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arXiv:2610.05521

Nonlinear quantum multi-spin dynamics without entanglement

Ivan P. Miranda, Erik Sjöqvist

quant-phcond-mat.mes-hallcond-mat.str-elSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
v1 · Oct 4, 2026v2 · Oct 18, 2026
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Abstract

We introduce a distinct route for extending classical atomistic spin dynamics into the quantum regime for arbitrary spin $s$, 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 $s\geq 1$ density operators, and nonlinear dynamics arise from both the mean-field interdependence of the local states and the state-dependent dissipative terms. This framework 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 ($s\to\infty$ and a large number of quantum spins $N$), and allows realistic $\mathcal{O}(N)$ simulations of quantum spin dynamics in materials.

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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, and nonlinear dynamics arise from both the mean-field interdependence of the local states and the state-dependent dissipative terms. This framework 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) 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 equation describes the time evolution of magnetic dipoles and can be derived by taking the classical limit of a quantum mechanical spin Hamiltonian. To take this limit, one constrains the many-body quantum state to a tensor product of coherent states, thereby neglecting entanglement between sites. Expectation values of the quantum spin operators produce the usual classical spin dipoles. One may also consider expectation values of polynomials of the spin operators, leading to quadrupole and higher-order spin moments, which satisfy a dynamical equation of motion that generalizes the Landau-Lifshitz dynamics [Zhang and Batista, Phys. Rev. B 104, 104409 (2021)]. Here we reformulate the dynamics of these ${N}^{2}\ensuremath{-}1$ generalized spin components as a mean-field Schr\"odinger equation on the NNN-dimensional coherent state. This viewpoint suggests efficient integration methods that respect the local symplectic structure of the classical spin dynamics.
Classical nonlinear theories are highly successful in describing far-from-equilibrium dynamics of magnets, encompassing phenomena such as parametric resonance, ultrafast switching, and even chaos. However, at ultrashort length and time scales, where quantum correlations become significant, these models inevitably break down. While numerous methods exist to simulate quantum many-body spin systems, they are often limited to near-equilibrium conditions, capture only short-time dynamics, or obscure the intuitive connection between nonlinear behavior and its geometric origin in the su(2) spin algebra. To advance nonlinear magnetism into the quantum regime, we develop a theory in which semiclassical spin correlations, rather than individual spins, serve as the fundamental dynamical variables. Defined on the bonds of a bipartite lattice, these correlations are inherently nonlocal, with dynamics following through a semiclassical mapping that preserves the original spin algebra. The resulting semiclassical theory captures nonlinear dynamics that are entirely nonclassical and naturally accommodates phenomenological damping at the level of correlations, which is typically challenging to include in quantum methods. As an application, we focus on Heisenberg antiferromagnets, which feature significant quantum effects. We predict nonlinear scaling of the mean frequency of quantum oscillations in the Néel state with the spin quantum number S. These have no classical analog and exhibit features reminiscent of nonlinear parametric resonance, fully confirmed by exact diagonalization. The predicted dynamical features are embedded in the geometric structure of the semiclassical phase space of spin correlations, making their physical origin much more transparent than in full quantum methods. With this, semiclassical spin-correlation dynamics provide a foundation for exploring nonlinear quantum magnetism.
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 examine the relation between the quantum Landau-Lifshitz equation (qqq-LL) [Phys. Rev. Lett. 110, 147201 (2013)] and quantum Landau-Lifshitz-Gilbert equation (qqq-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-12\frac{1}{2}21​ states. Here, we demonstrate that this equivalence breaks down even at the level of a single spin, provided s≥1s \geq 1s≥1. Focusing on a spin-1 particle in an anisotropic crystal field, we show that the qqq-LL and qqq-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.
Ground-state preparation of quantum many-body systems is a fundamental challenge across quantum science, with implications for quantum simulation, quantum computing, and the study of complex quantum matter. The exponential growth of Hilbert space and the complexity of entangled eigenstates motivate the development of new approaches that go beyond conventional strategies based on adiabatic evolution, variational optimization, imaginary-time methods, and engineered dissipative processes. Here, we demonstrate that the recently proposed quantum Landau-Lifshitz-Gilbert (QLLG) dynamics [Phys. Rev. Lett. {\bf 133}, 266704 (2024)] provides an intrinsic nonlinear dissipative mechanism for real-time ground-state preparation. Unlike engineered dissipative schemes, QLLG encodes relaxation directly into the quantum equation of motion while preserving the underlying quantum structure of the evolution. The QLLG evolution selectively suppresses excited-state contributions, driving the system toward the lowest-energy eigenstate within the accessible symmetry sector. We show that the dynamics admits a monotonic energy functional and a stable ground-state fixed point, and that for generic initial states in an NNN-qubit Hilbert space the characteristic convergence time scales linearly with NNN and inversely with the spectral gap. Numerical simulations of an interacting spin chain confirm our analytical predictions. These results establish nonlinear dissipative QLLG dynamics as a new avenue for quantum state preparation, opening a new route for quantum simulation, computation, and state engineering.