Topological quantum matter

Diary of the lectures

Lecture 1 (28/02/2019)

Overview of the course. The Quantum Hall Effect. Periodic Schrödinger operators. The Bloch-Floquet transform. Fibered operators. Spectral projections.

Lecture 2 (5/03/2019)

Definition and examples of vector bundles: trivial bundle, (co)tangent bundle, Bloch bundle. Sections and frames. A vector bundle admits a global frame iff it is trivial. Operations on bundles: Whitney sum, tensor product, dual bundle. The pullback bundle. Pullbacks via homotopic maps are isomorphic. Vector bundles over contractible base spaces are trivial.

Lecture 3 (21/03/2019)

The Grassmannian and its tautological and quotient bundles. Classifying maps and Gauss maps: every vector bundle over X arises as pullback of the tautological bundle via a classifying map from X to the Grassmannian. The infinite Grassmannian and its tautological bundle. Classification of isomorphism classes of vector bundles over X as homotopy classes of classifying maps from X to the infinite Grassmannian.

Lecture 4 (26/03/2019)

Vector bundles on spheres. Cohomology of the infinite Grassmannian, characteristic classes, Chern classes. Connections on vector bundles, connection matrix.

Lecture 5 (2/04/2019)

Curvature of a connection, curvature tensor. Invariant polynomials of the curvature define closed differential forms over the base space, whose cohomology class does not depend on the choice of a connection on the bundle. Chern classes as differential forms. First Chern class and Chern numbers as complete topological invariants over base spaces of dimension at most 3.

Lecture 6 (16/04/2019)

Grothendieck group of a semigroup. The K⁰-group of a manifold. Reduced K⁰-group and stable equivalence classes of vector bundles. Stable equivalence vs isomorphism; stable range condition. Topological constructions: wedge sum, smash product, reduced suspension. Computability of K⁰. Higher K-groups. Bott periodicity. Application: the K-theory of the d-dimensional torus.

Lecture 7 (30/04/2019)

States, observables and dynamics in quantum systems. Symmetries as transformation of states: Wigner's theorem. Symmetries as transformation of observables; dynamical symmetries. Projective representations, projective unitary representations, unitary representations of a group.

Lecture 8 (3/05/2019)

Examples of (groups of) quantum symmetries: (magnetic) translations, time-reversal symmetry, charge-conjugation/particle-hole symmetry, chiral/sublattice symmetry.

Lecture 9 (7/05/2019)

Translationally invariant systems: Gelfand transform, Fourier transform, Bloch-Floquet transform. Fiber reduction of translationally invariant Hamiltonians, band-gap spectrum. Definition of an abstract topological insulator. Spectral projections: the Riesz formula gives orthogonal projections .

Lecture 10 (14/05/2019)

Classification of topological phases of matter: homotopic and K-theoretic. Free and translationally invariant fermion ground states in class A. Discrete symmetries: the "ten-fold way". Periodic tables of topological insulators. Time-reversal, charge-conjugation and chiral symmetric bundles. The first Chern number of a time-reversal symmetric bundle vanishes.

Lecture 11 (21/05/2019)

Quantum Hall effect. Trace per unit volume of periodic operators: properties and expression in (modified) Bloch-Floquet representation. Diagonal and off-diagonal operators.

Lecture 12 (28/05/2019)

Liouvillian operator and its resolvent, invertibility over off-diagonal operators. The Hall conductivity is (e²/h) times the Chern number of the Bloch bundle.