Mathematical foundations · Quantum matter · Quantum information
Topology meets
quantum phenomena.
Why can a global property of a quantum wavefunction determine quantized conductance, protected edge states, fractional particles, and new ways to encode quantum information?
$$C=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_{xy}(k)\,d^2k\in\mathbb Z$$
Topology does not create quantum mechanics.
It organizes what quantum states can do.
Quantum phenomena arise from states, amplitudes, superposition, interference, measurement, and interactions. When quantum states vary across a closed parameter space, their global structure may be nontrivial: a phase may wind, a band may have nonzero Chern number, or quasiparticles may braid.
A spectral gap and the right symmetries make these global structures robust. This is why a local defect often cannot remove a chiral edge channel, although strong perturbations, gap closings, and decoherence can still change the physics.
Enter the subject.
Topic 01 · 8 conceptsFoundations of TopologyContinuity, homotopy, winding numbers, manifolds and bundles.Topic 02 · 6 conceptsGeometric Quantum MechanicsProjective Hilbert space, Berry phase, curvature and holonomy.Topic 03 · 12 conceptsTopological Band TheoryChern numbers, Hall conductance, edge states and topological insulators.Topic 04 · 8 conceptsInteracting Topological PhasesFractional quantum Hall physics, anyons, topological order and Majorana modes.Topic 05 · 8 conceptsQuantum Phenomena and ApplicationsTopological photonics, sensing, cold atoms and quantum error correction.
From topology to observation
01 / MathematicsSpaces & mapsManifolds, homotopy, winding, bundles.
02 / Quantum statesPhases & curvatureBerry connection, holonomy, geometric tensor.
03 / Physical evidenceTransport & modesHall quantization, edge channels, anyons, quantum codes.