We consider asymmetric and symmetric dimerized two-leg ladders, comprising of four different lattice points per unit cell, illuminated by circularly polarized light. in the asymmetric dimerized ladder case, rungs are not perpendicular to the ladder's legs whereas the rungs are perpendicular to the legs for the symmetric one. Using the Floquet theory, we obtain an effective Hamiltonian to study topological properties of the systems. Depending on the dimerization strength and driving amplitude, it is shown that topologically protected edge states manifest themselves not only as a zero-energy band within the gap between conduction and valence band but also as finite-energy curved bands inside the gap of subbands. The latter one can penetrate into bulk states and hybridize with the bulk states revealing hybridized floquet topological metal phase with delocalized edge states in the asymmetric ladder case. However, in the symmetric ladder, the finite-energy edge states while remaining localized can coexist with the extended bulk states manifesting floquet topological metal phase. Topological states of matter with intriguing properties have attracted a lot of attention in various fields of physics, particularly, solid-state physics 1. Because of robustness of such states against ubiquitous perturbations 2 , materials hosting topological states will be excellent candidates for sensitive electronic applications. Topological insulators 3 along with topological superconductors 4 exhibiting topologically nontrivial phases have been interesting topics from theoretical and experimental view points. However, the known topological systems in the equilibrium situation which can indeed be used to realistic applications are limited to a few cases leading to exploring topological quantum states out-of-equilibrium 5. Beside materials including static topological phases, engineering of exotic nontrivial phases of quantum materials 6 has been developed by means of externally applied dynamical fields. Such approach provides a flexible and practical way to produce desired phases which are absent in the static counterparts. For instance, periodic driving establishes dynamical topological states, known as topological Floquet states 7-9. An interesting characteristic of the Floquet theory 10,11 is to add extra dimension in a quantum system through continuous evolution over all times within the driving period 9,12,13 providing higher-dimensional systems effectively. In the opposite limit, i.e., stroboscopic picture 6,8 , periodic driving manipulates the system parameters expanding phase diagram to values that are not easily accessible in undriven systems. Both of these two features pave the way to turn trivial phases of the system into exotic ones, such as Floquet topological semimetals 14,15 , Floquet topological superconductors 16,17 , and Floquet topological insulators 7,18. There are a variety of techniques for exerting time periodicity and establishing topologically protected edge states such as shining a matter with light 7,19-21 , sha...

We theoretically report the finding of a new kind of topological phase transition between a normal insulator and a topological metal state where the closing-reopening of bandgap is accompanied by passing the Fermi level through an additional band. The resulting nontrivial topological metal phase is characterized by stable zero-energy localized edge states that exist within the full gapless bulk states. Such states living on a quasi-one-dimensional system with three sublattices per unit cell are protected by hidden inversion symmetry. While other required symmetries such as chiral, particle-hole, or full inversion symmetry are absent in the system.

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We investigate theoretically the topological properties of dimerized quasi-one-dimensional (1D) lattice comprising of multi legs (L) as well as multi sublattices (R). The system has main and subsidiary exchange symmetries. In the basis of latter one, the system can be divided into L 1D subsystems each of which corresponds to a generalized SSHR model having R sublattices and onsite potentials. Chiral symmetry is absent in all subsystems except when the axis of main exchange symmetry coincides on the central chain. We find that the system may host zero-and finite-energy topological edge states. The existence of zero-energy edge state requires a certain relation between the number of legs and sublattices. As such, different topological phases, protected by subsystem symmetry, including zero-energy edge states in the main gap, no zero-energy edge states, and zeroenergy edge states in the bulk states are characterized. Despite the classification symmetry of the system belongs to BDI but each subsystem falls in either AI or BDI symmetry class.

We report on the theoretical investigation of the topological properties of a periodically quenched one-dimensional dimerized lattice where a piece-wise constant Hamiltonian switches from h1 to h2 at a partition time tp within each driving period T . We examine different dimerization patterns for h1 and h2 and the interplay with the driving parameters that lead to the emergence of topological states both at zero energy and at the edge of the Brillouin-Floquet quasi-energy zone. We illustrate different phenomena, including the occurrence of both edge states in a semimetal spectrum, the topological transitions, and the generation of zero-energy topological states from trivial snapshots. The role of the different symmetries in our results is also discussed.