Physics / research
Topology and protected quantum response
Relate band topology and bulk-boundary correspondence to Hall transport, topological defects, anyons, and the limits of simplified classification models.
Your route through physics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 research
Topology and protected quantum response
Relate band topology and bulk-boundary correspondence to Hall transport, topological defects, anyons, and the limits of simplified classification models.
- 01
Berry curvature in crystal momentum
Define band geometry with explicit phase and orientation conventions and separate local curvature from a quantized global invariant.
31 min - 02
Chern numbers and quantized Hall conductance
Normalize the occupied-band curvature integral and connect its integer to a signed two-dimensional electrical response.
32 min - 03
SSH chain and winding number
Compute a chiral two-band winding invariant while keeping the unit cell, Zak-phase convention, and physical termination explicit.
31 min - 04
Time-reversal topological insulators
Connect spinful time reversal, Kramers pairs, and a two-dimensional Z₂ band invariant without assuming conserved spin.
32 min - 05
Bulk-boundary correspondence and edge modes
Derive an interface mode from a Dirac mass reversal and interpret protected net chirality under explicit locality and gap assumptions.
32 min - 06
Weyl semimetals and chiral nodes
Treat an isolated three-dimensional band crossing as an oriented curvature source and connect its local charge to lattice pairing and surface arcs.
33 min - 07
Fractional Hall liquids and Laughlin wavefunctions
Connect a correlated lowest-Landau-level wavefunction to exchange symmetry, filling, short-distance avoidance, and finite-size flux counting.
33 min - 08
Fractional charge and anyonic statistics
Separate fractional quasihole charge, exchange and full-winding phases, and non-Abelian fusion structure from ordinary path-dependent phases.
33 min - 09
Topological superconductors and Majorana models
Analyze the Kitaev chain's bulk criterion and end operators while keeping BdG redundancy, fermion parity, and logical-state counting explicit.
33 min - 10
Interacting topological order and ground-state degeneracy
Use the toric code to derive topology-dependent degeneracy and distinguish intrinsic order from symmetry-protected bands and ordinary symmetry breaking.
34 min