Optically induced magnetic inertia and magnons from non-Markovian extension of the Landau-Lifshitz-Gilbert equation

TL;DR

The paper addresses the limitations of phenomenological LLG descriptions for optically driven magnets by deriving a first-principles, non-Markovian extension of the LLG equation using Schwinger-Keldysh field theory (SKFT). By modeling itinerant electrons driven by light and integrating them out to obtain an effective spin action, the authors obtain a nonlocal memory kernel $\eta_{nn'}(t,t')$ and a light-induced field $\mathbf{B}^e_n$ that govern spin dynamics via a generalized equation for $\mathbf{S}_n(t)$. In the weak-light limit, this kernel yields an inertial term with time-dependent, spatially nonlocal prefactors ($\lambda_{nn'}(t)$ and $I_{nn'}(t)$), predicting optically induced magnetic inertia and the excitation of coherent magnons whose frequencies scale with the light frequency $\omega_L$ (e.g., peaks at $\omega^{(1)}_\pm = \omega_L \pm 0.23/\eta_1$ and $\omega^{(3)}_\pm = 3\omega_L \pm 0.23/\eta_1$). The memory kernel exhibits fractal structure in the $t$-$t'$ plane under strong fs-laser pumping, enabling sharp, optically controllable magnon modes and a band of incoherent magnons due to nutation; these insights provide a principled route to modeling optical magnetization switching and magnon generation in driven magnets. The framework promises improved simulation tools for magnonics and quantum information applications, while highlighting limitations for gapped-band systems where direct light-spin coupling is less effective.

Abstract

The Landau-Lisfhitz-Gilbert (LLG) equation has been the cornerstone of modeling the dynamics of localized spins, viewed as classical vectors of fixed length, within nonequilibrium magnets. When light is employed as the nonequilibrium drive, the LLG equation must be supplemented with additional terms that are usually conjectured using phenomenological arguments for direct opto-magnetic coupling between localized spins and (real or effective) magnetic field of light. However, direct coupling of magnetic field to spins is 1/c smaller than coupling of light and electrons; or both magnetic and electric fields are too fast for slow classical spins to be able to follow them. Here, we displace the need for phenomenological arguments by rigorously deriving an extended LLG equation via Schwinger-Keldysh field theory (SKFT). Within such a theory, light interacts with itinerant electrons, and then spin current carried by them exerts spin-transfer torque onto localized spins, so that when photoexcited electrons are integrated out we arrive at a spin-only equation. Unlike the standard phenomenological LLG equation with local-in-time Gilbert damping, our extended one contains a non-Markovian memory kernel whose plot within the plane of its two times variables exhibits fractal properties. By applying SKFT-derived extended LLG equation, as our central result, to a light-driven ferromagnet as an example, we predict an optically induced magnetic inertia term. Its magnitude is governed by spatially nonlocal and time-dependent prefactor, leading to excitation of coherent magnons at sharp frequencies in and outside of the band of incoherent (or thermal) magnons.

Figures (4)

• Figure 1: The memory kernel $\eta(t,t')$ [Eq. \ref{['eq:nonmarkovianKernel']}] of SKFT-derived extended LLG Eq. \ref{['eq:modifiedLLG']} plotted as a function of two-times $t$ and $t'$ for an example Sayad2015 of a single localized classical spin $\mathbf{S}(t)$ interacting with itinerant electrons in 1D. The kernel is Markovian ReyesOsorio2024 in (a), where light is absent; and non-Markovian in (b)--(d), where electrons are driven by fsLP of central frequency $\omega_L=0.3\gamma/\hbar$ and with its vector potential $|\mathbf{A}(t)|$ plotted at the bottom of each panel. The width of fsLP in panels (b) and (d) is $\lambda=200\hbar/\gamma$, whereas in panel (c) it is $\lambda=20\hbar/\gamma$. Finite values of the kernel away from the time diagonal $t=t'$ indicate greater memory effects. As the amplitude $z$ of fsLP increases in (b)--(d), the plot of $\eta(t,t')$ in $t$-$t'$ plane becomes a fractal characterized by noninteger dimension $d$.
• Figure 2: Time evolution of a single localized classical spin $\mathbf{S}(t)$ interacting with itinerant electrons within an infinite 1D TB chain, as computed from the non-Markovian LLG Eq. \ref{['eq:modifiedLLG']} in an external static magnetic field along the $z$-axis, $\mathbf{B}^{\rm ext}=0.01\gamma \mathbf{e}_z$, and with electrons driven by a train of fsLPs of amplitude (a) $z=1$ and (b) $z=5$, respectively. At $t=0$ spin is slightly tilted away from the $z$-axis, and for its time evolution in (a) and (b) we use the memory kernels from Figs. \ref{['fig:nmkernels']}(b) and \ref{['fig:nmkernels']}(d), respectively. Panels (c) and (d) plot FFT amplitude spectra of $S^x(t)$ from panels (a) and (b), respectively. Note that in the absence of fsLPs panel (c) and (d) contain only a single peak (dashed black line) whose position is determined by $|\mathbf{B}^{\rm ext}|$.
• Figure 3: (a) Magnon spectrum of a chain of 30 localized classical spins $\mathbf{S}_n(t)$, as extracted Kim2010Moreels2024 from FFT of $S^x_{n=15}(t)$ in the middle of the chain, which interact with itinerant electrons hopping along an infinite 1D TB chain. The dynamics of $\mathbf{S}_n(t)$, precessing around the $z$-axis, is computed from iLLG Eq. \ref{['eq:inertia']} where the nonequilibrium drive for electrons is CW light of frequency $\omega_L=5.5J/\hbar$ (also marked by vertical dashed line). The vectors $\mathbf{S}_n(t=0)$ are initially thermalized at $k_BT=0.02J$, so that their evolution via iLLG Eq. \ref{['eq:inertia']} with $I_{nn'}(t)=0$ and $\lambda_{nn'}$ being time-independent ReyesOsorio2024 in the absence of light produces a spectrum [enclosed within left gray box in panel (a)] of incoherent (or thermal) magnons Pirro2021De2024. In contrast, all four sharp peaks in panel (a) at frequencies $\omega^{(1)}_\pm=\omega_L\pm 0.23/\eta_1$ and $\omega^{(3)}_\pm=3\omega_L\pm 0.23/\eta_1$ are optically excited coherent magnons obtained from full iLLG Eq. \ref{['eq:inertia']}. Panel (b) shows FFT amplitude spectra for a range of $\omega_L$, revealing a linear relation between the frequency of optically excited coherent magnon peaks and $\omega_L$. Note that peaks from panel (a) are included in (b) as intersection of dashed red line at $\omega_L=5.5J/\hbar$ and tilted yellow traces.
• Figure 4: Wavevector of magnons in a chain of 30 localized spins externally driven by CW light of varying frequency $\omega_L$ [same setup employed in Fig. \ref{['fig:spectra']}] extracted from spatial FFT amplitude spectrum of $\mathbf{S}_n(t)$, on the proviso that chosen time $t$ is late enough so that only the long-lived light-generated coherent magnons [sharp peaks in Fig. \ref{['fig:spectra']}(a)] contribute to the spectrum. Darker curves correspond to higher $\omega_L$.