Discoveries of both altermagnets and topological states have profoundly reshaped our understanding of condensed-matter physics and materials. However, tunable topological states in altermagnets remain largely unexplored. Here, we put forward the realization of second-order topological insulator (SOTI) and topological crystalline insulator (TCI) in two-dimensional altermagnets, and, in particular, demonstrate the feasibility of achieving a tunable topological phase transition with different bulk-boundary correspondence, i.e., from SOTI to TCI. As a concrete example, we consider the ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer to test the proposed scheme. The ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer is a prototypical material for altermagnets, and under equilibrium condition, it is a SOTI distinguished by well-localized nontrivial corner states. Remarkably, armed with preserved mirror symmetry ${\mathcal{M}}_{\mathit{z}}$, biaxial strain provides an effective means to engineer a topological phase transition in the ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer from SOTI to TCI, which is unambiguously confirmed by the calculated mirror Chern number ${\mathcal{C}}_{\mathcal{M}}$ = 1 and emergence of gapless edge states. Moreover, after the topological phase transition, an almost quantized plateau of the spin Hall conductivity ${\sigma }_{xy}^S$ is observed in the altermagnetic ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer. Our work provides a concrete material platform for exploring the interplay between nontrivial band topology and altermagnetism, with promising implications for low-dissipation spintronics.

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• Altermagnets
• Monolayer films
• Topological materials
• Density functional calculations
• Strain engineering