Introduction to Condensed Matter Physics

Many-body methods for superconductors, magnets and topological matter

About this course

The many-body treatment of solids — second quantisation, Fermi liquids, screening and electron-phonon coupling — applied to superconductivity, magnetism, phase transitions and the renormalisation group, localisation, and quantum Hall and topological phases.

What you will be able to do

  • Use second quantisation and the tight-binding model to describe electrons in solids, including graphene
  • Explain Fermi-liquid quasiparticles and compute screening from linear response
  • Derive the BCS gap and use Ginzburg–Landau theory to describe vortices and Josephson junctions
  • Analyse phase transitions with order parameters, critical exponents and the renormalisation group
  • Explain the quantum Hall effects and topological insulators in terms of the Berry phase

What this course covers

  • Fock Space and Occupation-Number Representation
  • Creation, Annihilation Operators and Pauli Algebra
  • Jellium Hamiltonian
  • Hartree-Fock Energy of the Electron Gas
  • The Exchange Hole
  • Correlation Energy and Its Numerical Values
  • Two-Site Hubbard Model and Mott Localization
  • Cooper's Problem: Bound State for Any Attraction
  • BCS Ground State and the Gap Equation
  • BCS Wave Function and Particle-Number Superposition
  • Bogoliubov Quasiparticles and the Density of States
  • Condensation Energy and the Critical Field
  • Heat Capacity Jump and Coherence Length
  • Full Worked Example: Aluminum from λ to Four Measurements
  • Exchange interaction: Coulomb plus antisymmetry
  • The Heisenberg Hamiltonian and ferromagnetic ground state
  • Holstein-Primakoff bosons and spin-wave dispersion
  • Bloch's T^(3/2) law and Mermin-Wagner theorem
  • Antiferromagnetic spin waves and limits of the Heisenberg model
  • Stoner criterion for itinerant ferromagnetism
  • EuO worked example: exchange constants to TC, D, and moment loss
  • Anderson Localization and the Transfer Matrix
  • Dimensionality Dependence of Localization
  • Scaling Theory of Localization and the Beta Function
  • Mott's Criterion for the Metal–Insulator Transition
  • Variable-Range Hopping and Finite-Temperature Localization
  • The Mott Insulator and the Hubbard Dimer Gap
  • Worked Example: Phosphorus in Silicon
  • Landau mean-field critical exponents
  • Failure of mean-field below four dimensions
  • Onsager's exact 2D Ising solution
  • Monte Carlo simulation and Binder cumulant
  • Scaling relations and universality classes
  • Ginzburg criterion and mean-field validity
  • Van der Waals fluid as a mean-field critical point
  • Tight-binding Hamiltonian and Bloch diagonalization
  • 1D chain: band, effective mass, and density of states
  • Square lattice: band, van Hove singularity, and Fermi surface
  • Graphene: two-band Hamiltonian and Dirac cone
  • Graphene as a semi-metal: DOS and gate-controlled carriers
  • SSH chain: alternating bonds, bulk gap, and topological edge states
  • Graphene worked example: numbers from one hopping
  • Landau Levels and Filling Factor
  • Integer Quantum Hall Effect and Disorder
  • Laughlin's Gauge Argument for Exact Quantization
  • Chiral Edge States and Zero Longitudinal Resistance
  • Fractional Quantum Hall Effect and Laughlin's Wave Function
  • Worked Example: GaAs Hall Bar from Density to Plateaus
  • Topological Nature of the Quantized Hall Conductance
  • Ginzburg-Landau Free Energy Functional
  • Kappa and the Type-I / Type-II Boundary
  • The Vortex: Topology, Flux Quantum, and Line Energy
  • GL Theory: Scope, Limits, and Place in Condensed Matter
  • Lower and Upper Critical Fields and Abrikosov's Lattice
  • Josephson Effect and AC Voltage–Frequency Relation
  • Worked Example: NbTi MRI Magnet
  • Coarse graining and the renormalization group idea
  • Exact decimation of the Ising chain
  • Fixed points, critical points, and universality
  • Exponents as formulas in y_T and y_H
  • Gaussian fixed point and the upper critical dimension
  • Universality classes and superfluid helium
  • End-to-end worked example: 2D Ising by Migdal–Kadanoff
  • Fermi Liquid: Adiabatic Continuity Premise
  • Thermal Mass from Heat Capacity
  • Quasiparticle Lifetime: ε² Phase-Space Argument
  • Luttinger's Theorem: Fermi Surface Volume
  • Landau Parameters and Fermi Liquid Responses
  • T² Resistance and Kadowaki–Woods Ratio
  • Worked Example: Sodium's Thermal Mass Decomposed
  • Berry Phase and Berry Curvature
  • Quantization of Berry Phase and the Chern Number
  • SSH Chain and Winding Number
  • Bulk-Boundary Correspondence as a General Principle
  • Chern Number and the Quantum Anomalous Hall Effect
  • Z₂ Topological Insulators and Bulk-Boundary Correspondence
  • Polyacetylene as a Worked Topological Example
  • Thomas-Fermi Screening and Screened Coulomb Interaction
  • Lindhard Function and the 2k_F Kink
  • Friedel Oscillations from the 2k_F Kink
  • Plasma Oscillations and Plasmon Dispersion
  • Kubo Formula: Linear Response as a Correlation Function
  • Kramers-Kronig Relations and the F-Sum Rule
  • Sodium Worked Example: Screening, Friedel, and Plasmon
  • Hubbard Model at Half Filling and the Mott Gap
  • The Kondo Effect: Logarithm and Resistance Minimum
  • Kondo Temperature from Poor-Man Scaling
  • Heavy Fermions and the Kadowaki-Woods Ratio
  • Cuprate Phase Diagram and the Universal Superconducting Dome
  • Open Problems: Pseudogap and Quantum Criticality
  • Course Synthesis: Fourteen Numbers and Three Habits
  • Electron-phonon coupling overview and lambda
  • Bloch-Grüneisen phonon resistivity
  • Fröhlich's phonon-mediated electron attraction
  • The polaron: electron dressed in phonons
  • Migdal's theorem and the hidden small parameter
  • Peierls instability and lattice dimerization
  • McMillan's formula: lambda to T_c

Syllabus

  1. Many-Body Foundations

    1. L1 · Second Quantisation and the Many-Body Problem
    2. L2 · The Tight-Binding Model and Graphene
    3. L3 · Fermi Liquids and Quasiparticles
    4. L4 · Linear Response, Screening and Plasmons
    5. L5 · Phonons, Electron-Phonon Coupling and Polarons

  2. Superconductivity

    1. L6 · BCS Theory of Superconductivity
    2. L7 · Ginzburg-Landau Theory, Vortices and Josephson Junctions

  3. Magnetism and Phase Transitions

    1. L8 · Magnetism: Exchange, the Heisenberg Model and Spin Waves
    2. L9 · Phase Transitions, Order Parameters and Critical Exponents
    3. L10 · The Renormalisation Group and Universality

  4. Disorder, Topology and Strong Correlation

    1. L11 · Disorder, Localisation and the Mott Transition
    2. L12 · The Quantum Hall Effects
    3. L13 · Topological Insulators and Berry Phase
    4. L14 · Strongly Correlated Matter, and the Close