Condensed Matter

Condensed Matter in 29 chapters — from “Historical origins” to “The frontier”.

Chapters

  1. Historical origins
    • The puzzle of heat in solids
    • Einstein, 1907: a quantum solid
    • Sound, waves, and the phonon precursor
    • The puzzle of metals in 1900
    • Drude: the right answer for the wrong reasons
    • Sommerfeld: Fermi statistics restored
    • Bloch, Wilson, and the birth of band theory
    • How to read this book
  2. Lattices, reciprocal space, and the Brillouin zone
    • Bravais lattices and crystal structures
    • Reciprocal space
    • The Brillouin zone
    • Why reciprocal space?
    • Miller indices
    • Where this chapter fits
  3. Scattering and crystal-structure determination
    • The scattering geometry
    • The lattice sum: the Laue condition
    • The Ewald construction
    • The basis sum: structure factor and form factor
    • Powder diffraction and a worked NaCl example
    • What diffraction sees and what it does not
    • Inelastic scattering: a preview
    • Where this chapter fits
  4. The free electron gas
    • Drude, carefully
    • Sommerfeld: Fermi statistics restored
    • Pauli paramagnetism
    • Thermopower and the Mott formula
    • What Sommerfeld still misses
    • Heat capacity as the decisive test
    • Where this chapter fits
  5. Bloch’s theorem
    • The question, and why it’s hard
    • Bloch’s theorem, stated three ways
    • The proof, in two lines and then expanded
    • Reading the Bloch state
    • The empty-lattice construction: parabolas folded into the zone
    • Nearly-free electrons and the opening of gaps
  6. Nearly-free electrons and band gaps
    • From Bloch to a calculation
    • The gap at the zone boundary
    • Reading the gap formula
    • Fermi surfaces, free and nearly-free
    • Effective mass and band curvature
    • Direct, indirect, and the semiconductor zoo
    • Where this chapter fits
    • Problems
  7. Tight binding and graphene
    • The tight-binding picture
    • Multiple orbitals per site
    • Graphene: the structure
    • Graphene: the tight-binding Hamiltonian
    • Dirac cones
    • Reading the Dirac Hamiltonian
    • Where this chapter fits
    • Problems
  8. Semiclassical dynamics and the Berry phase (I)
    • The problem and the regime
    • The standard semiclassical equations
    • Bloch oscillations: the textbook equations applied
    • The Berry phase, introduced geometrically
    • Reading the anomalous velocity
    • Worked example: cyclotron motion in copper
    • Applications: three signatures of Berry curvature
    • Where this chapter fits
    • Problems
  9. Phonons in 1D
    • The monatomic chain
    • The diatomic chain
    • Counting modes in 3D and the Brillouin zone
    • Specific heat: the Debye model
    • Counting modes and what's next
    • Problems
  10. Specific heat
    • The problem: Dulong–Petit and its failure
    • Einstein's model
    • Phonons as quanta: the formal derivation
    • Debye's model
    • Comparing to experiment
    • Electron–phonon coupling
    • Resistivity from electron-phonon scattering
    • Problems
  11. Anharmonicity and thermal expansion
    • The harmonic approximation and its failures
    • The Grüneisen parameter
    • Phonon-phonon scattering and thermal conductivity
    • Magnetic response of atoms in solids: a first look
    • Where this chapter fits
    • Problems
  12. Second quantisationand the interacting electron gas
    • Why first quantisation is not enough
    • Creation and annihilation operators
    • One-body and two-body operators
    • The Hamiltonian of interacting electrons
    • Dimensional analysis: what scales win?
    • From Hubbard to Heisenberg: localised electrons
    • Sign of the exchange: ferromagnet or antiferromagnet?
    • Mean-field theory of the ferromagnet
    • Spin waves: the Holstein–Primakoff approach
    • Worked example: 1D Heisenberg ferromagnet
    • Where this chapter fits
    • Problems
  13. Screening and the RPA
    • The problem: Coulomb is long-ranged
    • The Lindhard function
    • Screening in the RPA
    • Two consequences: Friedel oscillations and plasmons
    • When screening fails: the Hubbard model
    • The Mott transition: band theory’s failure
    • Worked example: half-filled Hubbard at large U gives Heisenberg
    • Anderson localisation: insulators from disorder
    • Where this chapter fits
    • Problems
  14. Fermi liquid theory
    • The puzzle
    • Quasiparticles
    • The spectral function
    • Landau parameters
    • Heavy fermions: Fermi liquid theory at the extreme
    • Summary and bridge
  15. Electron-phonon coupling
    • The coupling, derived
    • The screened potential, in detail
    • Three consequences
    • Migdal's theorem and why this is tractable
    • Where this chapter fits
    • Problems
  16. Local moments and exchange
    • Where local moments come from
    • Exchange interaction
    • The Heisenberg ferromagnet at the simplest level
    • Itinerant electrons revisited: RPA, plasmons, and the Stoner instability
    • Where this chapter fits
    • Problems
  17. Antiferromagnets and spin waves
    • The Néel state is not the ground state
    • Antiferromagnetic spin waves
    • Frustration: when Néel order isn't even classical
    • Where this chapter fits
    • Forward look: from magnon exchange to Cooper pairs
    • Problems
  18. Itinerant magnetism and the Stoner criterion
    • The puzzle: ferromagnetism in a metal
    • The Stoner argument
    • Stoner enhancement and Fermi liquid connection
    • Spin fluctuations and quantum criticality
    • Where this chapter fits
    • A parallel instability: London electrodynamics and the Meissner effect
    • Ginzburg–Landau theory
    • Type I versus Type II superconductors
    • Worked example: λ and ξ for lead
    • Parallel with the Stoner instability
    • Problems
  19. The Kondo effect
    • The puzzle: rising resistivity at low T
    • Why a logarithm: a phase-space argument
    • The third-order calculation: where the log comes from
    • Poor man's scaling
    • The resistivity minimum
    • Worked example: TK​ for Fe in Cu
    • The Anderson model and Schrieffer–Wolff
    • The Kondo lattice and heavy fermions
    • Where this chapter fits
    • Problems
  20. Ginzburg–Landau theory
    • The order parameter
    • The Landau free energy
    • Spatial variations: the GL functional
    • Two-component order parameters and superconductivity
    • The Ginzburg criterion
    • Interlude: the Anderson and Kondo problems
    • The Schrieffer–Wolff transformation
    • Three regimes: Kondo, Anderson, mixed valence
    • Worked example: low-T crossover in heavy fermions
    • Where this chapter fits
    • Problems
  21. BCS superconductivity
    • The Cooper instability
    • The BCS Hamiltonian and gap equation
    • The temperature-dependent gap and thermodynamics
    • Tunnelling spectroscopy: the BCS density of states
    • Coherence factors and the Hebel-Slichter peak
    • Meissner effect
    • The isotope effect
    • Strong coupling: Eliashberg theory
  22. The Josephson effect and flux quantization
    • Two superconductors connected weakly
    • Flux quantization
    • The SQUID: applications of the Josephson effect
    • Where this chapter fits
    • Problems
  23. Superfluidity
    • Bose–Einstein condensation
    • The Landau criterion
    • The two-fluid model
    • Charged bosons in a magnetic field: a preview of Landau levels
    • Where this chapter fits
    • Problems
  24. Charge density waves
    • The Peierls instability
    • The Kohn anomaly and 2kF​ phonon softening
    • Pinning and sliding
    • Interlude: the Hall effect as a probe of broken-symmetry order
    • Semiclassical Hall effect in a band metal
    • The anomalous Hall effect
    • Worked example: the Hall coefficient of copper at 300 K
    • Where this chapter fits
    • Problems
  25. The integer quantum Hall effect
    • Landau levels
    • The Hall conductance and its quantisation
    • The Laughlin argument
    • The TKNN formula: Hall conductance from Kubo
    • Worked example: the ν=1 plateau
    • The role of disorder
    • Where this chapter fits
    • Problems
  26. The fractional quantum Hall effect
    • Why interactions matter
    • The Laughlin wavefunction
    • Fractional charge
    • Composite fermions
    • The 5/2 state and non-Abelian statistics
    • Where this chapter fits
    • Problems
  27. Topological insulators
    • Time reversal and Kramers pairs
    • The Z2​ invariant in 2D
    • The Kane-Mele model
    • The BHZ model and HgTe quantum wells
    • Three-dimensional topological insulators
    • Worked example: the Bi2​Se3​ surface state
    • Beyond the basic TI
  28. Weyl and Diracsemimetals
    • Berry phase and Berry curvature: a working toolkit
    • Weyl points
    • Berry monopoles
    • Fermi arcs and bulk-boundary correspondence
    • The chiral anomaly
    • Dirac semimetals
    • Where this chapter fits
    • Problems
  29. The frontier
    • Twisted bilayer graphene
    • Spin liquids
    • Strange metals and non-Fermi liquids
    • Quantum simulation with cold atoms
    • Topological superconductors
    • Topological quantum computation
    • Closing remarks