The course will take place on Tuesdays at 4pm, in Room 311 of Palazzina C (Department of Mathematics, Largo San Leonardo Murialdo 1, 00146 Rome).
The discovery of the quantum Hall effect and of topological insulators stimulated the interest of condensed matter physicists for the toolbox of topology and geometry, initiating a fruitful interplay between the two communities. This course will present a selection of topics from both mathematics (vector bundles, their invariants, K-theory) and physics (periodic tables of topological insulators, topological transport) in order to illustrate some basic aspects of the thriving research field on topological quantum matter.
The specifics of the program of the lectures will depend on the inclination, background and interests of the audience, and could include more advanced topics like K-theory for C*-algebras and applications to disordered topological insulators, or obstruction theory and constructive algorithms for Wannier functions.
Overview of the course. The Quantum Hall Effect. Periodic Schrödinger operators. The Bloch-Floquet transform. Fibered operators. Spectral projections.
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.
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.
Vector bundles on spheres. Cohomology of the infinite Grassmannian, characteristic classes, Chern classes. Connections on vector bundles, connection matrix.
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.
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.
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.
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 projections: the Riesz formula gives orthogonal projections .
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.
Quantum Hall effect. Trace per unit volume of periodic operators: properties and expression in (modified) Bloch-Floquet representation. Diagonal and off-diagonal operators.
Liouvillian operator and its resolvent, invertibility over off-diagonal operators. The Hall conductivity is (e²/h) times the Chern number of the Bloch bundle.
In due time, I will distribute some handwritten notes that I use to prepare the lectures. Topics from Lecture 1 were covered in more detail in Lectures 9 to 12.
Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lectures 11-12
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.
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).
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology. Graduate Texts in Mathematics 82 (Springer-Verlag, New York, 1982).
N. Kuiper, The homotopy type of the unitary group of Hilbert space. Topology 3 (1965), 19–30.
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. Proceedings 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.
A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002).
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.