The Renormalization Group and Condensed Matter Physics
Published on 27. May 2025
328 pages
978-0-691-27543-7 (ISBN)
Description
A graduate-level entrée to the application of renormalization group theory to condensed matter physics Renormalization group 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 "active matter"; and more.
- 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 entrée, suitable for both course use and self-study
- Features material ideal for graduate-level students as well as researchers
- Includes exercises throughout
- Offers a solutions manual for exercises (available only to instructors)
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David R. Nelson | Grace H. Zhang
The Renormalization Group and Condensed Matter Physics
Book
07/2025
Princeton University Press
€87.00
Shipment within 10-20 days
Persons
David R. Nelson is professor of physics and applied physics and the Arthur K. Solomon Professor of Biophysics at Harvard University. Grace H. Zhang obtained her PhD in theoretical condensed matter physics from Harvard University.
Content
- Cover
- Contents
- Preface
- Introduction
- 1. Critical Phenomena
- 1.1 Critical phenomena in magnets
- 1.1.1 Critical exponents
- 1.1.2 Models of magnetism
- 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's 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 field theory
- 1.5.2 Other functional integrals
- 1.6 First order phase transitions
- 1.6.1 First and second order phase transitions
- 1.6.2 Mean field theory and first order transitions in Ising models
- 1.6.3 First order transitions in nematic liquid crystals
- 1.7 Beyond mean field theory
- 1.7.1 T&Tc
- 1.7.2 The Ginzburg criterion: when are fluctuations 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's 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 Tc
- 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
- 3.3 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(3) 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 finite magnetic field
- 3.7.3 Correlation length critical exponent
- 3.7.4 Renormalization group flows near d=2
- 3.8 Renormalization group for quantum rotors at finite 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 fluctuating 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's argument
- 4.4.2 Continuum limit with self-avoidance
- 4.4.3 Self-avoiding polymers and the n0 limit of vector spins
- 4.4.4 High temperature series expansion
- 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 flat phase
- 5.1.3 Analogy with superconductivity
- 5.1.4 Simulation model
- 5.1.5 Low temperature statistical mechanics of the flat phase
- 5.2 In-plane elasticity of flat sheet polymers
- 5.2.1 Case I: finite 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: "liquid" 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 Föppl-von Kármán 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 fluid mechanics
- 6.1.1 Convective derivative
- 6.1.2 Mass conservation
- 6.1.3 Shear viscosity of Newtonian fluid
- 6.1.4 Experimental facts
- 6.1.5 Non-uniform shear stresses
- 6.2 Navier-Stokes equations
- 6.2.1 Pipe flow 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: ?=0
- 6.4.2 General case: ??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
- Bibliography
- Index
