Condensed Matter Theory, PHAS0059

  • Course Outline and Lecture Notes

    1. From the Harmonic Oscillator to Phonons

    • The Harmonic Oscillator
    • Two Coupled Oscillators
    • The Harmonic Chain
    • Specific Heat of Phonons in the Harmonic Crystal

    2. Second Quantisation

    • Identical Particles
    • Occupation Numbers and Fock Space
    • Creation and Annihilation Operators
    • Transformation between Bases
    • Single- and Two-Particle Operators
    • Unitary and Bogoliubov Transformations

    3. The Weakly-Interacting Bose Gas

    • Bose-Einstein Condensation
    • Mean Field Theory of Weakly Interacting Bose Gas
    • Landau's Critical Superfluid Velocity
    • Superfluid Fraction

    4. Quantum Magnets

    • The Heisenberg Model
    • Holstein-Primakoff Transformation
    • Linear Spin-Wave Theory
    • The Heisenberg Ferromagnet
    • The Heisenberg Antiferromagnet
    • One-Dimensional Spin Chains
    • Haldane's Conjecture
    • Jordan-Wigner Transformation for S=1/2 chains
    • The Haldane S=1 chain, AKLT Model

    5. The Free Fermi Gas

    • Fermi Function and Density of States
    • The Sommerfeld Expansion

    6. Landau Theory of Fermi Liquids

    • Lifetime of Quasiparticles
    • Relation between bare Fermions and Quasiparticles
    • Parametrising Excitation Energies
    • Measuring the Landau Parameters

    7. BCS Theory of Superconductivity

    • Electron-Phonon Interaction
    • The Cooper Problem
    • The Role of the Coulomb Repulsion
    • The BCS Wave Function
    • BCS Mean Field Theory

    8. Strong Correlations

    • Tight-Binding Approximation
    • The Hubbard Model
    • Mott Insulators and Antiferromagnetism
    • The Mott Transition
    • Magnetic Impurities in Metals
    • The Anderson Impurity Model
    • The Kondo Model
    • The Kondo Problem

    9. Topology in Condensed Matter

    • Topological Invariants
    • Integer Quantum Hall Effect and Chern Numbers
    • The Haldane Model
    • Bulk-Boundary Correspondence

  • Main Suggested Texts

    • See the lecture notes of Prof John Chalker with which this course has a large overlap

    • N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt-Sanders (1976).

    • A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer (1994).
      A good introduction to a range of current theoretical ideas.

    • S.K. Ma, Statistical Mechanics, World Scientific (1985).
      Strongly recommended book at a level suitable for first year graduate students.

    • J.P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, MIT (1986).
      A very thorough treatment of second quantisation, canonical transformations and
      self-consistent field approximations.

    • P.W. Anderson, Concepts in Solids, Benjamin (1963).
      A classic introduction to solid state physics at a graduate level.

    • D. Pines and P. Nozieres, The Theory of Quantum Liquid, Addison Wesley (1989).
      A standard account of Fermi liquids.

    • P.W. Anderson, Basic Notions in Condensed Matter Physics, Benjamin (1984).
      An advanced discussion of some of the most important ideas in the subject.

  • More Advanced Graduate Text

    • A. Altland and B.D. Simons Quantum Field Theory in Condensed Matter Physics,
      Cambridge University Press (2006).
      An accessible introduction to the subject.