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

TL;DR

The study develops a unified framework for dynamical metastability in driven-dissipative magnetic systems, bridging nonlinear quantum spin dynamics and classical magnetization dynamics. It shows that linear magnons in an interacting spin Lindbladian replicate a dynamical Hatano–Nelson chain with topologically protected edge modes (Dirac bosons), and that nonlinearities induce genuine phenomena such as spin dipping and transient attraction to unstable equilibria, with edge modes persisting under disorder. The LLGS analysis demonstrates that the same metastable and topological features reappear in realistic magnetic multilayers, while also enabling multistability and limit cycles absent in the quantum model. Together, these results establish dynamical metastability as a robust organizing principle for nonlinear magnetization dynamics with potential impact on spin-torque oscillators and magnonic devices, and they open avenues toward a quantum-classical open-system framework bridging semiclassical and fully quantum descriptions.

Abstract

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.

Figures (19)

• Figure 1: (a)-(b) Example rapidity spectra for the bosonic HN chain, described by Eq. \ref{['eq: bkEOM']}, in different dynamical stability regimes: $\Delta^\text{OBC}_N < \Delta^\text{Bulk} < 0$ for (a) and $\Delta^\text{OBC}_N<0< \Delta^\text{Bulk}$ for (b). In both cases, the rapidity band $A(k)$ is depicted in black, with black squares marking the discrete PBC rapidities $A\,(2m\pi/N)$, $m=0,\ldots,N-1$. The red disks correspond to the OBC rapidities in Eq. \ref{['eq: HNOBCRaps']}. The red (gray) dashed lines indicate the OBC (PBC) stability gaps $\Delta^\text{OBC}_N$ ($\Delta^\text{Bulk}$) with the gray region corresponding to the unstable half-plane $\text{Re}[\lambda]>0$. In (a), both gaps are negative, while in (b), the OBC gap is negative and the PBC gap is positive. In all cases, $J=D=\kappa_-=\Gamma =1$ and $N=10$. The frequency $\omega$ is arbitrary (see the discussion in Sec. \ref{['sec: frameTDM']} on rotating frames). In (a), $\kappa_+=0.8$, while for (b) $\kappa_+=1.3$. (c)-(d) Anomalous transient dynamics of the bulk site $|\langle a_{N/2}(t)\rangle|^2$ in each dynamical regime. The red curves correspond to the OBC dynamics, with darker curves featuring larger $N$, with $N$ increasing from 10 to 40 in steps of 6. The slope of the red dashed line is the OBC asymptotic relaxation rate $|\Delta^\text{OBC}_N|$, while the slope of the black dashed line is the PBC relaxation (amplification) rate in the case of (c) ((d)) $|\Delta^\text{Bulk}|$. Interestingly, the observed amplification timescale turns out to be much longer than the one predicted by Eq. \ref{['eq: amptime']}, which is only $\kappa_- \tau_\pi \sim 6.9$ for $N=100$. In both (c-d), the system is initialized in a state satisfying $\braket{b_k(0)} \propto \delta_{k\pi}$.
• Figure 2: (a) Example rapidity spectra for the HN QBL in the topologically metastable regime. (b) The evolution of the site-1 intensity $|\braket{a_1}|^2$ for the system described by Eq. \ref{['eq: HNQBL']} when prepared in the quasi-steady state $\rho_\text{ss}(z=1)$ defined in the main text. (c) The average value of the (normalized) DB expectation value $\braket{\alpha(t)}/\braket{\alpha(0)}$ averaged over 250 random initial conditions. In both (b) and (c), the black dashed line corresponds to the $N\to\infty$ exact trajectory whereby both $\rho_\text{ss}(z=1)$ is steady and $\alpha$ is an exact zero mode. The red dashed line indicates the characteristic OBC decay rate. In both plots, system size increases from $N=10$ to $40$ in steps of $6$ with the curve becoming darker for larger $N$. The model parameters chosen are $J=0.5$, $D=1.5$, $\kappa_-=1$, $\kappa_+=2$, and $\Gamma=1$. The frequency $\omega$ is arbitrary.
• Figure 3: A schematic representation of the spin Lindbladian described by Eqs. \ref{['eq: spinLind']}. Each spin precesses with a characteristic frequency $\omega$ about the $z$-axis. Nearest neighbors are coupled via real and imaginary exchange interactions of strength $J$ and $D$, respectively. Each spin dissipates into nearest-neighbor mutual reservoirs independently at a rate $\kappa_-$, with an additional correlated decay rate $\Gamma$. Additionally, each spin is pumped away from equilibrium at a rate $\kappa_+$.
• Figure 4: (a) The magnon rapidities for the spin Lindbladian models in Eqs. \ref{['eq: SC+']} and \ref{['eq: SC-']}, resulting from a Holstein-Primakoff expansion about the aligned spin state (${\cal L}_+$, left) and the antialigned spin state (${\cal L}_-$, right). For the aligned case, the OBC rapidities (red) are all strictly negative in real part, hence arbitrary nearby initial conditions converge to the aligned state. Meanwhile, under PBCs, convergence of bulk spin waves depends on whether $\text{Re}[A(k)]$ is positive or negative. In the antialigned case, the OBC rapidities are all strictly negative in real part, hence all nearby initial conditions diverge away. Meanwhile, under PBCs, those modes that previously were unstable around the aligned state, are now stable around the antialigned state, and vice-versa. The PBC rapidities of the modes $k=0$ and $k=\pi$ are indicated in each case. (b) The steady-state phase diagram for spin waves under PBCs, resulting from Eq. \ref{['eq: ReAk']}. As soon as the spin pumping $\kappa_+$ exceeds the spin damping $\kappa_-$, modes with wavevectors $k\sim \pm\pi$ begin to relax into the antialigned state. Once $\kappa_+$ exceeds $\kappa_-+4\Gamma$ (i.e., the decay rate of the $k=0$ mode), all spin waves relax to the antialigned state. (c) The vector fields specified by Eq. \ref{['eq: bulkeom']} defined on the sphere of radius $s$ for the three cases $\text{Re}[A(k)]<0$ , $\text{Re}[A(k)]=0$, and $\text{Re}[A(k)]>0$. In the first case, the north pole $s^z = +1$ is attractive, while in the last case, the south pole $s^z=-1$ is attractive. When $\text{Re}[A(k)]=0$, each circle of latitude at fixed $s^z$ is a closed trajectory with frequency dictated by Eq. \ref{['eq: closedlat']}. The chirality of each trajectory depends on the hemisphere in which it is located, with the equator $s^z = 0$ being a circle of fixed points.
• Figure 5: The hierarchy of EOM descendent from the full spin Lindbladian specified by Eqs. \ref{['eq: spinLind']}. When possible, an equation number for the relevant equation is appended.