Statistical Mechanics II

Landau–Ginzburg theory, the scaling hypothesis and the renormalisation group.

What this course covers

  • Statistical mechanics as microscopic-macroscopic bridge
  • Collective behavior from interactions
  • Statistical field theories and universality
  • Normal modes and lattice vibrations
  • Phonon dispersion and low-k linear behavior
  • Universal low-temperature heat capacity from phonons
  • Coarse graining and the continuum displacement field
  • Constructing an effective elastic field theory from symmetry
  • Isotropic elasticity in higher dimensions
  • Elasticity as a statistical field theory
  • Longitudinal and transverse modes in isotropic solids
  • Low-temperature heat capacity and universality
  • Phase transitions and the liquid-gas critical point
  • Critical exponents (α, β, γ, δ)
  • Critical opalescence and diverging correlation length
  • Landau-Ginzburg effective field theory
  • Landau-Ginzburg field theory construction
  • Saddle-point (mean-field) approximation
  • Mean-field critical exponents
  • Spontaneous symmetry breaking
  • Goldstone modes
  • Landau-Ginzburg theory for superfluidity
  • Phase fluctuation cost and Gaussian modes
  • d-dimensional Coulomb potential
  • Mermin-Wagner theorem and lower critical dimension
  • Landau-Ginzburg free energy functional
  • Scattering experiments and structure factor
  • Gaussian fluctuations around the mean-field state
  • Longitudinal vs. transverse susceptibility
  • Fourier-space decomposition and mode independence
  • Lorentzian scattering line shape and critical scattering
  • Real-space correlation function and correlation length
  • Divergence of the correlation length at T_c
  • Scaling relation between γ, ν, and η (Fisher relation)
  • Comparison of mean-field exponents with experiment
  • Statistical field theory for phase transitions
  • Critical exponent alpha and heat capacity singularities
  • Saddle point approximation for the partition function
  • Systematic expansion beyond saddle point
  • Fourier decomposition of fluctuation fields
  • Upper critical dimension d=4
  • Failure of mean-field theory below d=4
  • Ginzburg criterion for validity of mean-field theory
  • Critical points and universality
  • Failure of mean-field (saddle-point) exponents
  • Scaling hypothesis for the free energy
  • Gap exponent and crossover between singularities
  • Exponent identities from the scaling hypothesis
  • Scaling hypothesis for the correlation length
  • Josephson (hyperscaling) relation
  • Fisher identity and the exponent η
  • Scale invariance at the critical point
  • Critical points and universal scaling exponents
  • Scale invariance and power-law correlations
  • Renormalization group procedure: three steps
  • RG transformation of probability weights
  • RG flow equations and linearization near fixed point
  • Deriving thermodynamic scaling from RG exponents
  • General RG framework: parameter space and fixed points
  • Eigenvalues, eigendirections, and relevant vs. irrelevant operators
  • Basin of attraction and universality
  • Number of relevant directions and experimental tunability
  • Landau-Ginzburg statistical field theory
  • Fourier representation and mode decoupling
  • Gaussian model partition function and free energy
  • Singular behavior and upper critical dimension
  • Scaling form of the free energy and critical exponents
  • Coarse-graining step of the RG procedure
  • Rescaling and field renormalization steps
  • RG eigenvalues and relevant vs. irrelevant operators
  • Multiple fixed points and choice of rescaling factor z
  • Relevance of the m⁴ term and need for perturbative RG
  • Statistical field theory and partition function
  • Landau-Ginzburg Gaussian model
  • Fourier mode decomposition of the field
  • Gaussian propagator and two-point correlator
  • Wick's theorem for Gaussian averages
  • Perturbation theory for interacting field theory
  • Connected (cumulant) contractions via Wick's theorem
  • Symmetry constraints on the two-point correlator
  • Perturbative correction to susceptibility and shift of Tc
  • Breakdown of perturbation theory below four dimensions
  • Perturbation theory for partition functions
  • Divergence of perturbation theory below four dimensions
  • Momentum-shell coarse graining
  • Perturbative expansion in the interaction vertex
  • Wick's theorem and diagrammatic contractions
  • Renormalization of the parameter t at first order
  • Rescaling steps of the renormalization group
  • Differential RG flow equations
  • Fixed points and eigenvalue analysis
  • Wilson-Fisher fixed point and epsilon expansion
  • Partition function via statistical field theory
  • Renormalization group coarse graining step
  • Cumulant expansion of the effective Hamiltonian
  • Diagrammatic representation of perturbation terms
  • RG rescaling and renormalization steps
  • Differential RG flow equations
  • Gaussian fixed point and flow topology above d=4
  • Need for epsilon expansion below d=4
  • Second-order cumulant diagrams and u² corrections
  • Wilson-Fisher (O(N)) fixed point
  • Critical exponents from the O(N) fixed point
  • Universal critical exponents near phase transitions
  • Statistical field theory and order parameter symmetry
  • Failure of saddle-point and perturbation theory below four dimensions
  • Scaling hypothesis and exponent identities
  • Kadanoff's renormalization group idea and fixed points
  • Wilson's perturbative RG: integrating out shell modes
  • RG flow equations for t and u
  • Gaussian versus O(n) fixed points
  • Computing critical exponents from RG eigenvalues
  • Epsilon expansion and asymptotic series
  • Lattice models and microscopic degrees of freedom
  • Ising model
  • O(N) spin models (XY and Heisenberg)
  • Potts model and permutation symmetry
  • Approaches to solving lattice models
  • Position-space renormalization group in 1D
  • RG recursion relations and fixed points for 1D Ising
  • Correlation length and susceptibility near the 1D fixed point
  • Why exact decimation fails in higher dimensions
  • Transfer matrix method and exact solvability in 1D
  • Renormalization group in 1D Ising model
  • Challenges of RG on 2D lattices
  • Niemeijer–van Leeuwen cumulant approximation
  • Majority rule and cell-spin averaging
  • Recursion relation and fixed point (triangular lattice)
  • Critical exponents y_t and y_h from cell RG
  • Migdal–Kadanoff bond-moving approximation
  • Transfer matrix formulation of 1D models
  • Absence of phase transitions in 1D via Perron–Frobenius
  • Finite-size crossover and 2D phase transitions
  • Ising model as universal spin model
  • Low-temperature series expansion
  • Energy and heat capacity from series
  • Extracting critical exponents via ratio method
  • Alternating signs and variable transformations
  • High-temperature expansion and graph representation
  • 1D Ising model: exact solution via graphs
  • Ising model definition and setup
  • Low-temperature expansion of the Ising model
  • High-temperature expansion of the Ising model
  • Duality of the square-lattice Ising model
  • Self-duality and the exact critical point
  • Duality beyond the square lattice
  • Duality mapping to Z2 gauge theory in 3D
  • Elitzur's theorem: no spontaneous breaking of local gauge symmetry
  • Wilson loop as a non-local order parameter
  • Confinement and asymptotic freedom from Wilson loop behavior
  • High-temperature expansion of Ising partition function
  • Gas of non-intersecting vs. phantom loops
  • Counting random walks via transfer matrix
  • Diagonalising the connectivity matrix by Fourier transform
  • Spin–spin correlation function from walk counting
  • Divergence of correlation length at the critical point
  • Equivalence to the Gaussian (free-field) model
  • Fractal dimension of random walks and the upper critical dimension
  • Ising model partition function and graph expansion
  • Phantom loop approximation and its failure
  • Corrected loop sum with no-U-turn and crossing sign rules
  • Whitney's theorem: crossing parity via tangent winding
  • Oriented walk matrix with phase factors
  • Explicit 4×4 one-step transfer matrix T*
  • Fourier diagonalization to block-diagonal form
  • Partition function as determinant integral
  • High-temperature graphical expansion of Ising model
  • Phantom loop approximation and its failure
  • Exact loop expansion with no-backtrack and crossing sign
  • Transfer-matrix and Fourier diagonalization
  • Exact Ising partition function and its singularity structure
  • Kramers–Wannier duality and the critical temperature
  • Logarithmic divergence of the heat capacity (α = 0)
  • Critical exponents of the 2D Ising model
  • Fermionic (Grassmannian) interpretation of the 2D Ising model
  • Low-temperature expansion and Goldstone modes in n-component models
  • n-component unit spin vector model
  • Ground state and Goldstone modes
  • Integration measure and constraint on spin components
  • Non-linear sigma model Hamiltonian
  • Continuum limit and gradient expansion
  • Temperature-ordered perturbative expansion
  • Momentum-space Fourier representation and propagator
  • Perturbative renormalization group procedure
  • Diagrammatic evaluation of mode elimination
  • Renormalization of the coupling constant K
  • Physical interpretation: order vs. disorder in 2D
  • Non-linear sigma model and spin field
  • Transverse fluctuations and Goldstone modes
  • Renormalization group procedure for spin models
  • RG flow equation for coupling K and temperature T
  • Fixed points and the 2+epsilon expansion
  • Thermal and magnetic scaling exponents
  • Correlation length divergence in 2D Heisenberg model
  • Mermin-Wagner theorem and the n=2 puzzle
  • Spin-spin correlation functions: high vs low temperature
  • Topological defects (vortices) in the XY model
  • Vortex-antivortex binding and the KT transition
  • 2D XY model and its partition function
  • Topological defects and vortices
  • Decomposition of gradient into regular and topological parts
  • Coulomb interaction between topological defects
  • Core energy and single-defect partition function
  • Grand canonical Coulomb gas partition function
  • Dielectric screening by bound vortex pairs
  • Renormalization group recursion relations for k and y
  • XY model and low-temperature spin correlations
  • Topological defects (vortices) in the XY model
  • Energy and entropy balance for a single vortex
  • Coulomb gas representation of many vortices
  • Renormalization group (RG) flow equations for the Coulomb gas
  • RG flow diagram and the KT phase transition
  • Universal jump in superfluid density at the KT transition
  • Experimental verification via superfluid helium films
  • Essential singularity of the correlation length above Tc
  • Two-dimensional melting and dislocation unbinding
  • 2D elastic solid and strain tensor
  • Translational order parameter and quasi-long-range order
  • Dislocation unbinding and melting of translational order
  • Orientational order parameter and bond angles
  • Orientational correlations decay to a constant in the solid
  • Hexatic phase with quasi-long-range orientational order
  • Disclinations and orientational order melting
  • KTHNY two-transition melting scenario
  • Langevin equation and dissipative dynamics
  • Time-dependent Landau-Ginzburg equation
  • Critical slowing down and dynamic relaxation times
  • XY model vs. isotropic solid analogy
  • Goldstone modes and symmetry breaking
  • Elastic energy and elastic moduli
  • Fourier-space representation of elastic energy
  • Displacement fluctuation correlators
  • Long-range order and the Mermin–Wagner theorem
  • Power-law correlations in 2D and the KT transition
  • Lattice periodicity and the displacement ambiguity
  • Density-wave order parameter ρ_G
  • Landau-Ginzburg free energy functional
  • Dissipative dynamics and Brownian motion analogy
  • Time-dependent Landau-Ginzburg equation
  • Linearization and Fourier mode decomposition
  • Mode relaxation times and critical slowing down
  • Relationship between correlation length and relaxation time
  • Dynamic critical exponent z
  • Conserved versus non-conserved order parameter dynamics

Syllabus

  1. Lectures

    1. 1. Collective Behavior, from Particles to Fields Part 1
    2. 2. Lec 1 (continued); The Landau-Ginzburg Approach Part 1
    3. 3. The Landau-Ginzburg Approach Part 2
    4. 4. The Landau-Ginzburg Approach Part 3
    5. 5. The Landau-Ginzburg Approach Part 4
    6. 6. The Scaling Hypothesis Part 1
    7. 7. The Scaling Hypothesis Part 2
    8. 8. The Scaling Hypothesis Part 3
    9. 9. Perturbative Renormalization Group Part 1
    10. 10. Perturbative Renormalization Group Part 2
    11. 11. Perturbative Renormalization Group Part 3
    12. 12. Perturbative Renormalization Group Part 4
    13. 13. Position Space Renormalization Group, Part 1
    14. 14. Position Space Renormalization Group, Part 2
    15. 15. Series Expansions Part 1
    16. 16. Series Expansions Part 2
    17. 17. Series Expansions Part 3
    18. 18. Series Expansions Part 4
    19. 19. Series Expansions Part 5
    20. 20. Continuous Spins at Low Temperatures Part 1
    21. 21. Continuous Spins at Low Temperatures Part 2
    22. 22. Continuous Spins at Low Temperatures Part 3
    23. 23. Continuous Spins at Low Temperatures Part 4
    24. 24. Dissipative Dynamics
    25. 25. Continuous Spins at Low Temperatures Part 5
    26. 26. Continuous Spins at Low Temperatures Part 6