Fredholm Index Theory for Condensed Matter and Introduction to Spin Systems

First Part: Index theory with application to condensed matter systems,
mostly Integer Quantum Hall Effect

1. Recap of functional analysis: bounded operators on Hilbert spaces, with emphasis on compact and trace class. Paragraphs 4,5,6 of Chapter VI of Reed-Simon vol I.
2. Fredholm index theory, as in the first part of chapter 27 and 30.7 of Lax, Functional Analysis. In particular emphasizing the existence of a finite rank pseudo inverse for every Fredholm operator that allows to write its index as a difference among traces, as proven by Murphy in “Fredholm Index Theory and the Trace” https://www.jstor.org/stable/20489482?origin=JSTOR-pdf. All the statements with full proofs.
3. Example (with proof) of a Fredholm operator: (1 − P ) + P U P , with P a projection and U a unitary. Later on in the course P will be a Fermi projection of a Hamiltonian modelling an insulator in two dimensions, and U will be the “flux insertion” operator.
4. Introduction to the physics of the Integer Quantum Hall Effect. Sections I and II of “The noncommutative geometry of the quantum Hall effect” http://dx.doi.org/10.1063/1.530758.
5. Stating (without proof) the index theorem: Index((1-P) + PUP) = 2πiTr[P Λ_x P, P Λ_y P]
6. Index of a pair of projections, as in the Les Houches lecture notes of Avron, section 13.5 of https://phsites.technion.ac.il/avron/wp-content/uploads/sites/3/2013/08/leshouches.pdf Here Avron provides a simplified approach with respect to the original papers “The index of a pair of projections” and “Charge Deficiency, Charge Transport and Comparison of Dimensions” with Seiler and SImon.
7. Index of a pair of unitaries: the Bott index, as introduced by Hastings and Loring, is introduced to prove an index theorem in the context of insulators in two dimensions (with periodic boundary conditions) described within finite dimensional Hilbert spaces.
• Definition and properties of the Bott index, with proofs.
• Index theorem: proof of the equality with the transverse conductance.
Proofs are given following https://arxiv.org/abs/1708.05912.
8. Bulk-boundary correspondence for a finite 1-dimensional system with chiral symmetry: https://arxiv.org/abs/2203.17099. This paper establishes the equality of edge and bulk indices for the same finite system, while traditionally edge and bulk are treated for different geometries. Explaining the main ideas and results.

Second Part: Finite spin systems, introduction to the main models (Ising, XY and XXZ), proof of Lieb-Robinson bounds

1. Tensor product of finitely many Hilbert spaces: definition of partial traces and proof of its relation with Haar-averaging.
2. Discussion (at the level of theoretical physics) of the spin 1/2 system: Ising, XY and XXZ, with their phase diagrams.
3. Proof of the Lieb-Robinson bounds in the simplest setting of nearest neighbor one-dimensional spin Hamiltonian following appendix B of https://arxiv.org/pdf/2408.00743 and appendix B of https://arxiv.org/pdf/2509.02383.