Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

Kamppeter, T.; Mertens, Franz G.; Sánchez, Ángel; Bishop, A. R.; Domı́nguez-Adame, F.; Grønbech‐Jensen, Niels

doi:10.1007/s100510050653

Cited by 15 publications

(30 citation statements)

References 17 publications

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“…The cubic term in (24) straightforwardly derives from this integral. Similar superdiffusive behaviour was found in [32]. There the effects of noise on the position of a vortex in a two-dimensional easy-plane Heisenberg model were studied; the indirect effect of the noise on the position originated from an external force acting on the vortex.…”

Section: Stochastic Equations Of Motion Without Dampingsupporting

confidence: 59%

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Soliton diffusion on the classical, isotropic Heisenberg chain

2001

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Abstract. We investigate the diffusive motion of a solitary wave on a classical, isotropic, ferromagnetic Heisenberg spin chain with nearest-neighbour exchange interaction. The spins are coupled magnetically to Gaussian white noise and are subject to Gilbert damping. The noise induces a collective, stochastic time evolution of the solitary wave. Within a continuum version of the model we employ implicit collective variables to describe this stochastic behaviour. Thermally excited magnons are disregarded. We derive stochastic equations of motion for the collective variables and solve them numerically, in particular to obtain their variances as functions of time. These results are compared to data from spin dynamics simulations of a discrete chain. For some of the collective variables we find good agreement with respect to the long time behaviour, whereas for other variables the agreement is only qualitative; reasons for this are given. For shorter times we derive analytical expressions for the variances of the collective variables, which also agree well with spin dynamics.

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Section: Stochastic Equations Of Motion Without Dampingsupporting

confidence: 59%

“…In analogy to [33], along the lines of [26,32], we first take the vector product of (28) with ∂Sc ∂U k , then form the inner product of the result with S c and integrate over all space, obtaining…”

Section: Stochastic Equations Of Motion With Dampingmentioning

confidence: 99%

Soliton diffusion on the classical, isotropic Heisenberg chain

2001

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“…The systematic deviation of the theory for very long times (tտ2000) is due to a simplification, 21 namely keeping R 0 constant; in the simulation, R 0 slowly increases because of the damping. Results for other low temperatures compare equally well, the better the lower the temperature; for higher temperatures, our numerical estimates are less accurate, although qualitatively the results for both cases remain the same.…”

Section: Numerical Resultsmentioning

confidence: 97%

“…21 for the additive ͑thermal͒ noise apply to the multiplicative model presented here, specifically: ͑i͒ the existence of three different temperature regimes for the vortex propagation: a low temperature one, where the vortex motion follows essentially the third-order equation of motion with parameters independent of temperature; a middle temperature one, at which traces of the oscillations arising from the third-order equation are lost, and a high-temperature regime, which is not describable by a one-vortex approach because too many vortex-antivortex pairs arise in the system; and ͑ii͒ the dependence of the effective diffusion coefficient for the vortex on temperature. On the other hand, the experiments we have reported on here allow us to place on firmer ground that those are indeed the features of thermal vortex dynamics: The problem with the nonconstant spin length in a Langevin approach is now solved by the multiplicative approach, in which it is exactly conserved.…”

Section: Discussionmentioning

confidence: 99%

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Stochastic vortex dynamics in two-dimensional easy-plane ferromagnets: Multiplicative versus additive noise

et al. 1999

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We study how thermal fluctuations affect the dynamics of vortices in the two-dimensional classical, ferromagnetic, anisotropic Heisenberg model depending on their additive or multiplicative character. Using a collective coordinate theory, we analytically show that multiplicative noise, arising from fluctuations in the local-field term of the Landau-Lifshitz equations, and Langevin-like additive noise both have the same effect on vortex dynamics ͑within a very plausible assumption consistent with the collective coordinate approach͒. This is a nontrivial result, as multiplicative and additive noises usually modify the dynamics quite differently. We also carry out numerical simulations of both versions of the model finding that they indeed give rise to very similar vortex dynamics. ͓S0163-1829͑99͒03917-X͔

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Dynamics of Vortices in Two-Dimensional Magnets

Mertens¹,

Bishop²

Nonlinear Science at the Dawn of the 21st Century

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Theories, simulations and experiments on vortex dynamics in quasi-two-dimensional magnetic materials are reviewed. These materials can be modelled by the classical two-dimensional anisotropic Heisenberg model with XY (easy-plane) symmetry. There are two types of vortices, characterized by their polarization (a second topological charge in addition to the vorticity): Planar vortices have Newtonian dynamics (evenorder equations of motion) and exhibit strong discreteness effects, while non-planar vortices have non-Newtonian dynamics (odd-order equations of motion) and smooth trajectories. These results are obtained by a collective variable theory based on a generalized travelling wave ansatz which allows a dependence of the vortex shape on velocity, acceleration etc.. An alternative approach is also reviewed and compared, namely the coupling of the vortex motion to certain quasi-local spinwave modes. The influence of thermal fluctuations on single vortices is investigated. Different types of noise and damping are discussed and implemented into the microscopic equations which yields stochastic equations of motion for the vortices. The stochastic forces can be explicitly calculated and a vortex diffusion constant is defined. The solutions of the stochastic equations are compared with Langevin dynamics simulations. Moreover, noise-induced transitions between opposite polarizations of a vortex are investigated. For temperatures above the Kosterlitz-Thouless vortex-antivortex unbinding transition, a phenomenological theory, namely the vortex gas approach, yields central peaks in the dynamic form factors for the spin correlations. Such peaks are observed both in combined Monte Carlo-and Spin Dynamics-Simulations and in inelastic neutron scattering experiments. However, the assumption of ballistic vortex motion appears questionable.

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Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

Cited by 15 publications

References 17 publications

Soliton diffusion on the classical, isotropic Heisenberg chain

Soliton diffusion on the classical, isotropic Heisenberg chain

Stochastic vortex dynamics in two-dimensional easy-plane ferromagnets: Multiplicative versus additive noise

Dynamics of Vortices in Two-Dimensional Magnets

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