Exploring Topological Phases and Spin Group Protection in Two-Dimensional Altermagnets*

In magnetic materials containing light elements, where neglecting relativistic spin-orbit coupling (SOC) is a reasonable approximation, the crystalline symmetry groups are considerably larger than those of conventional magnetic materials. This expansion results from the decoupling of nontrivial spin degrees of freedom from the orbital part, which is known to be relevant in the newly proposed class of magnetic materials called 'altermagnets.' Here, we systematically study the description of spinful electrons in 2D magnetic lattices using spin groups, focusing on topological phases protected by order-two nonsymmorphic spin group elements in coplanar magnetic textures. Within the spin group framework, we explore intrinsic topological phases by considering one of the spin group elements that acts solely in spin space, effectively serving as a time-reversal symmetry in addition to the nonsymmorphic spin group element. This approach leads to the emergence of new topological phases, enriching the landscape of magnetic topological phases protected by magnetic space group symmetries. Furthermore, an important question to address is understanding the robustness of topological protection afforded by the spin group against SOC, which is inevitably present in all materials.