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arXiv:cond-mat/0502323

Classification of integrable clusters of classical Heisenberg spins

Marco Ameduri, Bogomil Gerganov, Richard A. Klemm

Published Feb 14, 2005• 2 citations
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Abstract

In this letter we present a general classification of integrable models of identical classical spins coupled via the isotropic Heisenberg Hamiltonian. Our constructive proof of integrability provides a solution scheme for the equations describing the time evolution of each spin. We show that for a broad class of Heisenberg couplings, it is possible to solve the equations of motion iteratively by building a hierarchy of spins, each of which precesses about the composite spin of the previous level. This leads to the identification of the frequencies of precession at each level.

DOI
10.48550/arxiv.cond-mat/0502323

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In this letter we present a general classification of integrable models of identical classical spins coupled via the isotropic Heisenberg Hamiltonian. Our constructive proof of integrability provides a solution scheme for the equations describing the time evolution of each spin. We show that for a broad class of Heisenberg couplings, it is possible to solve the equations of motion iteratively by building a hierarchy of spins, each of which precesses about the composite spin of the previous level. This leads to the identification of the frequencies of precession at each level.
Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.
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.
The integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher-order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower-order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...
The lowest Landau level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our paper provides a controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite N × N real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals. A type M family in our definition is formed by N – M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all type 1, 2 and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy-level crossings in type M integrable systems and analyze the taxonomy of types in the 1D Hubbard model.