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arXiv:2106.14125 · Physical review. B./Physical review. B

Classical spin dynamics based on SU ( N ) coherent states

Hao Zhang, Cristian D. Batista

Published Sep 7, 2021• 39 citations
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Abstract

We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of $\mathrm{SU}(N)$, where $N$ is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to $\mathrm{SU}(N)$, provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including $S\ensuremath{\ge}1$ systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion $S=1$ model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.

DOI
10.1103/physrevb.104.104409

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We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of SU(N)\mathrm{SU}(N)SU(N), where NNN is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to SU(N)\mathrm{SU}(N)SU(N), provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including S\ensuremath≥1S\ensuremath{\ge}1S\ensuremath≥1 systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion S=1S=1S=1 model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.
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.
Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.
We introduce a phase-space representation for qubits and spin models. The technique uses an SU(n)\mathrm{SU}(n)SU(n) coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-PPP function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of SU(n)\mathrm{SU}(n)SU(n) phase space is used to describe nnn-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.
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.
Abstract Quantum magnets admit more than one classical limit and N-level systems with strong single-ion anisotropy are expected to be described by a classical approximation based on SU(N) coherent states. Here we test this hypothesis by modeling finite temperature inelastic neutron scattering (INS) data of the effective spin-one antiferromagnet Ba2FeSi2O7. The measured dynamic structure factor is calculated with a generalized Landau-Lifshitz dynamics for SU(3) spins. Unlike the traditional classical limit based on SU(2) coherent states, the results obtained with classical SU(3) spins are in good agreement with the measured temperature-dependent spectrum. The SU(3) approach developed here provides a general framework to understand the broad class of materials comprising weakly coupled antiferromagnetic dimers, trimers, or tetramers, and magnets with strong single-ion anisotropy.