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Moscow University Physics Bulletin

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

Mariya Iv. Trukhanova, Pavel A. Andreev

Published Apr 1, 2024• 5 citations
Publisher

Abstract

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.

DOI
10.3103/s0027134924700255

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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.
Abstract We develop a novel model of the magnetized spin-1 Bose–Einstein condensate (BEC) of neutral atoms, using the method of many-particle quantum hydrodynamic (QHD) and propose an original derivation of the system of continual equations. We consider bosons with a spin-spin interaction and a short range interaction in the first order in the interaction radius, on the of basis of the self-consistent field approximation of the QHD equations. We demonstrate that the dynamics of the fluid velocity and magnetization is determined by a nontrivial modification of the Euler and Landau–Lifshitz equation, and show that a nontrivial modification of the spin density evolution equation contains the spin torque effect that arises from the self-interactions between spins of the bosons. The properties of the dispersion spectrum of collective excitations are described. We obtain the new contribution of the self-interaction of spins in the spin wave spectrum together with the influence of an external magnetic field and spin-spin interactions between polarized particles. The anisotropic spin wave instability is predicted.
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.
We develop the complete hydrodynamical description of the spin-1 partially polarized Bose–Einstein condensate (BEC), based on the many-particle quantum hydrodynamic formalism. We derive the continuity equation, momentum balance equation, spin evolution and nematicity evolution equations considering the spin–spin dipole long-range interactions and short-range interactions between partially spin-polarized atoms of condensate. The nematic or quadrupolar tensor plays an important role in the experiments of spin-nematic squeezing. We predict the influence of the spin–spin dipole interactions and spin torque force on the dynamics of nematic tensor. The collective modes for phonons and magnons are predicted for the ferromagnetic phase of spinor condensate, where the spin–spin and short-range interactions modify the dispersion equation.
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.
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The effect of spin exchange collisions on the internal variables (orientation, alignment, etc.) of colliding atoms is discussed, with particular emphasis on the role played by the Pauli principle. The calculation presented is valid for distinguishable or identical particles, and the initial introduction of an effective exchange hamiltonian for the spins is avoided. The one-atom density matrix of the internal variables is calculated after collision, in an operatorial form which enables the discussion of various physical effects. The first term in the density matrix evolution is a transfer term which does not originate from particle indistinguishability. The following terms are indeed introduced by the Pauli antisymmetrization principle applied to electrons and nuclei, and they can be described as electronic or nuclear apparent magnetic field (or electric gradient). In addition, terms arising from the combined exchange of electrons and nuclei are obtained, in particular from the exchange of identical whole atoms (bosons in the case of hydrogen). The evolution equations for the density operators are applied to a few particular cases in order to evaluate spin exchange effects in various situations : weak or strong hyperfine decoupling, Zeeman or hyperfine resonances, etc...