Integrable and nonintegrable classical spin clusters

The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.     

Integrable vs. nonintegrable geodesic soliton behavior

We study confined solutions of certain evolutionary partial differential equations (PDE) in 1+1 space–time. The PDE we study are Lie–Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler–Poincaré equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from theL² norm of the velocity. These PDE possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call “pulsons”. We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. The results for head-on antisymmetric collisions of pulsons tend to be singularity formation. Numerical simulations of these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the same type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture this behavior occurs because the integrable two-pulson interactions dominate the dynamics on the invariant pulson manifold, and this dynamics dominates the PDE initial value problem for most choices of confined pulses and initial conditions of finite extent.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Percolation for low energy clusters and discrete symmetry breaking in classical spin systems

We consider classical lattice systems in two or more dimensions with general state space and with short-range interactions. It is shown that percolation is a general feature of these systems: If the temperature is sufficiently low, then almost surely with respect to some equilibrium state there is an infinite cluster of spins trying to form a ground state. For systems having several stable sets of symmetry-related ground states we show that at low temperatures spontaneous symmetry breaking occurs because in a two-dimensional subsystem there is a unique infinite cluster of this type.     

Integrable quantum systems and classical Lie algebras

We have obtained six new infinite series of trigonometric solutions to triangle equations (quantumR-matrices) associated with the nonexceptional simple Lie algebras:sl(N),sp(N),o(N). TheR-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). TheseR-matrices provide an exact integrability of anisotropic generalizations ofsl(N),sp(N),o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.     

Integrable classical systems in higher dimensions

A general method for the construction of the second constant of motion (up to second order) for higher-dimensional classical systems is carried out. Correspondingly, the first- and the second-order potential equations are obtained whose solutions can directly provide the integrable systems.     

Precession of classical spin and spin-orbit splitting

We present solutions of a fermion-boson model based on the supersymmetric (Susy) extension of theU(N) σ-models with the Wess-Zumino-Witten (WZW) term. We study some properties of these solutions. We point out that the obtained solutions are related to the components of the energy-momentum tensor of the purely bosonicU(N)σ-model and that some classes of these solutions are traceless.     

On a classical spin glass model

A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperatureT is lowered, the spin-glass phase is reached via an equilibrium phase transition atT=T _f . The spin-glass magnetization exhibits a distinctS-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions.

1. A mixed phase is found where spin-glass and ferromagnetism coexist.
2. The zero-field susceptibility has a flat plateau for 0≦T≦T _f and a Curie-Weiss behaviour forT>T _f .
3. At low temperatures the magnetic specific heat is linearly dependent on the temperature.

The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.     

Inequalities between different classical spin models

We apply correlation inequalities to show that in a multicomponent spin model some quantities (e.g. magnetisation) decrease from a one-component model to a two-component, or from two to n > 2.     

The classical limit ofn-vector spin models

It is proved that the free energy of a system ofn-dimensional spins with Kac type potential is equal, in the infinite range zero strength limit, to the free energy of the corresponding Curie-Weiss system in which every spin interacts equally with every other spin.     

Subdynamics and nonintegrable systems

Work on the application of Poincaré's theorem to large classical or quantum systems with a continuous spectrum is continued. In situations where it is applicable, Poincaré's theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant. In contrast, the theory of subdynamics as developed by the Brussels group permits the construction of a unique set of projectors , giving up the requirement of hermiticity which is replaced by “star-hermiticity”.

The theory of subdynamics is presented in a new self-contained way, starting from the commutation relation , where L_H is the Liouvillian. This presentation is far more direct, and avoids some of the lengthy discussions associated with previous presentations (based mainly on the resolvent of the Liouvillian).

Subdynamics appears to be of interest from many points of view. It generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed Hamiltonian. In contrast, degrees of freedom are lost when going to the spectral representation (e.g. in the Friedrichs model). Subdynamics permits us to solve the initial value problem associated with the Liouville equation retaining the “non-Markovian” contributions which appear in the standard presentation. Finally, it introduces a classification of large dynamical systems, classical or quantum, into integrable and nonintegrable ones. It is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems, such as the interaction of matter with light. The applications of this technique to these problems will be worked out in subsequent papers.     

Study of squeezing in spin clusters

The conditions under which spin squeezing occurs in an asymmetric chain of spins are discussed. The time evolution of the system is calculated for different initial conditions. The effects of the use of spin coherent states to model the initial condition are analyzed.     

Evolution of spin clusters from the spin glass region inAuFe

Zero field Mössbauer spectra (4.2–40 K) have been obtained onAuFe alloys containing 5 and 10 at %Fe in differing metallurgical states. Graphs of the reduced magnetic hyperfine field versus reduced temperature have been compared With Brillouin curves. This comparison shows that the heat treated 5 at %Fe sample is essentially a homogeneous spin glass whereas the behaviour of the heat treated 10 at %Fe sample and alloys in an as rolled (atomically clustered) state can be better described by the additional presence of magnetic clusters of spins.     

Magnetic properties of magnetoactive spin clusters

A simple model is proposed for describing magnetic properties of magnetoactive nanoclusters, which permits exact analytic solution. Exact expressions are obtained for thermodynamic characteristics of the model, which hold in the entire range of temperatures, magnetic fields, and interaction parameters. It is found that in the case of easy-axis anisotropy, the field dependence of magnetization of a nanocluster consisting of N particles with a spin of 1/2 has [N/2] fractional plateaus ([…] is the integer part) corresponding to polarized phases with ruptures singlet pairs. A nonmonotonic behavior observed for the magnetic susceptibility of an easy-plane cluster is typical of gap magnets. The spin gap between the ground state and excited states is proportional to the anisotropy parameter.     

Dual symmetry in classical and quantum spin theory

Apropos the continuing discussion of the observability of the magnetic charge (the Dirac monopole [1]), there is interest in investigating the dual symmetry of the spin equations of motion in classical and quantum theory. It is known that in dual symmetry theory the presence of a magnetic charge is a simple consequence of the dual invariance of the equations of electrodynamics [2], The matter of the observability of a magnetic charge is not tackled there, and becomes only experimentally verifiable [2, p. 64].Translated from Izvestiya Vysshikh Uchebnkh, Zavedenii, Fizika, No. 7, pp. 56–59, July, 1932.     

Activated scaling of classical and quantum spin glasses

A model for thermally activated dynamics in disordered systems shows that the linear and nonlinear susceptibility follows a generic exponential form with a "critical rounding," chi(1) proportional to chi(3) proportional to [T ln(t/tau(0)')/K](gamma/b phi) exp - [Tt(g)(phi b)ln(t/tau(0)'/K)](nu/b) (T=temperature, t=time, K=barrier constant, t(g) = 1 - T(SG)/T, and T(SG) = transition temperature; gamma>0 for chi(3) and <0 for chi(1)). This model, also valid in the presence of resonant tunneling states at energies K(0) < K [provided that K is replaced by K(0)+2T ln (1/Gamma(0)), where Gamma(0)(2) proportional, variant tunnel splitting of a spin S=1], is potentially applicable to a wide variety of systems opening the way for the study of thermally activated quantum phase transitions. The famous spin-glass system LiHo(x)Y(1-x) seems to follow this model.     

On clusters and critical dimensions in spin glasses

We discuss the problem of critical dimensions in the Sherrington-Kirkpatrick model of a spin glass, using the presence of clusters above T_c. It is argued that these clusters are ramified. This fact accounts for the different (upper) critical dimensions of the spherical and Ising models. The special role of dimension four is pointed out.     

Longitudinal magnetic excitations in classical spin systems

Using spin dynamics simulations we predict the splitting of the longitudinal spin-wave peak in all antiferromagnets with single site anisotropy into two peaks separated by twice the energy gap at the Brillouin zone center. This phenomenon has yet to be observed experimentally but can be easily investigated through neutron scattering experiments on MnF2 and FeF2. We have also determined that for all classical Heisenberg models the longitudinal propagative excitations are entirely multiple spin wave in nature.     

Vortex dynamics in classical non-Abelian spin models

We discuss the Abelian vortex dynamics in the Abelian projection approach to non-Abelian spin models. We show numerically that in the three-dimensional SU(2) spin model in the maximal Abelian projection the Abelian off-diagonal vortices, unlike the diagonal vortices, are not responsible for the phase transition. A generalization of the Abelian projection approach to SU(N) spin models is briefly discussed. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 8, 526–530 (25 April 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.