Rotation topological states: theory and material realization

The conventional characterization of topological materials relies on topological invariants calculated from the entire set of occupied bands. However, when a system possesses rotational symmetry, the occupied Hilbert space can be decomposed into multiple subspaces labeled by distinct rotation eigenvalues. We show that this decomposition reveals hidden topological states characterized by a novel $\mathbb{Z}_2^n$ topological invariant, where $n$ is the number of subspaces, while the conventional $\mathbb{Z}_2$ invariant may fail to detect the topology hidden in the rotation subspaces. Remarkably, time-reversal symmetry pairs conjugate rotation eigenvalues and guarantees that the two subspaces have the same $\mathbb{Z}_2$ invariants, making the topology always hidden from the conventional global invariant. We formulate the theory of rotation-subspace topology and demonstrate its material realization in bulk CsCl. Using first-principles calculations and symmetry analysis, we show that bulk CsCl, which is diagnosed as topologically trivial by the conventional approach, features a nontrivial $\mathbb{Z}_2^3$ invariant along the $\Gamma$-R path and a nontrivial $\mathbb{Z}_2^4$ invariant along the $\Gamma$-Z and M-R paths, leading to double Weyl points on the (111) and (001) surfaces, respectively. The subspace $\mathbb{Z}_2^n$ invariant proposed here serves as a necessary refinement for symmetry-protected topological phases and will facilitate the identification of a large class of topological states overlooked by existing diagnostics.