Holstein polaron in a pseudospin-1 quantum spin Hall system: First- and second-order topological phase transitions

We investigate the effects of Holstein polarons formed due to the electron-phonon (e-p) coupling, on the quantum spin Hall (QSH) phase of a pseudospin-1 fermionic $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice. The parameter $\ensuremath{\alpha}$ and the e-p coupling strength $\ensuremath{\lambda}$ have an interesting interplay, which demonstrates that at smaller values of $\ensuremath{\alpha}$, there is a single transition from a topological to a trivial phase as a function of $\ensuremath{\lambda}$, while the larger $\ensuremath{\alpha}$ values host two gap-closing transitions, namely, trivial-topological-trivial transitions, accompanied by a narrow semimetallic phase in-between. The topological properties are characterized by computing the ${\mathbb{Z}}_{2}$ invariant which confirms the existence of topological (trivial) phases, which are hence verified against the presence (absence) of counterpropagating helical edge modes in a nanoribbon. Furthermore, on the introduction of a planar magnetic field into the system, emergence of a second-order topological phase is observed. The in-plane field causes gapping out of the first-order edge states while maintaining the topological phase of the bulk intact, subsequently leading to the emergence of robust corner modes under suitable open boundary conditions. This resultant phase is adequately designated by computing the projected spin Chern number, a well-established invariant for the time-reversal symmetry-broken QSH phase. Further, we show that the e-p coupling yields a complete disruption of the corner modes as we tune it beyond a certain critical strength, giving rise to a second-order topological phase transition.