The course will take place on Thursdays at 4pm, starting on March 1st, 2018, in Room 311 of Palazzina C (Department of Mathematics, Largo San Leonardo Murialdo 1, 00146 Rome).
In the last decades, solid state physics has witnessed a plethora of phenomena that have a topological origin: from the pioneering works of Thouless and collaborators to explain the quantization of the transverse conductivity in the quantum Hall effect, to the thriving field of topological insulators and superconductors. This field of research rapidly attracted the attention of mathematical physicists as well.
The methods involved in the theoretical understanding of these phenomena, at least in the one-particle approximation, rely on the theory of vector bundles and K-theory. The course will therefore be divided in two parts:
Some basic differential geometry and a first course in the mathematics of quantum mechanics will be assumed as prerequisites.
Depending on the inclinations, background and interests of the students, and if time permits, more advanced topics could also be covered, including for example:
Overview of the course. Crystalline systems and periodic Schrödinger operators. Bloch-Floquet and Bloch-Floquet-Zak transforms. Spectral projections and the associated Bloch bundle.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The Quantum Hall Effect. Kubo formula. Trace per unit volume. Quantization of the Hall conductivity.
These are the handwritten notes that I used to prepare the lectures. Proceed with care!
Notes to Lecture 2, Lecture 3, Lectures 4-5, Lecture 6, Lectures 7-8, Lectures 9-10, Lecture 11. The material covered in Lecture 1 has been reprised in Lecture 11, see also this paper.
The list of references will be updated as the course progresses. Papers and books cited here are often just one possible starting point for a bibliographic research.
J.E. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159 (1994), 399–422.
J. Bellissard, A. van Elst, and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35 (1994), 5373.
G.M. Graf, Aspects of the integer quantum Hall effect. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, ed. by F. Gesztesy, P. Deift, C. Galvez, P. Perry, and W. Schlag. Proceedings of Symposia in Pure Mathematics, vol. 76 (American Mathematical Society, Providence, RI, 2007), pp. 429–442.
P. Kuchment, An overview of periodic elliptic operators. Bull. Amer. Math. Soc. 53 (2016), 343–414.
D. Monaco and G. Panati, Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Acta App. Math. 137 (2015), 185–203.
D. Husemoller, Fibre Bundles. 3rd edition. Graduate Texts in Mathematics 20 (Springer-Verlag, New York,1994).
J.W. Milnor and J.D. Stasheff, Characteristic classes. Annals of Mathematical Studies 76, Princeton University Press, Princeton, New Jersey (1974).
S.J. Avis and C.J. Isham, Quantum field theory and fibre bundles in a general spacetime. In: Recent Developments in Gravitation – Cargèse 1978, ed. by M. Lévy and S. Deser. Proceeding of the 1978 NATO Advanced Study Institute (Plenum Press, New York, 1979), pp. 347–401.
G. Panati, Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poicaré 8 (2007), 995–1011.
V. Bargmann, Note on Wigner's theorem on symmetry operations. J. Math. Phys. 5 (1964), 862–868.
W. Moretti, Spectral Theory and Quantum Mechanics. 2nd edition. UNITEXT 110 (Springer International Publishing, Cham, 2017).
A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conference Proceedings 1134 (2009), 22.
S. Ryu, A.P. Schnyder, A. Furusaki, and A.W.W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12 (2010), 065010.