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arXiv:2406.10613 · New Journal of Physics

Landau–Lifshitz damping from Lindbladian dissipation in quantum magnets

Götz Silvester Uhrig

Published Oct 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract As of now, the phenomenological classical Landau–Lifshitz (LL) damping of magnetic order is linked to a quantum theory of dissipation of Lindbladian form only in sophisticated systems via complicated calculations in certain limits. This 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 rather simple way using a local mean-field theory for weak external fields. The necessary assumptions are (i) the validity of a local mean-field approach, (ii) a local dissipator, and (iii) a very rapid bath adapting to the non-equilibrium H ( t ) instantaneously. The derivation also extends the LL dynamics beyond the orientation m → / | m → | to the length | m → | of the magnetization, thereby transcending the classical description.

DOI
10.1088/1367-2630/ae115c

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Abstract As of now, the phenomenological classical Landau–Lifshitz (LL) damping of magnetic order is linked to a quantum theory of dissipation of Lindbladian form only in sophisticated systems via complicated calculations in certain limits. This 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 rather simple way using a local mean-field theory for weak external fields. The necessary assumptions are (i) the validity of a local mean-field approach, (ii) a local dissipator, and (iii) a very rapid bath adapting to the non-equilibrium H ( t ) instantaneously. The derivation also extends the LL dynamics beyond the orientation m → / | m → | to the length | m → | of the magnetization, thereby transcending the classical description.
Abstract Magnetic systems can be described by the classical Landau–Lifshitz (LL) equation or the fully quantum open generalized Heisenberg model. Using the Lindblad master equation and the mean-field approximation, we demonstrate that the open generalized Heisenberg model is reduced to a generalized LL equation. The open dynamic is modeled using spin-boson interactions with a common bosonic reservoir at thermal equilibrium. By tracing out the bosonic degrees of freedom, we obtain two different decoherence mechanisms: on-site dissipation and an effective spin–spin interaction mediated by bosons. Using our approach, we perform hysteresis calculations, closely connected with the Stoner–Wohlfarth theory. We compare the exact numerical master equation and the mean-field model, revealing the role of correlations originated by non-local interactions. Our work opens new horizons for the study of the LL dynamics from an open quantum formalism.
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 entanglement of many localized spins (LSs) within solid magnetic materials is a topic of great basic and applied interest, particularly after becoming amenable to experimental scrutiny where recent neutron scattering experiments have witnessed macroscopic entanglement in the ground state (GS) of antiferromagnets persisting even at elevated temperatures. On the other hand, spintronics and magnonics studies assume that LSs of antiferromagnets are in unentangled N\'eel GS, as well as that they evolve, when pushed out of equilibrium by current or external fields, according to the Landau-Lifshitz-Gilbert (LLG) equation viewing LSs as classical vectors of fixed length. The prerequisite for applicability of the LLG equation is zero entanglement in the underlying many-body quantum state of LSs. In this study, we initialize quantum Heisenberg ferro- or antiferromagnetic chains hosting S=1/2,S=1S=1/2, S=1S=1/2,S=1, or S=5/2S=5/2S=5/2 LSs into an unentangled pure state and then evolve them by quantum master equations (QMEs) of Lindblad or non-Markovian type, derived by coupling LSs weakly to a bosonic bath (due to phonons in real materials) or by using additional ``reaction coordinate'' in the latter case. The time evolution is initiated by applying an external magnetic field, and entanglement of the ensuing mixed quantum states is monitored by computing its negativity. We find that non-Markovian dynamics never brings entanglement to zero, in the presence of which the vector of spin expectation value changes its length to render the LLG equation inapplicable. Conversely, Lindbladian (i.e., Markovian) dynamics ensures that entanglement goes to 0, thereby enabling quantum-to-classical transition in all cases---S=1/2,S=1S=1/2, S=1S=1/2,S=1, and S=5/2S=5/2S=5/2 ferromagnets or S=5/2S=5/2S=5/2 antiferromagnets---except for S=1/2S=1/2S=1/2 and S=1S=1S=1 antiferromagnets. Finally, we investigate the stability of an entangled antiferromagnetic GS upon suddenly coupling it to bosonic baths.
We introduce a distinct route for extending classical atomistic spin dynamics into the quantum regime for arbitrary spin sss, based on a variational product-state \textit{ansatz} that includes both pure and mixed local states. By construction, intersite entanglement -- expected to remain negligible even on very short time scales in many real materials -- is excluded, while local quantum effects are retained through the nonclassical structure of s≥1s\geq 1s≥1 density operators. Nonlinear dynamics arise from both the mean-field interdependence of the local states and the state-dependent dissipative terms. The framework recovers classical spin dynamics for s=12s=\frac{1}{2}s=21​ while allowing genuinely quantum local dynamics for higher spins. It clarifies the variational and geometric origin of the inequivalence between quantum Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics, provides a criterion for their equivalence under a common time rescaling, establishes connections with the classical theory through two qualitatively distinct limits (s→∞s\to\inftys→∞ and a large number of quantum spins NNN), and allows realistic O(N)\mathcal{O}(N)O(N) simulations of quantum spin dynamics in materials.
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.