Dynamical metastability and transient topological magnons in interacting driven-dissipative magnetic systems

Metastability, i.e., partial relaxation to long-lived, quasi-stationary states before true asymptotic equilibrium sets in, emerges ubiquitously in classical and quantum dynamical systems as a result of timescales separation. In open quantum systems, an intrinsically nonequilibrium analogue, dynamical metastability, can originate from the spectral geometry of a non-Hermitian operator. In noninteracting models, this mechanism produces boundary-sensitive anomalous relaxation, transient amplification, and topologically mandated long-lived edge modes, all of which are enhanced as system size grows. Here we extend dynamical metastability into the nonlinear, interacting regime and identify magnetic heterostructures as a natural platform for its exploration. We introduce an interacting spin Lindbladian whose linearized magnon dynamics map onto a dynamically metastable Hatano-Nelson chain, and show that dynamical metastability in the noninteracting limit seeds genuinely nonlinear phenomena, including size-dependent spin dipping and anomalous attraction to unstable equilibria. Long-lived edge states associated to topologically mandated Dirac bosons persist under nonlinearities and disorder. We further analyze the magnetization dynamics in magnetic multilayers within the classical Landau-Lifshitz-Gilbert-Slonczewski framework, identifying Dzyaloshinskii-Moriya interaction, nonlocal damping, and spin-transfer torque as control parameters governing bulk-boundary stability mismatch and band topology. While all the distinctive dynamical phenomena previously identified reappear in this experimentally relevant setting, the LLGS framework also supports multistability and limit cycles that are absent in the quantum model. Our results constitute the first systematic study of dynamical metastability in nonlinear dynamics, directly relevant to spin-torque oscillator arrays, magnonic devices, and beyond.