Comparison between two approaches to classify topological insulators using K-theory
Scaglione Lorenzo
TL;DR
This work bridges two K-theory-based frameworks for classifying symmetry-protected topological phases: Kellendonk’s abstract C*-algebra approach and Zirnbauer’s concrete BdG/Nambu-space construction. Using Van Daele’s K-theory, it demonstrates that the physically motivated Zirnbauer setup is a special case of the more general Kellendonk framework, with the Tenfold Way providing a consistent mapping of symmetries to K-groups. A key insight is the role of charge conservation in reducing the algebra and spin rotation symmetry in enabling quaternionic factorization, which shifts K-theory indices. The results establish a coherent, cross-validated classification scheme applicable to disordered and interacting-free-fermion systems, unifying complex and real K-theory descriptions across all ten symmetry classes. This alignment has practical implications for predicting topological invariants in realistic materials and for interpreting experimental signatures within a unified mathematical framework.
Abstract
We compare two approaches which use K-theory for C*-algebras to classify symmetry protected topological phases of quantum systems described in the one particle approximation. In the approach by Kellendonk, which is more abstract and more general, the algebra remains unspecified and the symmetries are defined using gradings and real structures. In the approach by Alldridge et al., the algebra is physically motivated and the symmetries implemented by generators which commute with the Hamiltonian. Both approaches use van Daele's version of K-theory. We show that the second approach is a special case of the first one. We highlight the role played by two of the symmetries: charge conservation and spin rotation symmetry.