Quantum Spin Hall Effect with Extended Topologically Protected Features in Altermangetic Multilayers

Conventional topological classification theory dictates that time-reversal symmetry confines the quantum spin Hall (QSH) effect to a $\mathbb{Z}_2$ classification, permitting only a single pair of gapless helical edge states. Here, we utilize the recently discovered altermagnetism to circumvent this fundamental constraint. We demonstrate the realization of a unique QSH phase possessing multiple pairs of gapless helical edge states in altermagnetic multilayers. This exotic QSH phase, characterized by a mirror-spin Chern number, emerges from the interplay of spin-orbit coupling and $d$-wave altermagnetic ordering. Moreover, using first-principles calculations, we identify altermagnetic Fe$_2$Se$_2$O multilayers as promising material candidates, in which the number of gapless helical edge states scales linearly with the number of layers, leading to a correspondingly large, exactly quantized, and experimentally accessible spin-Hall conductance. Our findings unveil a new mechanism for stabilizing multiple pairs of gapless helical edge states, significantly expanding the scope of QSH effects, and provide a blueprint for utilizing altermagnetism to engineer desired topological phases.