The Landau-Lifshitz-Gilbert (LLG) equation provides the fundamental description of dissipative magnetization dynamics. A recently proposed quantum analog, the quantum Landau-Lifshitz-Gilbert (QLLG) equation, extends this framework to quantum spin systems. However, whether the QLLG equation recovers the LLG equation in the classical large-spin limit for interacting spin systems remains an open question. Here, we demonstrate this connection by deriving the LLG equation as the classical large-spin limit of the QLLG equation. We establish this correspondence analytically and verify it numerically for an interacting spin system. These results provide a quantum-to-classical description of dissipative spin dynamics, offering a microscopic foundation for the LLG equation. From numerical calculations provided here, we also conclude that quantum effects become much less noticeable for systems with larger number of spins in the Hamiltonian.
The classical Landau-Lifshitz equation has been derived from quantum mechanics. Starting point is the assumption of a non-Hermitian Hamilton operator to take the energy dissipation into account. The corresponding quantum mechanical time dependent Schrödinger, Liouville and Heisenberg equation have been described and the similarities and differences between classical and quantum mechanical spin dynamics have been discussed. Furthermore, a time dependent Schrödinger equation corresponding to the classical Landau-Lifshitz-Gilbert equation and two ways to include temperature into the quantum mechanical spin dynamics have been proposed.
We examine the relation between the quantum Landau-Lifshitz equation (
q-LL) [Phys. Rev. Lett. 110, 147201 (2013)] and quantum Landau-Lifshitz-Gilbert equation (
q-LLG) [Phys. Rev. Lett. 133, 266704 (2024)]; two non-linear purity preserving master equations that extend classical atomistic spin dynamics into the quantum regime. While the classical LL and LLG counterparts for any number of spins are known to be equivalent, i.e., give identical spin trajectories up to a rescaling of the time parameter, the quantum formulations are equivalent only in certain cases, such as for pure states or for arbitrary single spin-
21 states. Here, we demonstrate that this equivalence breaks down even at the level of a single spin, provided
s≥1. Focusing on a spin-1 particle in an anisotropic crystal field, we show that the
q-LL and
q-LLG equations generate inequivalent time evolution. We introduce temporal rescaling misfits that quantify the inequivalence of the two types of dynamics. Although our results highlight fundamental differences in dissipation mechanisms encoded in these equations, the resulting trajectories remain qualitatively similar for this system.
The Landau-Lifshitz-Gilbert (LLG) and Landau-Lifshitz (LL) equations play an essential role for describing the dynamics of magnetization in solids. While a quantum analog of the LL dynamics has been proposed in [Phys. Rev. Lett. 110, 147201 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.147201], the corresponding quantum version of LLG remains unknown. Here, we propose such a quantum LLG equation that inherently conserves purity of the quantum state. We examine the quantum LLG dynamics of a dimer consisting of two interacting spin-1/2 particles. Our analysis reveals that, in the case of ferromagnetic coupling, the evolution of initially uncorrelated spins mirrors the classical LLG dynamics. However, in the antiferromagnetic scenario, we observe pronounced deviations from classical behavior, underscoring the unique dynamics of becoming a spinless state, which is nonlocally correlated. Moreover, when considering spins that are initially entangled, our study uncovers an unusual form of revival-type quantum correlation dynamics, which differs significantly from what is typically seen in open quantum systems.