arxiv: 2604.09900 · v1 · submitted 2026-04-10 · 🪐 quant-ph

Recognition: unknown

Inequivalence of Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics for a single quantum spin

Yuefei Liu , Olle Eriksson , Erik Sj\"oqvist

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:55 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum spin dynamicsLandau-Lifshitz equationLandau-Lifshitz-Gilbert equationmaster equationsspin-1 particleanisotropic crystal fielddissipationquantum magnetism

0 comments

The pith

Quantum extensions of the Landau-Lifshitz and Landau-Lifshitz-Gilbert equations produce inequivalent time evolution for a single spin-1 particle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether two quantum versions of classical spin dynamics equations remain equivalent in the quantum regime. Classically the Landau-Lifshitz and Landau-Lifshitz-Gilbert equations give identical trajectories up to a rescaling of time for any number of spins. In the quantum case this equivalence survives only for pure states and for arbitrary states of spin-1/2 particles. For a single spin-1 particle in an anisotropic crystal field the two equations generate distinct paths, which the authors measure with temporal rescaling misfits. The result matters because it shows that the modeling choice changes the predicted dissipation even in the simplest quantum spin systems.

Core claim

The q-LL and q-LLG equations generate inequivalent time evolution for a spin-1 particle in an anisotropic crystal field. Although the resulting trajectories remain qualitatively similar for this system, the two equations encode distinct dissipation mechanisms, as shown by the introduction of temporal rescaling misfits that measure the deviation from perfect equivalence under time rescaling.

What carries the argument

Temporal rescaling misfits that quantify the mismatch in time evolution produced by the two equations for the same initial quantum state.

Load-bearing premise

The chosen mathematical forms of the quantum Landau-Lifshitz and quantum Landau-Lifshitz-Gilbert equations correctly extend the classical dynamics to the quantum domain.

What would settle it

Numerical integration of the density-matrix evolution under both equations for the spin-1 system in an anisotropic field, followed by checking whether an optimal time rescaling makes the two state trajectories identical or leaves a nonzero residual difference.

Figures

Figures reproduced from arXiv: 2604.09900 by Erik Sj\"oqvist, Olle Eriksson, Yuefei Liu.

**Figure 2.** Figure 2: FIG. 2. (Color online.) The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (Color online.) The total variance [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online.) The temporal rescaling misfits of (a) spin [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Color online.) The temporal rescaling misfits of spin com [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (a) The [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

read the original abstract

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-$\frac{1}{2}$ states. Here, we demonstrate that this equivalence breaks down even at the level of a single spin, provided $s \geq 1$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows q-LL and q-LLG are inequivalent for single spins with s >= 1, with a concrete misfit measure and numerics for the spin-1 case.

read the letter

The key point is that these two quantum spin dynamics equations, which are equivalent classically and for s=1/2, stop being equivalent once you go to a single spin with s >=1. The authors demonstrate this for a spin-1 particle in an anisotropic crystal field by defining temporal rescaling misfits and showing they stay nonzero under numerical evolution. They also confirm the equivalence cases hold as stated. That gives a clear, falsifiable distinction between the dissipation mechanisms in the 2013 q-LL and 2024 q-LLG forms. The numerics and the misfit definition are the parts that actually do new work here. The trajectories remain qualitatively close, so the difference is not catastrophic for every application, but it is measurable. The limitation is the single-spin restriction. Most atomistic simulations involve coupled spins, and the paper does not check whether the inequivalence survives or averages out in that setting. The specific equation forms are taken from the cited papers without re-deriving them, so the result stands or falls with those choices. Within those bounds the evidence looks direct and free of obvious gaps or circular fitting. This is mainly for people already running quantum spin simulations or choosing dissipation models for s>1/2 systems. It flags a modeling decision that can matter. I would bring it to a reading group to talk through the misfit construction and whether it generalizes. It deserves peer review because the central claim is testable and they supply the supporting equations and data.

Referee Report

0 major / 1 minor

Summary. The manuscript claims that the quantum Landau-Lifshitz (q-LL) and quantum Landau-Lifshitz-Gilbert (q-LLG) equations are inequivalent for single spins with s ≥ 1, in contrast to their classical counterparts which are equivalent up to time rescaling. This is shown for a spin-1 particle in an anisotropic crystal field by introducing temporal rescaling misfits that quantify the difference in time evolution, while equivalence is preserved for pure states and s = 1/2.

Significance. If valid, this result reveals fundamental differences in the dissipation mechanisms of the two quantum master equations even for a single spin, with implications for atomistic spin dynamics in quantum materials. The provision of explicit numerical evidence for the spin-1 case and the definition of misfits strengthens the claim, offering a concrete way to test the inequivalence.

minor comments (1)

[Abstract] Abstract: the statement about equivalence for pure states could be expanded with a reference to the relevant section or equation where this is shown.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the accurate summary of our claims regarding the inequivalence of q-LL and q-LLG dynamics for single spins with s ≥ 1, the recognition of the role of temporal rescaling misfits, and the recommendation for minor revision. We appreciate the constructive feedback on the significance of the results for quantum atomistic spin dynamics.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper imports the q-LL master equation from the 2013 PRL reference and the q-LLG equation from the 2024 PRL reference as external definitions, then performs a direct numerical comparison of their generated trajectories for a spin-1 system in an anisotropic crystal field. Inequivalence is quantified via explicitly defined temporal rescaling misfits, with no parameter fitting, no redefinition of inputs as outputs, and no reliance on a uniqueness theorem or ansatz from the authors' own prior work. The stated equivalence for pure states and all s=1/2 cases is presented as a known property of the imported equations rather than a derived claim internal to this manuscript. The derivation chain consists of solving the two independent differential equations and measuring their difference on an external test system, rendering the result self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the prior definitions of the q-LL and q-LLG master equations from the two cited papers and on the choice of the spin-1 anisotropic-field Hamiltonian as the test case; no new free parameters or entities are introduced.

axioms (1)

domain assumption q-LL and q-LLG are purity-preserving nonlinear master equations that extend classical atomistic spin dynamics to the quantum regime
Invoked in the abstract as the foundation for comparing the two dynamics.

pith-pipeline@v0.9.0 · 5541 in / 1128 out tokens · 46869 ms · 2026-05-10T16:55:01.394269+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Inequivalence of Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics for a single quantum spin

equation. Notably, the LL and LLG equations are equiv- alent, i.e., give identical spin trajectories up to a rescaling of the time parameter, for any number of spins [5]. More recently, quantum analogs of these equations have been explored. The quantum Landau-Lifshitz (q-LL) equa- tion, introduced in Ref. [6], is formulated as a non-linear mas- ter equati...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

V . P. Antropov, M. I. Katsnelson, B. N. Harmon, M. Van Schilf- gaarde, and D. Kusnezov, Spin dynamics in magnets: Equation of motion and finite temperature effects, Phys. Rev. B54, 1019 (1996)

1996
[3]

Eriksson, A

O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,Atom- istic Spin Dynamics: Foundations and Applications, (Oxford University Press, Oxford, UK, 2017)

2017
[4]

L. D. Landau and E. M. Lifshitz, On the theory of the disper- sion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion8, 153 (1935)

1935
[5]

T. L. Gilbert, A phenomenological theory of damping in ferro- magnetic materials, IEEE Trans. Mag.40, 3443 (2004)

2004
[6]

Lakshmanan and K

M. Lakshmanan and K. Nakamura, Landau-Lifshitz Equation of Ferromagnetism: Exact Treatment of the Gilbert Damping, Phys. Rev. Lett.53, 2497 (1984)

1984
[7]

Wieser, Comparison of Quantum and Classical Relaxation in Spin Dynamics, Phys

R. Wieser, Comparison of Quantum and Classical Relaxation in Spin Dynamics, Phys. Rev. Lett.110, 147201 (2013)

2013
[8]

Y . Liu, I. P. Miranda, L. Johnson, A. Bergman, A. Delin, D. Thonig, M. Pereiro, O. Eriksson, V . Azimi-Mousolou, and E. Sj¨oqvist, Quantum Analog of Landau-Lifshitz-Gilbert Dynam- ics, Phys. Rev. Lett.133, 266704 (2024)

2024
[9]

Pivetta, F

M. Pivetta, F. Patthey, I. Di Marco, A. Subramonian, O. Eriks- son, S. Rusponi, and H. Brune, Measuring the Intra-Atomic Exchange Energy in Rare-Earth Adatoms, Phys. Rev. X10, 031054 (2020)

2020
[10]

Jelezko, T

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, Observation of Coherent Oscillations in a Single Electron Spin, Phys. Rev. Lett.92, 076401 (2004)

2004
[11]

To see this, we combine ϱt ≥0⇒ ⟨ϱ t⟩= 1 3 (1+⟨m t ·S⟩)≥0 7 and⟨m t ·S⟩ ≥ − |mt|
[12]

Thus, a spin-types=1 state can never be pure

By using the identity Tr S jS k = s(s+1)(2s+1) 3 s=1 δ jk =2δ jk and the inequality |mt| ≤1, we obtain Trϱ2 t = 1 3 + 2 9 |mt|2 ≤ 5 9 . Thus, a spin-types=1 state can never be pure. One can show that this result is true for alls≥1
[13]

R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A: Math. Theor.41, 235303 (2008)

2008
[14]

Arvind, K. S. Mallesh, and N. Mukunda, A generalized Pan- charatnam geometric phase formula for three-level quantum systems, J. Phys. A: Math. Gen.30, 2417 (1997)

1997
[15]

H. M. Bharath, M. Boguslawski, M. Barrios, L. Xin, and M. S. Chapman, Exploring Non-Abelian Geometric Phases in Spin-1 Ultracold Atoms, Phys. Rev. Lett.123, 173202 (2019)

2019
[16]

Mirzaei, B

D. Mirzaei, B. Hashemi, and V . Azimi-Mousolou, LREI: A fast numerical solver for quantum Landau-Lifshitz equations, arXiv:2508.21200

work page arXiv
[17]

Anticoherent

J. Zimba, “Anticoherent” Spin States via the Majorana Repre- sentation, Electron. J. Theor. Phys.3, 143 (2006)

2006
[18]

Peres,Quantum Theory: Concepts and Methods(Kluwer, Dordrecht, 1995), pp

A. Peres,Quantum Theory: Concepts and Methods(Kluwer, Dordrecht, 1995), pp. 328–329

1995