Obeying non-Abelian statistics, Majorana fermions holds a promise to implement fault-tolerant quantum computing. It was found that Majorana fermions can be simulated by the zero-energy excitation in a nanowire with strong spin-orbit coupling interacting with an s-wave superconductor under a magnetic field. However, the signal of Majorana fermion in that system is obscured by the disorder in the nanowire and the confinement potential at the wire end. Thus, more controllable platforms are desired to simulate Majorana fermions. We here propose an alternative scheme to simulate the Majorana fermions in a trapped-ion system. Our dimerized-ion configuration permits us to generate the Majorana modes not only at zero energy but also at the nonzero ones, which enlarge the family of Majorana modes and supply another qubit carrier for quantum computing. We also investigate the controllability of the Majorana modes by Floquet engineering. It is found that a widely tunable number of Majorana modes are created on demand by applying a periodic driving on the trapped-ion system. Enriching the platforms for simulating Majorana fermions, our result would open another avenue for realizing fault-tolerant quantum computing.

Real-space topological invariants were widely used to characterize chiral-symmetric higher-order topological phases (HOTPs). However, a momentum-space characterization to these HOTPs, which essentially reveals their intrinsic bulk-boundary correspondence and facilitates their detection in quantum simulation systems, is still lacking. Here, we propose an experimentally observable momentum-space characterization to the chiral-symmetric HOTPs by the concept of polarized topological charges. It makes a unified description to the HOTPs caused by the closing and reopening of band gap not only of the bulk states but also the edge states. Remarkably, these polarized topological charges can also be identified by measuring the pseudospin structures. A feasible scheme to detect the HOTPs in the $^{87}$Rb atomic system is given. Our work opens an avenue for characterization and experimental detection of the HOTPs in momentum space.

Nodal-line semimetals are commonly believed to exist in $\mathcal{PT}$ symmetric or mirror-rotation symmetric systems. Here, we find a flux-induced parameter-dimensional second-order nodal-line semimetal (SONLS) in a two-dimensional system without $\mathcal{PT}$ and mirror-rotation symmetries. It has coexisting hinge Fermi arcs and drumhead surface states. Meanwhile, we discover a flux-induced second-order topological insulator (SOTI). We then propose a Floquet engineering scheme to create exotic parameter-dimensional hybrid-order nodal-line semimetals with abundant nodal-line structures and widely tunable numbers of corner states in a SONLS and SOTI, respectively. Our results break the perception of SONLSs and supply a convenient way to artificially synthesize exotic topological phases by 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.