Magnon-magnon interaction in monolayer MnBi$_2$Te$_4$

MnBi$_2$Te$_4$, the first confirmed intrinsic antiferromagnetic topological insulator, has garnered increasing attention in recent years. Here we investigate the energy correction and lifetime of magnons in MnBi$_2$Te$_4$ caused by magnon-magnon interaction. First, a calculation based on the density functional theory was performed to get the parameters of the magnetic Hamiltonian of MnBi$_2$Te$_4$. Subsequently, the perturbation method of many-body Green's function was employed and the first-order self-energy [$\Sigma^{(1)}(\bm k)$] and second-order self-energy [$\Sigma^{(2)}(\bm k,\varepsilon_{\bm k})$] of magnon were obtained. Numerical computations reveal that the corrections from both $\Sigma^{(1)}(\bm k)$ and $\Sigma^{(2)}(\bm k,\varepsilon_{\bm k})$ strongly rely on momentum and temperature, with the energy renormalization near the Brillouin zone (BZ) boundary being significantly more pronounced than that near the BZ center. Furthermore, our findings indicate the occurrence of dip structures in the renormalized magnon spectrum near the $\rm K$ and $\rm M$ points. These dip structures are determined to be attributed to the influence of $\Sigma^{(2)}(\bm k,\varepsilon_{\bm k})$.