Topological Phases of Matter
Beyond Landau's symmetry-breaking paradigm, topological phases are characterized by global invariants that are robust against smooth deformations. The Berry phase provides the geometric foundation, while topological invariants like Chern numbers and Z₂ indices classify distinct quantum phases.
The integer quantum Hall effect was the first topological phase discovered, with its Hall conductance quantized as $\sigma_{xy} = \nu e^2/h$. Topological insulators generalize this to time-reversal invariant systems with protected surface states.
Chapters
1. Berry Phase in Solids
Geometric phase, Berry connection and curvature, Chern numbers, and applications to band theory.
2. Quantum Hall Effect
Landau levels, integer quantum Hall effect, TKNN invariant, edge states, and fractional quantum Hall effect.
3. Topological Insulators
Z₂ invariant, Kane-Mele model, surface Dirac cones, and topological protection by time-reversal symmetry.