; the dynamics associated with the Navier-Stokes equations and simplified models of randomly stirred fluids; the properties of 鈥渁ctive matter鈥�; and more. Exercises are included throughout.

Explores the broad applicability of renormalization groups to condensed matter
Covers critical phenomena in different dimensions, quantum critical points, polymer physics and flexural phonons in free-standing graphene, nonequilibrium fluid dynamics, and more
Provides a modern, physics-centered entr茅e, suitable for both course use and self-study
Features material ideal for for graduate-level students as well as researchers
Includes exercises throughout
Offers a solutions manual for exercises (available only to instructors)

Preface

Introduction
1 Critical Phenomena
- 1.1 Critical phenomena in magnets
  - 1.1.1 Critical exponents
  - 1.1.2 Models of magnetism
    - 1.1.2.1 Simpli铿乧ation of the Heisenberg model
- 1.2 Ising model and liquid/gas critical point: lattice gas model
  - 1.2.1 Antiferromagnetic version of the lattice gas
- 1.3 Exact solution of the 1d Ising model
  - 1.3.1 Spin-spin correlation function
  - 1.3.2 Spontaneous magnetization
  - 1.3.3 Spin susceptibility
  - 1.3.4 Ising antiferromagnet in 1d
  - 1.3.5 Landau鈥檚 argument for no phase transition in 1d
- 1.4 2d Ising model
  - 1.4.1 Peierls argument
  - 1.4.2 Duality theorem for 2d Ising model
  - 1.4.3 Transfer matrix solution for the 2d Ising Model
- 1.5 The coarse-grained Landau-Ginzburg partition function
  - 1.5.1 Mean 铿乪ld theory
    - 1.5.1.1 Case I: T > T_c (F₂ >0)
    - 1.5.1.2 Case II: T < T_c (F₂ <0)
    - 1.5.1.3 Case III: T= T_c (F₂ = 0)
  - 1.5.2 Other functional integrals
    - 1.5.2.1 Super铿倁id Helium 4
    - 1.5.2.2 Quantum spin models
- 1.6 First order phase transitions
  - 1.6.1 First and second order phase transitions
  - 1.6.2 Mean 铿乪ld theory and 铿乺st order transitions in Ising models
  - 1.6.3 First order transitions in nematic liquid crystals
- 1.7 Beyond mean 铿乪ld theory
  - 1.7.1 T > T_c
  - 1.7.2 The Ginzburg criterion: when are 铿倁ctuations important?
- 1.8 Exercises

2 The Renormalization Group and the Epsilon Expansion
- 2.1 Correlation functions and the magnetic susceptibility
  - 2.1.1 Fourier analysis
  - 2.1.2 Magnetization correlation function
  - 2.1.3 Generating function for higher order correlations (Wick鈥檚 theorem)
- 2.2 Perturbation theory
  - 2.2.1 Graphical rules
  - 2.2.2 Vertex renormalization
  - 2.2.3 Fixing perturbation theory
- 2.3 Renormalization group for the Gaussian model
  - 2.3.1 Correlation length
  - 2.3.2 Susceptibility
- 2.4 Renormalization group with interactions
  - 2.4.1 Other critical exponents
  - 2.4.2 Corrections to scaling
  - 2.4.3 Other irrelevant variables
- 2.5 Direct calculation of the susceptibility near four dimensions
- 2.6 Dangerous irrelevant variables: XY model susceptibility below T_c
- 2.7 Exercises

3 Quantum Critical Phenomena
- 3.1 Path integrals and quantum statistical mechanics
  - 3.1.1 Monte Carlo simulations of path integrals
- 3.2 Particle in a quantum double-well: analytic treatment
  - 3.2.1 1d classical Ising chain: transfer matrix and the double-well quantum oscillator
    - 3.2.1.1 From the 1d classical Ising model to a single quantum spin
    - 3.2.1.2 Connection with the 1d Ising model at low temperatures
- 3.3 Two-component O(2) quantum rotors
- 3.4 Three-component O(3) quantum rotors
  - 3.4.1 Realization of quantum rotors from quantum Heisenberg spins
- 3.5 Path integral for rigid rotors
  - 3.5.1 XY rotor path integrals
  - 3.5.2 O(N) rotor path integrals
- 3.6 Quantum ordered phase
- 3.7 Finite temperature rotors
  - 3.7.1 Statistical mechanics of the classical ordered phase
  - 3.7.2 Generalization to a 铿乶ite magnetic 铿乪ld
  - 3.7.3 Correlation length critical exponent
  - 3.7.4 Renormalization group 铿俹ws near d= 2
    - 3.7.4.1 Correlation length divergence in d= 2 dimensions
- 3.8 Renormalization group for quantum rotors at 铿乶ite temperature
  - 3.8.1 Consequences of the quantum recursion relations for n= 3 rotors
  - 3.8.2 Crossovers for quantum rotors at d= 2
- 3.9 Exercise

4 Linear Polymer Chains
- 4.1 The random walking polymer
- 4.2 Freely-jointed chain model: polymers with independently 铿倁ctuating segments
- 4.3 Worm-like chain model: polymers with interacting segments
  - 4.3.1 Zero force partition function
  - 4.3.2 Spontaneous mass generation in polymer chains
- 4.4 Polymers with self-avoidance
  - 4.4.1 Flory鈥檚 argument
  - 4.4.2 Continuum limit with self-avoidance
  - 4.4.3 Self-avoiding polymers and the n鈫�0 limit of vector spins
  - 4.4.4 High temperature series expansion
    - 4.4.4.1 Partition function
    - 4.4.4.2 Spin-spin correlation function
- 4.5 Alternative epsilon expansions for linear polymers
- 4.6 Exercise

5 Statistical Mechanics of Freely Fluctuating Elastic Sheets
- 5.1 Statistical mechanics of sheet polymers
  - 5.1.1 Connection with a metric tensor
  - 5.1.2 Goldstone modes of the 铿俛t phase
  - 5.1.3 Analogy with superconductivity
  - 5.1.4 Simulation model
  - 5.1.5 Low temperature statistical mechanics of the 铿俛t phase
- 5.2 In-plane elasticity of 铿俛t sheet polymers
  - 5.2.1 Case I: 铿乶ite momentum modes
  - 5.2.2 Case II: zero momentum modes
  - 5.2.3 Response of planar crystals to external forces
- 5.3 Pure out-of-plane elasticity: 鈥渓iquid鈥� sheets (e.g., lipid bilayers)
- 5.4 General treatment
  - 5.4.1 Integrating out the in-plane phonons
  - 5.4.2 Self-consistent theory
- 5.5 Renormalization group treatment
- 5.6 Overview of the statistical mechanics of atomically thin plates
  - 5.6.1 Nonlinear F枚ppl-von K谩rm谩n equations at T= 0
  - 5.6.2 Thin solid shells and structures
  - 5.6.3 A more sophisticated renormalization group for thermalized membranes
  - 5.6.4 Experiments on graphene cantilevers
  - 5.6.5 Path integrals for long graphene ribbons
- 5.7 Exercises

6 Renormalization and Fluid Dynamics
- 6.1 Basics of 铿倁id mechanics
  - 6.1.1 Convective derivative
  - 6.1.2 Mass conservation
  - 6.1.3 Shear viscosity of Newtonian 铿倁id
  - 6.1.4 Experimental facts
  - 6.1.5 Non-uniform shear stresses
- 6.2 Navier-Stokes equations
  - 6.2.1 Pipe 铿俹w at low Reynolds number
  - 6.2.2 Flow past a sphere (dry water)
  - 6.2.3 Flow past a sphere (wet water)
  - 6.2.4 Energy transfer in Navier-Stokes equations
- 6.3 Mesoscopic turbulence: Navier-Stokes with random stirring
  - 6.3.1 Time correlation functions
  - 6.3.2 Linear theory in terms of a Reynolds number
- 6.4 Noisy Burgers equation
  - 6.4.1 Linear case: 位= 0
  - 6.4.2 General case: 位鈮�0
- 6.5 Dynamical renormalization group
  - 6.5.1 Noise renormalization
  - 6.5.2 Vertex renormalization
  - 6.5.3 Recursion relations
  - 6.5.4 Other nonlinear terms are irrelevant variables
  - 6.5.5 Surface growth
- 6.6 Flocking and active matter
- 6.7 Exercises

7 Appendices

7.1 Path integral expression for the density matrix
7.2 Path integral for O(3) quantum rotors
7.3 Mapping quantum operator correlation functions onto equal time classical path integrals

References
Index

"This is one of the very best physics books I have ever read. I like the clarity of thought, the physical intuition that has been masterfully woven into the text, the pedagogical structure that builds one thought upon another, the clear figures, and the choice of topics that start from basic knowledge and end with the most modern applications of the techniques introduced. The attractive writing style also makes the text easy and fun to read. A truly outstanding introduction and the textbook that the field has been waiting for.”—Karin Dahmen, University of Illinois, Urbana–Champaign

“David Nelson is a leading condensed matter theorist, and this textbook reflects the wide and deep contributions that he has made for several decades. It explains how to use the renormalization group to extract the universal properties of continuous phase transitions in a wide range on physically important systems. I am not aware of any other book with the scope of this one.”—Eduardo Fradkin, University of Illinois, Urbana–Champaign

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The Renormalization Group and Condensed Matter Physics

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