Classical Heisenberg Spin Chain Research Articles

Invariant Measures for Integrable Spin Chains and an Integrable Discrete Nonlinear Schrödinger Equation

We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise under focusing cubic nonlinear Schrödinger equation. These continuum models are completely integrable and connected by the Hasimoto transform; correspondingly, we focus our attention on discretizations that are also completely integrable and also connected by a discrete Hasimoto transform. We consider these models on the infinite lattice ${\mathbb{Z}}$. Concretely, for a completely integrable variant of the classical Heisenberg spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin) we prove the existence and uniqueness of solutions for initial data following a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we prove invariance of a measure that we will show is the appropriate discrete analogue of white noise. We also include a thorough discussion of the Poisson geometry associated to the discrete Hasimoto transform introduced by Ishimori that connects the two models studied in this article.

Energy and spin diffusion in the one-dimensional classical Heisenberg spin chain at finite and infinite temperatures.

The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.

Parity-time symmetry breaking in spin chains

We investigate nonequilibrium phase transitions in classical Heisenberg spin chains associated with spontaneous breaking of parity-time ($\mathcal{PT}$) symmetry of the system under the action of Slonczewski spin-transfer torque (STT) modeled by an applied \textit{imaginary} magnetic field. We reveal the STT-driven $\mathcal{PT}$ symmetry-breaking phase transition between the regimes of precessional and exponentially damped spin dynamics and show that its several properties can be derived from the distribution of zeros of the system's partition function, the approach first introduced by Yang and Lee for studying equilibrium phase transitions in Ising spin chains. The physical interpretation of imaginary magnetic field as describing the action of non-conservative forces opens the possibility of direct observations of Lee-Yang zeros in nonequilibrium physical systems.

Excitations of breathers and rogue wave in the Heisenberg spin chain

We study the excitations of breathers and rogue wave in a classical Heisenberg spin chain with twist interaction, which is governed by a fourth-order integrable nonlinear Schrödinger equation. The dynamics of these waves have been extracted from an exact solution. In particular, the corresponding existence conditions based on the parameters of perturbation wave number K, magnon number N, background wave vector ks and amplitude c are presented explicitly. Furthermore, the characteristics of magnetic moment distribution corresponding to these nonlinear waves are also investigated in detail. Finally, we discussed the state transition of three types nonlinear localized waves under the different excitation conditions.

Energy repartition in the nonequilibrium steady state

The concept of temperature in nonequilibrium thermodynamics is an outstanding theoretical issue. We propose an energy repartition principle that leads to a spectral (mode-dependent) temperature in steady-state nonequilibrium systems. The general concepts are illustrated by analytic solutions of the classical Heisenberg spin chain connected to Langevin heat reservoirs with arbitrary temperature profiles. Gradients of external magnetic fields are shown to localize spin waves in a Wannier-Zeemann fashion, while magnon interactions renormalize the spectral temperature. Our generic results are applicable to other thermodynamic systems such as Newtonian liquids, elastic solids, and Josephson junctions.

Coming full circle: constructing a [Gd6] wheel dimer by dimer and the importance of spin topology.

The syntheses, structures, magnetic and thermodynamic properties of three related triethanolamine-based GdIII complexes are described. The smallest, a dimer ([Gd2]), can be viewed as the subunit from which the two larger complexes, a linear tetramer ([Gd2]2) and a cyclic hexamer ([Gd2]3), are composed by further deprotonation of the triethanolamine ligand. In all cases, nearest neighbour magnetic ions are weakly correlated by antiferromagnetic isotropic exchange, whose strength does not change significantly from one complex to another; J ranging from -0.10 to -0.13 cm-1. Therefore, rather than the strength of the coupling, it is the spin topology that is the dominant factor in determining the differences between the physical properties - specifically, the nuclearity and the transition from open (dimer and tetramer) to cyclic (hexamer) boundary conditions. Indeed the hexanuclear wheel reaches the continuum limit of classical Heisenberg spin chains. In terms of the magnetocaloric properties, the smaller the nuclearity, the larger the magnetic entropy and adiabatic temperature changes.

A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic.

Topological Monodromy of an Integrable Heisenberg Spin Chain

We investigate topological properties of a completely integrable system on $S^2\times S^2 \times S^2$ which was recently shown to have a Lagrangian fiber diffeomorphic to $\mathbb{R} P^3$ not displaceable by a Hamiltonian isotopy [Oakley J., Ph.D. Thesis, University of Georgia, 2014]. This system can be viewed as integrating the determinant, or alternatively, as integrating a classical Heisenberg spin chain. We show that the system has non-trivial topological monodromy and relate this to the geometric interpretation of its integrals.

Heisenberg model in pseudo-Euclidean spaces

We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system) on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light--like cone leads to a new example of integrable discrete contact system.

Thermal rectification and negative differential thermal resistance in a driven two segment classical Heisenberg chain

Using computer simulation we investigate thermal transport in a two segment classical Heisenberg spin chain with nearest neighbor interaction and in the presence of an external magnetic field. The system is thermally driven by heat baths attached at the two ends and transport properties are studied using energy conserving dynamics. We demonstrate that by properly tuning the parameters thermal rectification can be achieved—the system behaves as a good conductor of heat along one direction but becomes a bad conductor when the thermal gradient is reversed, and crucially depends on nonlinearity and spatial asymmetry. Moreover, suitable tuning of the system parameters gives rise to the counterintuitive and technologically important feature known as negative differential thermal resistance (NDTR). We find that the crucial factor responsible for the emergence of NDTR is a suitable mechanism for impeding the current in the bulk of the system.

Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation

In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.

A τ-function on the inhomogeneous lattice and the classical isotropic Heisenberg spin chain

A τ-function on the inhomogeneous lattice is introduced by a deformation of that of the Toda lattice hierarchy. An associated discrete-time integrable system is constructed in Sato and Lax formalisms. It is shown that a one-dimensional reduction of the system gives a Bäcklund transformation of the classical isotropic Heisenberg spin chain, which is equivalent to the RII-chain proposed by Spiridonov and Zhedanov.

Partition functions of classical Heisenberg spin chains with arbitrary and different exchange

The classical Heisenberg model has been effective in modelling exchange interactions in molecular magnets. In this model, the partition function is important as it allows the calculation of the magnetization and susceptibility. For an ensemble of N-spin sites, this typically involves integrals in 2N dimensions. Here, for two-, three- and four-spin nearest neighbour open linear Heisenberg chains these integrals are reduced to sums of known functions, using a result due to Gegenbauer. For the case of the three- and four-spin chains, the sums are equivalent in form to the results of Joyce. The general result for an N-spin chain is also obtained.

Continuous and Discrete (Classical) Heisenberg Spin Chain Revised

Most of the work done in the past on the integrability structure of the Classical Heisenberg Chain (CHSC) has been devoted to studying the su(2) case, both at the continuous and at the discrete level. In this paper we address the problem of constructing integrable generalized Spin Chains models, where the relevant field variable is represented by aN◊N matrix whose eigenvalues are theN th roots of unity. To the best of our knowledge, such an extension has never been systematically pursued. In this paper, at first we obtain the continuous N ◊ N generalization of the CHSC through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit some explicit, and hopefully interesting, examples for 3 ◊ 3 and 4 ◊ 4 matrices; then, we discuss the much more difficult discrete case, where a few partial new results are derived and a conjecture is made for the general case.

Periodic weak solutions for a classical one-dimensional isotropic biquadratic Heisenberg spin chain

In this paper, we consider the model proposed by Lakshmanan, Porsezian and Daniel [M. Lakshmanan, K. Porsezian, M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A 133 (1988) 483–488] describing the continuum isotropic biquadratic Heisenberg spin chain in one dimensional case. Following the results of integrability in [K. Porsezian, M. Daniel, M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic Heisenberg spin chain, J. Math. Phys. 33 (5) (1992) 1807–1816], we prove the existence of periodic weak solutions to this model.

The Cauchy problem for the hyperbolic–elliptic Ishimori system and Schrödinger maps**Any opinions, findings and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

We show an improved local in time existence and uniqueness result for Schrödinger maps and for the hyperbolic–elliptic nonlinear system proposed by Ishimori in analogy with the two-dimensional classical continuous isotropic Heisenberg spin (2d-CCIHS) chain. The proof uses fairly standard gauge geometric tools and energy estimates in combination with Kenig's version of the Koch–Tzvetkov method, to obtain a priori estimates for classical solutions to certain dispersive equations.

Integrability and soliton in a classical one-dimensional site-dependent biquadratic Heisenberg spin chain and the effect of nonlinear inhomogeneity

The integrability of one-dimensional classical continuum inhomogeneous biquadratic Heisenberg spin chain and the effect of nonlinear inhomogeneity on the soliton of an underlying completely integrable spin model are studied. The dynamics of the spin system is expressed in terms of a higher order generalized nonlinear Schrodinger equation through a differential geometric approach which becomes integrable for a particular choice of the biquadratic exchange interaction and for linear inhomogeneity. The effect of nonlinear inhomogeneity on the spin soliton is studied by carrying out a multiple scale perturbation analysis.

Dynamics of twists on antiferromagnetic spin chains: Theory

The equation of motion of twists on classical antiferromagnetic Heisenberg spin chains are derived. It is shown that twists interact via position- and velocity-dependent long-range two-body forces. A quiescent regime is identified wherein the system conserves momentum. With increasing kinetic energy the system exits this regime and momentum conservation is violated due to walls annihilation. A bitwist system is shown to be integrable and its exact solution is analysed. Many-twist systems are discussed and novel periodic twist lattice solutions are found on closed chains. The stability of these solutions is discussed.

Two-time Green function technique in Tsallis’ classical statistical mechanics

The two-time Green function technique is formulated in nonextensive classical statistical mechanics within the optimal Lagrange multiplier framework. We introduce also the nonextensive classical version of the spectral density method, for a direct calculation of the Green functions, and its effectiveness is tested exploring the equilibrium properties of the classical ferromagnetic Heisenberg spin chain.

Discretizations of the Landau-Lifshits equation

The relation between the Sklyanin chain and the Backlund transformations for the Landau-Lifshits equation is established. The stationary solutions of the chain determine an integrable mapping, which is a kind of classical Heisenberg spin chain. Some multifield generalizations are found.