We propose a theoretical model to investigate the interplay between altermagnetism and $p$-wave superconductivity, with a particular focus on topological phase transitions in a two-dimensional $p$-wave superconductor, considering both chiral and helical phases. Our study reveals that the emergence of helical and chiral Majorana states can be tuned by the amplitude of a $d-\text{wave}$ altermagnetic order parameter, with the outcome depending on the nature of the superconducting state. In the helical superconductor, such an altermagnet can induce a topological phase transition into a gapless topological superconductor hosting Majorana flat edge modes. On the other hand, in the chiral superconductor, the topological transition takes place between a topologically nontrivial gapped phase and a gapless nodal-line superconductor, where the Bogoliubov quasiparticle bands intersect at an isolated line in momentum space. Remarkably, we show that when such an altermagnet is coupled to a mixed-pairing superconductor, with both chiral and helical components, a hybrid topological phase emerges, featuring dispersive Majorana edge modes that coexist with nearly flat Majorana edge states. Our findings therefore establish a platform for controlling and manipulating Majorana modes in unconventional superconductors with vanishing total magnetization.

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• Topological superconductors
• Altermagnets
• Unconventional superconductors
• Bogoliubov-de Gennes equations