Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.32405

Quantum Landau Lifshitz Gilbert equation in ferromagnetic conductors and magnetic insulators

Ji Yang Zhu, Zheng Chuan Wang

cond-mat.mes-hallquant-phSubmitted Oct 23, 2026• Announced Oct 26, 2026
arXivHTMLPDF

Abstract

Understanding the microscopic origin of magnetic damping is essential for describing magnetization dynamics in nanoscale magnetic systems. In this work, we develop a microscopic framework to derive Gilbert damping terms from specific interactions in ferromagnetic conductors and magnetic insulators. In ferromagnetic conductors, magnetic damping is dominated by the s d exchange interaction between the localized magnetic moments and the itinerant electrons, which is effectively equivalent to an electronmagnon interaction. From the spin diffusion equation and the spin transfer torque therein, we obtain the corresponding quantum LLG equation. In magnetic insulators, where free electrons are absent, spin phonon interactions become the primary source of the damping.Starting from an effective spin phonon interaction and using linear response theory, we derive a quantum LLG equation with a Gilbert damping tensor analytically. Our results demonstrate that different microscopic interactions lead to qualitatively different structures of magnetic damping. Instead of being described by a universal scalar parameter, Gilbert damping can exhibit spatial dependence and tensorial characteristics determined by the underlying microscopic interactions.

AI Summary

The paper addresses how microscopic interactions determine magnetic damping in ferromagnetic conductors and magnetic insulators, and how these mechanisms enter a quantum Landau–Lifshitz–Gilbert (LLG) equation. It considers whether the damping can be represented by a scalar Gilbert parameter or instead requires spatially dependent or tensorial descriptions. For ferromagnetic conductors, the proposed framework treats the $s$–$d$ exchange interaction between localized magnetic moments and itinerant electrons as the dominant damping mechanism, expressed effectively as an electron–magnon interaction. A quantum LLG equation is obtained from the spin-diffusion equation and its spin-transfer-torque contribution. For magnetic insulators, the framework instead takes spin–phonon coupling as the primary damping mechanism. Starting from an effective spin–phonon interaction, linear-response theory is used to derive analytically a quantum LLG equation containing a Gilbert damping tensor. The stated result is that the structure of damping depends on the microscopic interaction: it can have spatial dependence and tensorial character rather than being captured by a universal scalar parameter. Only the abstract is available, so the explicit damping kernels, parameter dependences, assumptions underlying the effective interactions, and regimes of validity cannot be assessed. The abstract gives no numerical predictions or bounds, experimental comparisons, or explicit limitations, and does not specify how the resulting equations relate quantitatively to existing microscopic damping theories.

Little sign of AI-generated textUnclear noveltyMissing comparison

Written by gpt-6.1-sol from the abstract only. Automated: it can be wrong.

Comments
2 figures

Related papers

From Quantum to Classical Dissipative Spin Dynamics

Alireza Ataei, Leila Ataei, Olle Eriksson +1 more

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.

quant-pharXiv:2609.35112•Sep 29, 2026

Quantum Analog of Landau-Lifshitz-Gilbert Dynamics

Yuefei Liu, I. P. Miranda, Lee Johnson +7 more

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.

arXiv:2403.09255•Physical Review Letters•Dec 30, 2024•8 citations

Numerical solution of quantum Landau-Lifshitz-Gilbert equation

Vahid Azimi Mousolou, Davoud Mirzaei

The classical Landau–Lifshitz–Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems, yet its classical nature limits its applicability to inherently quantum phenomena such as entanglement and nonlocal correlations. Inspired by the need to incorporate quantum effects into spin dynamics, recently a quantum generalization of the LLG equation is proposed [Phys. Rev. Lett. 133, 266704 (2024)] which captures essential quantum behavior in many-body systems. In this work, we develop a robust numerical methodology tailored to this quantum LLG framework that not only handles the complexity of quantum many-body systems but also preserves the intrinsic mathematical structures and physical properties dictated by the equation. We apply the proposed method to a class of quantum systems with a moderate number of spins that host host topological states of matter, and demonstrate rich quantum behavior, including the emergence of long-time entangled states. This approach opens a pathway toward reliable simulations of quantum magnetism beyond classical approximations, potentially leading to new discoveries.

arXiv:2506.19594•Computer Physics Communications•Oct 28, 2025•1 citation

Fate of entanglement in magnetism under Lindbladian or non-Markovian dynamics and conditions for their transition to Landau-Lifshitz-Gilbert classical dynamics

Federico Garcia-Gaitan, Branislav K. Nikolić

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 $S=1/2, S=1$, or $S=5/2$ 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---$S=1/2, S=1$, and $S=5/2$ ferromagnets or $S=5/2$ antiferromagnets---except for $S=1/2$ and $S=1$ antiferromagnets. Finally, we investigate the stability of an entangled antiferromagnetic GS upon suddenly coupling it to bosonic baths.

arXiv:2303.17596•Physical review. B./Physical review. B•May 16, 2024•12 citations

Ferroelectrical Switching as a Probe of Quantum Damping in Magnetic Spin Systems

Yuefei Liu, Anna Delin, Olle Eriksson +3 more

While damped spin dynamics is important for the understanding of magnetic materials, clear signatures of \emph{quantum corrections} to the Gilbert damping mechanism remain elusive. We propose a route to distinguish quantum and classical Gilbert spin damping using ferroelectric control of a magnetic dimer. Ab initio calculations for dimers on ferroelectric substrates show that polarization reversal switches the inter-spin exchange between ferromagnetic and antiferromagnetic regimes. We formulate a magnetization-based diagnostic that relates magnetization traces to entanglement dynamics, which enables ferroelectrical on/off control of dimer entanglement. Material-informed quantum Landau-Lifshitz-Gilbert simulations illustrate how the signature of magnetization dynamics can, in principle, be used to infer the existence of quantum Gilbert spin damping. This minimal and non-volatile platform connects first-principles modeling to experimentally accessible observables and provides a starting point for voltage-controlled quantum entanglement in magnetic spin networks.

quant-pharXiv:2606.07378•Jun 8, 2026

A New Microscopic Representation of Spin Dynamics in Quantum Systems with Biquadratic Exchange Interactions

Mariya Iv. Trukhanova, Pavel A. Andreev

There is a version of the Landau–Lifshitz equation that takes into account the Coulomb exchange interactions between atoms, expressed by the term $${\sim}\mathbf{s}\times\triangle\mathbf{s}$$ . On the other hand, ions in magnetic materials have several valence electrons on the $$d$$ -shell, and, therefore, the Hamiltonian of many-electron atoms with spins $$S>1$$ should include a biquadratic exchange interaction. We propose a novel fundamental microscopic derivation of the spin density evolution equation with an explicit form of biquadratic exchange interaction using the method of many-particle quantum hydrodynamics. The equation for the spin density evolution is obtained from the many-particle Schrödinger–Pauli equation and contains the contributions of the usual Coulomb exchange interaction and the biquadratic exchange. Furthermore, the derived biquadratic exchange torque in the spin density evolution equation is proportional to the nematic tensor for the medium of atoms with spin $$S=1$$ . Our method may be very attractive for further studies of the magnetoelectric effect in multiferroics.

Moscow University Physics Bulletin•Apr 1, 2024•5 citations
Understanding the microscopic origin of magnetic damping is essential for describing magnetization dynamics in nanoscale magnetic systems. In this work, we develop a microscopic framework to derive Gilbert damping terms from specific interactions in ferromagnetic conductors and magnetic insulators. In ferromagnetic conductors, magnetic damping is dominated by the s d exchange interaction between the localized magnetic moments and the itinerant electrons, which is effectively equivalent to an electronmagnon interaction. From the spin diffusion equation and the spin transfer torque therein, we obtain the corresponding quantum LLG equation. In magnetic insulators, where free electrons are absent, spin phonon interactions become the primary source of the damping.Starting from an effective spin phonon interaction and using linear response theory, we derive a quantum LLG equation with a Gilbert damping tensor analytically. Our results demonstrate that different microscopic interactions lead to qualitatively different structures of magnetic damping. Instead of being described by a universal scalar parameter, Gilbert damping can exhibit spatial dependence and tensorial characteristics determined by the underlying microscopic interactions.
The paper addresses how microscopic interactions determine magnetic damping in ferromagnetic conductors and magnetic insulators, and how these mechanisms enter a quantum Landau–Lifshitz–Gilbert (LLG) equation. It considers whether the damping can be represented by a scalar Gilbert parameter or instead requires spatially dependent or tensorial descriptions.
For ferromagnetic conductors, the proposed framework treats the sss–ddd exchange interaction between localized magnetic moments and itinerant electrons as the dominant damping mechanism, expressed effectively as an electron–magnon interaction. A quantum LLG equation is obtained from the spin-diffusion equation and its spin-transfer-torque contribution. For magnetic insulators, the framework instead takes spin–phonon coupling as the primary damping mechanism. Starting from an effective spin–phonon interaction, linear-response theory is used to derive analytically a quantum LLG equation containing a Gilbert damping tensor.
The stated result is that the structure of damping depends on the microscopic interaction: it can have spatial dependence and tensorial character rather than being captured by a universal scalar parameter. Only the abstract is available, so the explicit damping kernels, parameter dependences, assumptions underlying the effective interactions, and regimes of validity cannot be assessed. The abstract gives no numerical predictions or bounds, experimental comparisons, or explicit limitations, and does not specify how the resulting equations relate quantitatively to existing microscopic damping theories.
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 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.
The classical Landau–Lifshitz–Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems, yet its classical nature limits its applicability to inherently quantum phenomena such as entanglement and nonlocal correlations. Inspired by the need to incorporate quantum effects into spin dynamics, recently a quantum generalization of the LLG equation is proposed [Phys. Rev. Lett. 133, 266704 (2024)] which captures essential quantum behavior in many-body systems. In this work, we develop a robust numerical methodology tailored to this quantum LLG framework that not only handles the complexity of quantum many-body systems but also preserves the intrinsic mathematical structures and physical properties dictated by the equation. We apply the proposed method to a class of quantum systems with a moderate number of spins that host host topological states of matter, and demonstrate rich quantum behavior, including the emergence of long-time entangled states. This approach opens a pathway toward reliable simulations of quantum magnetism beyond classical approximations, potentially leading to new discoveries.
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.
While damped spin dynamics is important for the understanding of magnetic materials, clear signatures of \emph{quantum corrections} to the Gilbert damping mechanism remain elusive. We propose a route to distinguish quantum and classical Gilbert spin damping using ferroelectric control of a magnetic dimer. Ab initio calculations for dimers on ferroelectric substrates show that polarization reversal switches the inter-spin exchange between ferromagnetic and antiferromagnetic regimes. We formulate a magnetization-based diagnostic that relates magnetization traces to entanglement dynamics, which enables ferroelectrical on/off control of dimer entanglement. Material-informed quantum Landau-Lifshitz-Gilbert simulations illustrate how the signature of magnetization dynamics can, in principle, be used to infer the existence of quantum Gilbert spin damping. This minimal and non-volatile platform connects first-principles modeling to experimentally accessible observables and provides a starting point for voltage-controlled quantum entanglement in magnetic spin networks.
There is a version of the Landau–Lifshitz equation that takes into account the Coulomb exchange interactions between atoms, expressed by the term ∼s×△s{\sim}\mathbf{s}\times\triangle\mathbf{s}∼s×△s . On the other hand, ions in magnetic materials have several valence electrons on the ddd -shell, and, therefore, the Hamiltonian of many-electron atoms with spins S>1S>1S>1 should include a biquadratic exchange interaction. We propose a novel fundamental microscopic derivation of the spin density evolution equation with an explicit form of biquadratic exchange interaction using the method of many-particle quantum hydrodynamics. The equation for the spin density evolution is obtained from the many-particle Schrödinger–Pauli equation and contains the contributions of the usual Coulomb exchange interaction and the biquadratic exchange. Furthermore, the derived biquadratic exchange torque in the spin density evolution equation is proportional to the nematic tensor for the medium of atoms with spin S=1S=1S=1 . Our method may be very attractive for further studies of the magnetoelectric effect in multiferroics.