Topological phases of non-interacting systems: A general approach based on states
Giuseppe De Nittis
TL;DR
The work introduces a state-based framework for classifying topological phases using $C^*$-bundles over a quantum parameter space $X$ and homotopy classes of configurations $F:X\to\mathcal{P}_{\mathscr{A}}$, extending beyond spectral projections to accommodate interacting settings. It proves that localizable topological phases correspond to $[X,\mathcal{P}_{\mathscr{O}}]$ and shows that, in the canonical fiber case $\mathscr{O}=\mathscr{K}(\mathcal{H})$, these reduce to familiar topological invariants like $\mathrm{Pic}(X)\cong H^2(X,\mathbb{Z})$ and, for $\dim X\le 3$, to $\widetilde{K}^0(X)$, thereby linking to $K$-theory. The paper demonstrates the framework via concrete applications to crossed products, Bloch algebras, $C^*$-dynamical systems, and the non-commutative torus, reproducing the standard type A TI classifications and quantum Hall-type results from an algebraic-topological perspective. This unifies multiple topological phases under a common state-centric paradigm and provides a pathway to address interacting and more general systems within a rigorous $C^*$-algebraic setting.
Abstract
In this work we provide a classification scheme for topological phases of certain systems whose observable algebra is described by a trivial $C^*$-bundles. The classification is based on the study of the homotopy classes of \emph{configurations}, which are maps from a \emph{quantum parameter space} to the space of pure states of a reference \emph{fiber} $C^*$-algebra. Both the quantum parameter space and the fiber algebra are naturally associated with the observable algebra. A list of various examples described in the last section shows that the common classification scheme of non-interacting topological insulators of type A is recovered inside this new formalism.