Classical Heisenberg Spin Research Articles

Finite quantum Heisenberg spin models and their approach to the classical limit

We determine the temperature range over which classical Heisenberg spin models closely reproduce the zero-field susceptibility of the corresponding quantum Heisenberg models for a finite number $N$ of interacting quantum spins $s$. Using mostly quantum and classical Monte Carlo methods, as well as analytical methods where applicable, we have explored a variety of geometries, including polygons, open chains, and all Platonic and several Archimedean polytopes. These systems range in size from $N=2$ to 120, and we have considered values of $s$ from $1∕2$ to 50 for both antiferromagnetic and ferromagnetic exchange. Particular attention is devoted to quantifying the slow convergence of the large $s$ quantum data to the limiting classical data. This is motivated by the desire to define conditions where classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.

Monte Carlo simulations of a ferromagnetic-FeF 2 system

In this work, we perform Monte Carlo simulations to study the magnetization reversal mechanism in ferromagnetic thin films on FeF 2. In particular, we emulate a bilayer AFM/FM structure, where the AFM interface corresponds to an uncompensated (1 0 0) plane. The magnetic moments are modeled by classical Heisenberg spin variables. Our analysis focus on the role of the exchange interaction J AF between the FM spins and the spins belonging to the AFM interface on the reversal mechanisms of the magnetization. By simulating hysteresis loops we study the effect of temperature on the bias field.

The effect of uniaxial stress to spin reorientation transition in magnetic thin-films: Monte Carlo investigation

Abstract In this work, we studied magnetic thin-films with thicknesses ranging from 2 to 4 layers via Monte Carlo simulations. The magnetic systems were modelled using anisotropic classical Heisenberg spins under the influence of mechanical uniaxial stresses. The study was performed with a spin-flip algorithm to investigate how the averaged magnetic properties, including their microscopic orientation, depend on temperatures, thicknesses and the magnitude of the mechanical uniaxial stress. From our results, it is found that the applied stress reduces the spin reorientation temperatures (from the out-of-plane to the in-plane direction) in a same way as increasing the number of thicknesses. The observations are in good agreement with related experiments.

Thermal conductivity of a classical one-dimensional Heisenberg spin model

We investigate the thermal conductivity of classical spin chains through a Green-Kubo approach using both Monte Carlo and Langevin simulations as well as by direct numerical determination of the microscopic heat transfer processes. All three approaches show that magnetic heat conductivity is normal for all finite temperatures and magnetic fields. In ferromagnets, heat conduction grows linearly with the magnetic field while in antiferromagnets this holds only at high fields. At low fields a quadratic decrease leading to a minimum is attributed to spin mode velocity suppression.

Peculiar `from-Edge-to-Interior' Spin Freezing in a Magnetic Dipolar Cube

By molecular dynamics simulation, we have investigated classical Heisenberg spins, which are arrayed on a finite simple cubic lattice and interact with each other only by the dipole-dipole interaction, and have found its peculiar it from-Edge-to-interior freezing process. As the temperature is decreased, spins on each edge predominantly start to freeze in a ferromagnetic alignment parallel to the edge around the corresponding bulk transition temperature, then from each edges grow domains with short-range orders similar to the corresponding bulk orders, and the system ends up with a unique multi-domain ground state at the lowest temperature. We interpret this freezing characteristics is attributed to the anisotropic and long-range nature of the dipole-dipole interaction combined with a finite-size effect.

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.

Competing Spin Phases in Geometrically Frustrated Magnetic Molecules

We identify a class of zero-dimensional classical and quantum Heisenberg spin systems exhibiting anomalous behavior in an external magnetic field B similar to that found for the geometrically frustrated kagome lattice of classical spins. Our calculations for the isotropic Heisenberg model show the emergence of a pronounced minimum in the differential susceptibility dM/dB at B(sat)/3 as the temperature T is raised from 0 K for structures based on corner-sharing triangles, specifically the octahedron, cuboctahedron, and icosidodecahedron. As the first experimental evidence we note that the giant Keplerate magnetic molecule {Mo(72)Fe(30)} (Fe(3+) ions on the 30 vertices of an icosidodecahedron) exhibits this behavior. For low T when B approximately B(sat)/3 two competing families of spin configurations exist of which one behaves magnetically "stiff" leading to a reduction of dM/dB.

Stochastic dynamics and the dynamic phase transition in thin ferromagnetic films

The dynamic phase behavior of a classical Heisenberg spin system with a bilinear exchange anisotropy in a planar thin film geometry has been investigated by Monte Carlo simulations using different forms for the stochastic dynamics. In simulations of the dynamic phase transition (DPT) in films subject to a pulsed oscillatory external field with competing surface fields, both Glauber and Metropolis dynamics show a continuous DPT. But while the field amplitude dependence of the DPT is similar in both cases, the transition region for the DPT as a function of temperature is more extended with Metropolis dynamics. The difference arises from a decoupling of the surface and bulk responses of the film near the dynamic phase transition with Metropolis dynamics that is not evident for Glauber dynamics.

Magnetisation switching in a ferromagnetic Heisenberg nanoparticle with uniaxial anisotropy: a Monte Carlo investigation

We investigate the thermal-activated magnetisation reversal in a single ferromagnetic nanoparticle with uniaxial anisotropy using Monte Carlo simulations. The aim of this work is to reproduce the reversal magnetisation by uniform rotation at very low temperature in the high-energy barrier hypothesis, that is to realize the Néel–Brown model. For this purpose we have considered a simple cubic nanoparticle where each site is occupied by a classical Heisenberg spin. The Hamiltonian is the sum of an exchange interaction term, a single-ion anisotropy term and a Zeeman interaction term. Our numerical data of the thermal variation of the switching field are compared to an approximated expression and previous experimental results on Co nanoparticles.

Exact time correlation functions forNclassical Heisenberg spins in the squashed equivalent neighbour model

We present exact integral representations of the time-dependent spin-spin correlation functions for the classical Heisenberg N-spin `squashed' equivalent neighbor model, in which one spin is coupled via the Heisenberg exchange interaction with strength $J_1$ to the other N-1 spins, each of which is coupled via the Heisenberg exchange coupling with strength $J_2$ to the remaining N-2 spins. At low temperature T we find that the N spins oscillate in four modes, one of which is a central peak for a semi-infinite range of the values of the exchange coupling ratio. For the N=4 case of four spins on a squashed tetrahedron, detailed numerical evaluations of these results are presented. As $T\to\infty$, we calculate exactly the long-time asymptotic behavior of the correlation functions for arbitrary N, and compare our results with those obtained for three spins on an isosceles triangle.

Phase transitions in ferro–antiferromagnetic bilayers with a stepped interface

We have studied magnetic ordering in ferro/antiferromagnetic (F/AF) bilayers using Monte Carlo simulations of classical Heisenberg spins. For both flat and stepped interfaces we observed order in the AF above the Neel temperature, with the AF spins aligning collinearly with the F moments. In the case of the stepped interface there is a transition from collinear to perpendicular alignment of the F and AF spins at a lower temperature.

Magnetic properties of helimagnetic–ferromagnetic superlattices: a Monte Carlo study

Using the classical Heisenberg spin model, we have investigated the magnetic properties of hexagonal compact (HCP) A/B superlattices where the materials A and B exhibit, respectively, a helimagnetic and a ferromagnetic phase at low temperature. In order to model the helimagnetic phase, the magnetic phase diagram at 0 K of the HCP structure has been determined for two exchange interaction ranges. Using Monte Carlo simulations, we have simulated the magnetisation thermal variation of such a superlattice as a function of the applied field. The corresponding thermal evolution of the magnetic structures has been analysed to explain the magnetisation curves. The effect of the A–A exchange interactions on the magnetisation thermal variation has also been investigated.

Exchange anisotropy and the dynamic phase transition in thin ferromagnetic Heisenberg films.

Monte Carlo simulations have been performed to investigate the dependence of the dynamic phase behavior on the bilinear exchange anisotropy of a classical Heisenberg spin system. The system under consideration is a planar thin ferromagnetic film with competing surface fields subject to a pulsed oscillatory external field. The results show that the films exhibit a single discontinuous dynamic phase transition (DPT) as a function of the anisotropy of the bilinear exchange interaction in the Hamiltonian. Furthermore, there is no evidence of stochastic resonance associated with the DPT. These results are in marked contrast to the continuous DPT observed in the same system as a function of temperature and applied field strength for a fixed bilinear exchange anisotropy.

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.

Classical ground states of symmetric Heisenberg spin systems

We investigate the ground states of classical Heisenberg spin systems which have point group symmetry. Examples are the regular polygons (spin rings) and the seven quasi-regular polyhedra including the five Platonic solids. For these examples, ground states with special properties, e.g. coplanarity or symmetry, can be completely enumerated using group-theoretical methods. For systems having coplanar (anti-) ground states with vanishing total spin we also calculate the smallest and largest energies of all states having a given total spin S. We find that these extremal energies depend quadratically on S and prove that, under certain assumptions, this happens only for systems with coplanar S = 0 ground states. For general systems the corresponding parabolas represent lower and upper bounds for the energy values. This provides strong support and clarifies the conditions for the so-called rotational band structure hypothesis which has been numerically established for many quantum spin systems.

Effect of interfacial coupling on the magnetic ordering in ferro-antiferromagnetic bilayers

Monte Carlo simulations have been used to study magnetic ordering in coupled anisotropic ferro/antiferromagnetic (FM/AFM) films of classical Heisenberg spins. We consider films with flat interfaces that are fully uncompensated as well as rough interfaces that are compensated on average. For both types of interfaces above the “Néel temperature” we observed order in the AFM with the AFM spins aligning collinearly with the FM moments. In the case of rough interfaces there is a transition from collinear to perpendicular alignment of the FM and AFM spins at a lower temperature.

Dynamic phase transitions in thin ferromagnetic films

Monte Carlo simulations have been used to investigate the dynamic phase behavior of a classical Heisenberg spin system with a bilinear exchange anisotropy \ensuremath{\Lambda} in a planar thin film geometry. Studies of the field amplitude, frequency, and temperature dependence show dynamic phase transitions in films subject to a pulsed oscillatory external field. Thin films with competing surface fields show separate and distinct dynamic phase transitions for the bulk and surface layers of the film. Between the two transitions, a mixed state with coexisting dynamically ordered and dynamically disordered phases is observed in the film. In contrast, the free film with no surface fields show a single dynamic phase transition as in a bulk system.

Exchange striction model for the spin configuration in the antiferromagnetic YMn 2 with the cubic Laves phase structure

By making use of the Luttinger–Tisza method for the classical spin Heisenberg model, we examine how the exchange striction affects the highly frustrated pyrochlore-type spin system in YMn 2 with the cubic Laves phase (C15) structure. The distance dependence of the exchange parameter between the nearest neighboring spin pair is taken into account. The complicated spin configuration observed in YMn 2 is deduced through energy change due to the exchange striction.

Nonlinear dynamics of the classical isotropic Heisenberg antiferromagnetic chain: The sigma model sector and the kink sector

We identify two distinct low-energy sectors in the classical isotropic antiferromagnetic Heisenberg spin-S chain. In the continuum limit, we show that two types of rotation generators arise for the field in each sector. Using these, the Lagrangian for sector I is shown to be that of the nonlinear sigma model. Sector II has a null Lagrangian; Its Hamiltonian density is just the Pontryagin term. Exact solutions are found in the form of magnons and precessing pulses in I and moving kinks in II. The kink has `spin' S. Sector I has a higher minimum energy than II.

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.