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Recently, we presented a two-dimensional (2D) model of a weak topologicalinsulator formed by stacking an $N$ number of Su-Schrieffer-Heeger (SSH) chains\cite{Agrawal_2022-02}. We now study the influence of periodic driving on thetopological properties of this system, which has all the fundamentalsymmetries, by shining it with circularly polarized light (CPL). The CPL ischosen because it breaks the time-reversal symmetry, which induces more exotictopological properties in the system. We investigate two different formationsof the $N$ stacked SSH chains: all the SSH chains are topologically trivial inone formation and nontrivial in the other one. In contrast to the undriven orstatic case, both formations exhibit distinct topological behaviors. Here, weparticularly derive the Floquet or the effective Hamiltonian using the replicamethod, which facilitates the study of high- and low-frequency regimes. We havediscovered that this model exhibits laser-induced Floquet topological phaseswith higher Chern numbers. This system has nonlinear dispersion along bothdirections with additional $k_x-k_y$ coupling terms, which made the dispersionof this system {\it unconventional}. We closely study the role of thisunconventional dispersion in the system at the low-energy limit and itsresponse to periodic driving. The low-energy Hamiltonian also reveals ahierarchy in the gaps of the neighboring Floquet bands. Interestingly, thoughthis model has nonlinear quasi-energy dispersion, it still shows somesignatures of hierarchy, which was observed in the system with lineardispersion like graphene. Furthermore, we study the effect of linearlypolarized light (LPL) on the topological properties of the system. In responseto the LPL driving, the band-touching point either opens up or splits into twoband-touching points.

Laser induced Flqouet topological phases in a 2D weak topological insulator with unconventional nonlinear dispersion