Floquet generation of hybrid-order topology and $\mathbb{Z}_2$-like bipolar localization

Abstract

Higher order topology, in the form of the emergence of corner modes, is observed in two dimensions when crystalline symmetries are superposed on the Altland-Zirnbauer classification of topological insulators. It occurs in Benalcazar-Bernevig-Hughes (BBH) model on a 2D square lattice, which owing to an embedded $\mathbb{Z}_2$ gauge field, features a bulk quadrupole moment with localized zero-energy corner states. Further, as a dividend, the BBH model transmutes the general notion of the space-time inversion ($\mathcal{PT}$) symmetry and behaves as a spinful system, without having to invoke `real' spin degrees of freedom. A two-fold engineering of the model, namely a periodic drive, followed by a non-reciprocal hopping render intriguing consequences. As a first, the drive activates first-order topology, and the resulting Floquet phase hosts a coexistence of first-order conducting edges at both zero and $\pi$ quasienergies along with higher order corner states, which qualifies the coexisting state to be denoted as a hybrid-order topological phase. Further, inclusion of non-reciprocal couplings features a $\mathbb{Z}_2$-like skin effect which demonstrates a drive-induced transition from a unipolar to a bipolar localized phase, and is evidenced via the generalized Brillouin zone (GBZ) theory. While depiction of a GBZ in 2D is challenging, a corresponding 1D map is still possible and can be implemented by exploiting the mirror symmetry. We further uncover conditions under which the skin effect is completely suppressed in our system. Putting together, our results manifest an efficient technique to dynamically engender and control Hermitian and non-Hermitian topological features, that remain otherwise masked in a static scenario.