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In recent decades, topology related problems of electron systems such as topological insulators and topological superconductors in condensed matter physics have been attracting much of interests of theorists and experimentalists for the novel features and potential applications ranging from topological quantum computation to semiconductor spintronics.
Triggered by the study of topological aspects of Quantum Hall effect in 1980s, a new and distinctive phase of matter, topological phase which cannot be categorized within the framework of spontaneously broken symmetry breaking has been found. Topological properties of matters characterized by topological order beyond the Landau's formalism are independent of their detailed geometry and the presence of disorder and remain preserved unless we change materials or structural properties.
Efforts for manipulating the topological properties of matters by external resorts have been considerably devoted for various purposes, including the provocation of the topological phase transitions in the periodically driven system.
Floquet topological insulator (FTI) is referred as to the one that is driven by an external periodic perturbation, where Floquet theory comes into play. The bulk of works have been done regarding the observations and theoretical predictions for instance, for criticality, linear response, and optical phenomena in FTI and the Floquet topological phases in various systems. Among those, Floquet engineering is one of the most celebrating areas in the line of studying FTIs and enables us to manipulate the phases of matter which was regarded to be utterly impossible. Lindner et al showed that a topological phase can be induced by irradiation with microwave frequencies in HgTe/CdTe heterostructures, identified by the emergence of helical edge states, gap closing, and Chern number. Inoue et al studied photo-induced phase transition from normal insulator to Chern insulator in the Haldane model under the presence of irradiating circularly polarized laser field. The system is half-filled and the artificial magnetic flux piercing hexagonal parquet is still present for breaking time-reversal symmetry. They found the critical intensity of laser field to cause the topological phase transition, together with a phase diagram with respect to hopping parameters, time-reversal symmetry breaking phase, and the laser intensity. Kitagawa et al also showed the appearance of the quasi energy spectrum in topological phases accompanied by edge states emerged from trivial phase in a periodically driven two dimensional honeycomb lattice tight binding model. The second nearest neighbor hopping terms are neglected in the Hamiltonian. Moreover, in order to find the effective Hamiltonian that dominates the photo-induced dynamical transition in honeycomb lattice, they used a very rough approximation by dividing a full period of the evolution into three stages, each of which is represented by a different hopping terms.
All these studies demonstrate that the topological robustness that are independent of microscopic details of the system is not only unique in the static systems but still holds for the nonequilibrium systems driven by external periodic perturbation. By adopting Floquet formalism, the time periodicity of the Hamiltonian of the system makes it possible to find the quasi energy spectrum and the corresponding eigenstates ("Floquet states") even in nonequilibrium condition. It gives us an opportunity to convert a conventional insulator into a quantum spin Hall system by laser tuning and in particular, from the perspective of engineering of material function, it greatly fertilizes the ground of topological materials since topological characteristics are tuned by external perturbation as we wish.
Recently, we have studied the change of the appearance of the quasi energy spectrum in honeycomb nanoribbon when exposed to the external field.
We found that the topologically non trivial phase can be derived by means of the periodic external perturbation which is circularly polarized laser field in our case. The emergence of the topologically non trivial phases is recognized by two crossing edge states in quasi energy spectrum accompanied by the gap closing and the sudden raise in the value of the first Chern number from 0 to 1.
All the results that we have shown confirm again the stability of the existence of topological characters in nonequilibrium condition.
Our results can contribute to shed light on the applications such as engineering of the topological materials by manipulating the topological properties by means of the external field.
Our work has been published in "International Journal of Modern Physics B"(Vol. 35, No. 17, 2150180) in 2021 under the title of "Induction of floquet topological phases in honeycomb nanoribbon irradiated by circularly polarized laser field" (https://doi.org/10.1142/S0217979221501800).