K-theory in condensed matter physics

Diary of the lectures

Lecture 1 (1/03/2018)

Overview of the course. Crystalline systems and periodic Schrödinger operators. Bloch-Floquet and Bloch-Floquet-Zak transforms. Spectral projections and the associated Bloch bundle.

Lecture 2 (15/03/2018)

Definition of vector bundle. Examples: trivial bundle, (co)tangent bundle, Bloch bundle. Sections and frames. There exist non-trivial bundles: the Möbius bundle. Operations on bundles: direct (Whitney) sum, tensor product, dual bundle, pullback bundle. Pullback bundles via homotopic maps are isomorphic.

Lecture 3 (23/03/2018)

The Grassmannian, the tautological bundle and the quotient bundle. Any bundle over a manifold X is the pullback of the tautological map via a map from X to the Grassmannian. Infinite Grassmannian. Classification theorem: isomorphism classes of vector bundles of fixed rank are in 1:1 correspondence with homotopy classes of maps to the Grassmannian. Application: vector bundles over spheres.

Lecture 4 (29/03/2018)

Proof of the classification theorem. Characteristic classes. Chern classes. Connections, connection matrix, change of the connection matrix under the change of local frame. Curvature, curvature tensor, change of the curvature tensor under the change of local frame.

Lecture 5 (05/04/2018)

The Grassmann connection on the tautological bundle. Invariant polynomials P over the algebra of matrices. The form P(K), evaluated at the curvature tensor, is closed and its cohomology class is independent of the choice of a connection on the bundle. Chern classes as differential forms. Example: Berry connection, Berry curvature and first Chern class of a bundle. Chern classes are not complete invariants, except in low dimensions.

Lecture 6 (12/04/2018)

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 n-dimensional torus.

Lecture 7 (19/04/2018)

Axioms of classical and quantum mechanics: states, observables, dynamics. Transformations of rays and quantum symmetries. Wigner's theorem: every quantum symmetry is (anti)unitarily implemented on the Hilbert space of the system.

Lecture 8 (03/05/2018)

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

Lecture 9 (10/05/2018)

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 projection.

Lecture 10 (17/05/2018)

The Riesz formula gives orthogonal projections. Classification schemes for topological phases of matter: gapped deformations give isomorphic Bloch bundles, addition of trivial bands give stably equivalent Bloch bundles. Examples: free and translationally invariant ground states in class A. Discrete symmetries of topological insulators: time-reversal, charge-conjugation, and chiral symmetric bundles. Periodic tables of topological insulators.

Lecture 11 (24/05/2018)

The Quantum Hall Effect. Kubo formula. Trace per unit volume. Quantization of the Hall conductivity.