In this work, a thorough exploration has been carried out to unravel the role of electron-phonon interaction (EPI) in a Bernevig-Hughes-Zhang quantum spin Hall (QSH) insulator subjected to a time-periodic step drive. It is observed that upon inclusion of the EPI, the system demonstrates emergent Floquet QSH phases and several topological phase transitions thereof, mediated solely by the interaction strength. Such transitions are indicated by the discontinuities in the projected spin Chern number. Furthermore, to employ a topological invariant that can characterize the system in an energy-resolved manner, a spectral localizer ($\mathcal{SL}$) is introduced, which distinctly ascertains the nature of the (zero or $\pi$) edge modes obtained within the corresponding topological gap. Quite intriguingly, we also observe the emergence of phases hosting a topological zero ($\pi$) energy gap while its $\pi$ (zero) energy sector is gapless. With the Chern number being found to be deficient in characterizing such coexistent phases, a real-space Chern marker (following the $\mathcal{SL}$ prescription) computed by us further provides support to such a gapless Floquet topological scenario. Our results may be realized in optical setups that may underscore the importance of EPI-induced Floquet features.

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• Dynamical phase transitions
• Edge states
• Polarons
• Quantum spin Hall effect
• Spin-orbit coupling
• Topological insulators
• Topological phase transition
• Topological phases of matter
• Floquet systems