Time reversal symmetry broken quantum spin hall effect in pseudospin-1 Dirac-Rashba system

The Quantum spin Hall (QSH) phase is conventionally understood to be protected by time-reversal symmetry (TRS). Here, we theoretically investigated the fate of the QSH phase in a pseudospin-1 fermionic $\alpha-\mathcal{T}_3$ system in the presence of a TRS-breaking ferromagnetic exchange field and spin-nonconserving Rashba spin-orbit coupling. Despite broken TRS, the QSH phase survives over a finite parameter regime and is characterised by a non-zero projected spin-Chern number $C_\sigma (\sigma = \uparrow, \downarrow)$, protected by a spin-spectral gap. In the absence of Rashba coupling, the QSH phase remains robust up to an $\alpha$-dependent critical exchange field. Rashba SOC qualitatively reshapes the phase diagram by driving transitions into two distinct quantum anomalous Hall (QAH) phases: a $C=2$ phase, irrespective of $\alpha$-values, and a $C=1$ phase for $\alpha \neq 0,1$, which is further identified as a valley-polarized QAH phase arising from a single valley. Rotating the magnetization to in-plane gaps out the first-order helical edge states and gives rise to second-order topological insulator (SOTI) phases that host localized corner states in suitable finite geometry. We further identify a topological phase transition between two different SOTI phases, mediated by nanoribbon edge states at an exchange field equal to $\alpha$. These results establish spin-resolved topology in a higher pseudospin system as well as the $\alpha-\mathcal{T}_3$ lattice as a versatile platform for engineering and controlling multiple topological phases through magnetic exchange and spin-orbit coupling.