| Résume | In [1] it was shown that irreducible free fermion systems with symmetries can be classified into 10 distinct classes, based on the presence and behaviour of chiral, time-reversal and charge-conjugation symmetries. Each class is composed of spaces of free equivariant Hamiltonians, and in [2] it was shown that the spaces of time-evolution operators generated by each class is one of the 10 symmetric spaces of compact type classified by Cartan.
The symmetric spaces in this classification scheme form the underlying spaces of KU and KO, the spectra that represent complex and real topological K-theory. Thus, topological K-theory can be represented by spaces of time-evolution operators of free fermion systems with symmetries. This is at the heart of the K-theoretical classification of crystalline topological insulators and superconductors [3, 5, 4].
In this seminar I will present how the irreducible Nambu space models of free fermion systems with symmetries are classified by symmetric spaces of compact type. I will also present the basics of topological K-theory, and how the symmetric spaces in the classification form the spectra KU and KO which represent these cohomology theories. This leads to the classification of stable ground states of crystalline materials via equivariant K-theory. As an application I will sketch the derivation of the bulk and boundary invariants of the Su-Schriefer-Heeger model of a polyacetylene wire, and how the bulk-boundary correspondence leads to the presence of conducting boundary states. This is the main feature making this model a topological insulator.
I will also show how me and my collaborator, Lucas Müssnich, extend this classification scheme to take into account weak interactions between fermions. We introduce a geometric definition of weakly interacting time evolution operators in terms of the notion of cut locus of submanifolds, and show how this leads to the construction of spectra KUwi and KOwi that deformation retract to KU.
[1] Altland, Alexander; Zirnbauer, Martin R. Nonstandard symmetry classes in mesoscopic
normal-superconducting hybrid structures. Physical Review B 55.2 (1997)
[2] Heinzner, Peter, A. Huckleberry, and Martin R. Zirnbauer. Symmetry classes of disordered
fermions. Communications in mathematical physics 257 (2005)
[3] Freed, Daniel S., and Moore, Gregory W. Twisted equivariant matter. Annales Henri Poincar´e.
Vol. 14. No. 8. Basel: Springer Basel, 2013.
[4] Prodan, Emil; Schulz-Baldes, Hermann. Bulk and boundary invariants for complex topological
insulators: from K-theory to physics. Springer (2016)
[5] Cornfeld, Eyal; Carmeli, Shachar. Tenfold topology of crystals: Unified classification of crys-
talline topological insulators and superconductors. Physical Review Research 3.1 (2021)
[6] Müssnich, Lucas CPAM, and Renato Vasconcellos Vieira. The weakly interacting tenfold way.
arXiv preprint arXiv:2603.16799 (2026). |
