Landau-Lifshitz damping from Lindbladian dissipation in quantum magnets

TL;DR

The paper addresses the lack of a quantum foundation for Landau-Lifshitz damping by deriving LL dynamics from Lindbladian open-quantum-system theory within a local mean-field framework for weak external fields. It shows that LL behavior emerges in the weak-field limit $h_0\ll J$ from a fully quantum Lindblad evolution and further extends the description to allow relaxation of the magnetization length $|\vec{m}|$, yielding a generalized LL equation with a length-dynamics term. Finite-temperature corrections are also derived, modifying the damping rate to $\widetilde{\gamma}=\gamma(1+n_B/S)$ and reducing the saturation length to $m_{\infty}=S/(1+n_B/S)$, thus establishing a coherent quantum-to-classical correspondence across temperatures. The findings provide a principled route to compute damping parameters from microscopic reservoir models and enable integrating quantum-dissipative effects into atomistic spin simulations, including extensions to antiferromagnets and nontrivial spin textures.

Abstract

As of now, the phenomenological classical Landau-Lifshitz (LL) damping of magnetic order is not conceptually linked to the quantum theory of dissipation of the Lindbladian formalism which is unsatisfactory for the booming research on magnetic dynamics. Here, it is shown that LL dynamics can be systematically derived from Lindbladian dynamics in a local mean-field theory for weak external fields. The derivation also extends the LL dynamics beyond the orientation $\vec{m}/|\vec{m}|$ to the length $|\vec{m}|$ of the magnetization. A key assumption is that the Lindbladian dissipation adapts to the non-equilibrium $H(t)$ instantaneously to lower its expectation value.

Figures (3)

• Figure 1: Solutions (solid lines) of Lindbladian \ref{['eq:lindblad-fm']} with the mean-field self-consistency \ref{['eq:eff-field']} for the initial length $S$ of the order parameter $\vec{m}$; the initial direction of the magnetization is tilted by $\pi-0.1$ relative to the $z$ direction. The other couplings are $S=1/2$, $\lambda =0.2$, $J$ is given in the panels, and $\gamma=\lambda J$. The dashed curves represent the solution for the same parameters of the LL equation \ref{['eq:ll']}.
• Figure 2: Solutions (solid lines) of Lindbladian \ref{['eq:lindblad-fm']} with the mean-field self-consistency \ref{['eq:eff-field']} for the initial length of $\vec{m}$ given in the panels; the initial direction of the magnetization is tilted by $\pi-0.1$ relative to the $z$ direction. The other couplings are $S=1/2$, $J=8h_0$, $\lambda =0.2$ so that $\gamma=\lambda J= 8/5.$ The dashed curves represent the solution for the same parameters of equation \ref{['eq:analytic1']} which results in the weak-field limit $h_0/J\to 0$ in linear order.
• Figure 3: Solutions (dashed lines) of the Landau-Lifshitz equation \ref{['eq:ll']} for $\lambda=\gamma/J=0.2$ and $S=1/2$. Solutions (solid lines) of the Lindbladian equation \ref{['eq:lindblad-fm']}. The initial direction of the magnetization of length $S$ is tilted by $\pi-0.1$ with respect to the $z$ direction.