Anomalous multi-gap topological phases in periodically driven quantum rotors

We demonstrate that periodically driven quantum rotors provide a promising and broadly applicable platform to implement multi-gap topological phases, where groups of bands can acquire topological invariants due to non-Abelian braiding of band degeneracies. By adiabatically varying the periodic kicks to the rotor we find nodal-line braiding, which causes sign flips of topological charges of band nodes and can prevent them from annihilating, indicated by non-zero values of the %non-Abelian patch Euler class. In particular, we report on the emergence of an anomalous Dirac string phase arising in the strongly driven regime, a truly out-of-equilibrium phase of the quantum rotor. This phase emanates from braiding processes involving all (quasienergy) gaps and manifests itself with edge states at zero angular momentum. Our results reveal direct applications in state-of-the-art experiments of quantum rotors, such as linear molecules driven by periodic far-off-resonant laser pulses or artificial quantum rotors in optical lattices, whose extensive versatility offers precise modification and observation of novel non-Abelian topological properties.

💡 Research Summary

The authors investigate periodically driven quantum rotors—systems such as linear molecules or synthetic rotors in optical lattices subjected to ultrashort, far‑off‑resonant laser pulses—and demonstrate that they constitute a versatile platform for realizing multi‑gap topological phases. Starting from the free‑rotor Hamiltonian (H_0=\pi \hat L^2/\tau_B), they introduce a sequence of kicks described by a potential ( \hat V(P_1,P_2)=P_1\cos\hat\theta+P_2\cos2\hat\theta). By arranging three kicks per Floquet period (a “triple‑kick” protocol) they obtain a Floquet operator (U_{\rm TKR}) with (N\ge3) quasienergy bands, where (N) is set by the number of kicks within one revival time and can be chosen arbitrarily.

Because the protocol respects combined parity‑time (PT) symmetry, the effective Floquet Hamiltonian can be brought to a real form. Consequently, the eigenstates form an orthonormal real frame (a “dreibein”) that lives on the coset space (O(3)/\mathbb Z_2^3\simeq SO(3)/D_2). Band‑touching points (nodal points) in adjacent gaps carry non‑Abelian quaternion charges ({ \pm1,\pm i,\pm j,\pm k}). When a nodal point is braided around another belonging to a neighboring gap, its charge flips sign, reflecting the non‑commutative multiplication rules of the quaternion group. This non‑Abelian braiding requires a two‑dimensional parameter space; the authors achieve it by adiabatically modulating the kick strengths along a closed loop parameterized by (\alpha\in