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Physical review. B, Solid state

Quasiclassical Spin Dynamics for the Heisenberg Model

Yao-Chung Chang, George C. Summerfield, D. M. Kaplan

Published May 1, 1971• 9 citations
Publisher

Abstract

The Wigner method is employed to develop explicit relationships among quantum-mechanical and classical equations of motion and quantum-mechanical and classical pair-correlation functions of spin operators. In this method, quantum corrections to the classical theory can be accurately estimated. The formalism is applied to a spin-\textonehalf{} Heisenberg model at high temperatures. The most attractive feature of this method is that many of the results are intuitively satisfying. Finally, we make a connection between our results and the classical calculations made by several other authors.

DOI
10.1103/physrevb.3.3052

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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.

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The Wigner method is employed to develop explicit relationships among quantum-mechanical and classical equations of motion and quantum-mechanical and classical pair-correlation functions of spin operators. In this method, quantum corrections to the classical theory can be accurately estimated. The formalism is applied to a spin-\textonehalf{} Heisenberg model at high temperatures. The most attractive feature of this method is that many of the results are intuitively satisfying. Finally, we make a connection between our results and the classical calculations made by several other authors.
A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.
Abstract The Heisenberg spin chain is considered in ϕ 4 model approximation. Quantum corrections to classical solutions of the one-dimensional ϕ 4 model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.
We present a strategy for mapping the dynamics of a fermionic quantum system to a set of classical dynamical variables. The approach is based on imposing the correspondence relation between the commutator and the Poisson bracket, preserving Heisenberg's equation of motion for one-body operators. In order to accommodate the effect of two-body terms, we further impose quantization on the spin-dependent occupation numbers in the classical equations of motion, with a parameter that is determined self-consistently. Expectation values for observables are taken with respect to an initial quasiclassical distribution that respects the original quantization of the occupation numbers. The proposed classical map becomes complete under the evolution of quadratic Hamiltonians and is extended for all even order observables. We show that the map provides an accurate description of the dynamics for an interacting quantum impurity model in the coulomb blockade regime, at both low and high temperatures. The numerical results are aided by a novel importance sampling scheme that employs a reference system to reduce significantly the sampling effort required to converge the classical calculations.
Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where six{{s}_{i}}^{x}si​x, siy{{s}_{i}}^{y}si​y, and siz{{s}_{i}}^{z}si​z are components of the unit vector si{\mathbf{s}}_{i}si​, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions Jijx=Jijy=J{{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=JJij​x=Jij​y=J and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.
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.