Flux Response of Rotation-Invariant Topological Insulators

Threading magnetic flux into topological phases can induce bound states that reveal intrinsic properties of the ground state. In a 3D $\mathbb{Z}_2$ topological insulator, a quantized $\pi$ flux traps a pair of 1D helical modes, whereas a trivial insulator hosts none. In this work, we show that in the presence of even-fold rotation symmetry $C_n$, a 3D band insulator features a refined $\mathbb{Z}_2 \times \mathbb{Z}_2$ classification of the flux response. Specifically, it can host two distinct types of helical flux-bound modes that are distinguished by their angular momentum. When both types of flux modes coexist, the system is not a strong topological insulator, but a $C_n$-protected topological crystalline insulator. Building on this result, we propose that flux-threaded nanowires of such topological phase provide a natural platform for realizing 1D crystalline topological superconductors with multiple $C_n$-protected Majorana modes.