We study the formation of magnon-polaron excitations and the consequences of different time scales between the magnon and lattice dynamics. The spin-spin interactions along the 1D lattice are ruled by a Heisenberg Hamiltonian in the anisotropic form XXZ, in which each spin exhibits a vibrational degree of freedom around its equilibrium position. By considering a magnetoelastic coupling as a linear function of the relative displacement between nearest-neighbor spins, results provide an original framework for achieving a hybridized state of magnon-polaron. Such state is characterized by high cooperation between the underlying excitations, where the traveling or stationary formation of magnon-polaron depends on the effective magnetoelastic coupling. A systematic investigation reveals the critical amount of the magnon-lattice interaction ($chi_c$) necessary to emergence of the stationary magnon-polaron quasi-particle. Different characteristic time scales of the magnon and the vibrational dynamics unveiled the threshold between the two regimes, as well as a limiting value of critical magnetoelastic interaction, above which the magnon velocity no longer interferes at the critical magnetoelastic coupling capable of inducing the stationary regime.
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