Competitive orders in altermagnetic chiral magnons

The magnons in altermagnets exhibit chiral splitting even in the absence of spin-orbit coupling and external magnetic fields. Typically, this chiral splitting behavior can be well described by alternating isotropic spin exchanges (ISE) near the zero temperature. However, its finite-temperature dynamics, particularly when incorporating spin-orbit coupling effects, remains elusive. In this study, we reveal that, when including magnon-magnon interactions, long-range anisotropic spin exchange (ASE) can also induce chiral splitting of magnons at a finite temperature. Crucially, the chiral splitting induced by ASE competes with that arising from ISE, leading to a pronounced temperature-dependent modulation of the altermagnetic chiral splitting. Moreover, this competition is intimately connected to spin fluctuations, and can reverse the spin current driven by the band splitting as temperature increases. Our work uncovers the intrinsic competition governing collective spin excitations in altermagnets, providing new insights into their finite-temperature dynamical behavior.