Engineering second-order nodal-line semimetals by breaking $\mathcal{PT}$ symmetry and periodic driving

Hosting unique drumhead surface states enclosed by nodal lines, topological nodal-line semimetals exhibit novel transport phenomena. Thus, the exploration of topological semimetals with different nodal-line structures has attracted much attention. In this paper, we first find a second-order nodal line semimetal (SONLS), which has coexisting hinge Fermi arcs and drumhead surface states, in a $\mathcal{PT}$-symmetry broken system. Then, without changing the intrinsic parameters, we artificially create exotic hybrid-order nodal-line semimetals hosted by different quasienergy gaps and rich nodal-line structures including nodal chains, crossing ring nodal nets, crossing line nodes, and nodal nets by applying a periodic driving on our SONLS. Enriching the classification of topological semimetals, such Floquet engineered high tunability of the orders and nodal-line structures of the SONLS sets up a foundation for exploring its further applications.