A topological quantum system consists of a Hilbert space \(\mathcal H\) (quantum phase space), a closed manifold \(X\) (parameter space), and a continuous map \(X \ni x \mapsto H(x) = H(x)^*\) taking values in compact self-adjoint operators \(H(x) : \mathcal H \to \mathcal H\) (the \(x\)-dependent Hamiltonian). The reason for the monicker “topological” is that important physical features correspond to topological properties of the family. Perhaps the most important case is when the spectrum of the family is gapped: when there are \(m\) eigenvalues (aka energy bands) \(\lambda_1(x),\dots,\lambda_m(x)\) of \(H(x)\) that stay more than some “gap” \(C_g > 0\) away from all other eigenvalues for all values of \(x \in X\). When this happens, there is a corresponding rank-\(m\) complex vector bundle on \(X\) spanned by the corresponding eigenvectors. Topologically inequivalent vector bundles correspond to physically inequivalent systems.
As such, the classification of complex vector bundles is central in condensed matter. Fortunately, this classification is quite well studied in the mathematical literature, with all kinds of homotopy-theoretic and differential-geometric tools.
The present paper extends such analysis to the case of real (in the sense of Atiyah) vector bundles, which correspond to topological quantum systems with a time reversal symmetry. (Quaternionic vector bundles are also briefly discussed, with full details awaiting a future paper by the authors.) By definition, a time reversal symmetry for a topological quantum system consists of a complex conjugation \(C\) on \(\mathcal H\) (i.e. an antiunitary involution), an involution \(\tau\) on \(X\), and a continuous family of unitary maps \(x \mapsto J(x)\), equivariant in the sense that \(J(x)^* = CJ(\tau x) C\), and providing an equivariance for \(H\) in the sense that \(J(x)^* H(\tau x) J(x) = C H(x) C\). When such a system is gapped, the corresponding vector bundle is real (for the involution \(\tau\)).
It is hard for this reviewer to evaluate any paper from the point of view of a condensed matter theorist. But from the point of view of a mathematician, the paper is very good, and obviously achieves its two goals: to develop the (new) homotopy-theoretic and differential-geometric tools necessary to analyze topological quantum systems with time reversal symmetry, and to provide a largely self-contained presentation of all materials. In particular, the paper is written in the language of mathematics, defining physical notions in a mathematically-understandable way, but nevertheless contains brief introductions to everything from connections and Chern-Weil forms to spectral sequences, so that non-experts can follow.