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Despite the intense amount of attention and huge potential of two-dimensional(2D) magnets for applications in novel magnetic, magneto-optical,magneto-thermal and magneto-electronic devices, there has yet to be a robuststrategy developed to systematically understand magnon-magnon (MMI)interactions at finite temperature. In this paper, we present afirst-principles theoretical method to introduce the finite temperaturemagnon-magnon interaction into Heisenberg Hamiltonian through a nonlinearcorrection energy. The Wick theorem is used to decouple the four-magnonoperators to two-magnon order. We demonstrate the capabilities of this methodby studying the strength of MMI in Cr2Ge2Te6 (CGT) monolayer. The spin wavespectrum at finite temperature and the time-dependent spin autocorrelationfunction are explored. It is found that the magnon relaxation time due tomagnon-magnon scattering increases with temperature because of the reduction inmagnon energy, while decreases with wavevector and external magnetic field. Ourresults provide a new insight to understand the magnon damping and energydissipation in two-dimensional ferromagnetic materials.

Magnon-magnon interaction and magnon relaxation time in ferromagnetic Cr2Ge2Te6 monolayer