Three routes to chaos in a Heisenberg chain of spins

Journal of Magnetism North-Holland

and Magnetic

Materials

104-107

B-d

M

(1992) 867-868

M z r

M

Three routes to chaos in a Heisenberg

chain of spins

Adam Jaroszewicz a and Andrzej Sukiennicki b ’ Imtituteof Physics, Polish Academy of Sciences, ALLotnikdw 32 /46, ’ Institute of Physics, Warsaw Technical Uniuersity, Kosqkowa Three routes to chaos in a chain of classical, damped intermittency, (ii) chaos from torus T2 via phase-locked

We considered

a damped,

anisotropic

chain

02-668 Warsaw, Poland 75, 00.662 Warsaw, Poland

and periodically driven spins were found. torus and (iii) chaos from torus T’.

T------

of N =

The driving oscillating magnetic field B(t) = B,,cos(w,,t) e, was applied. The motion of the system is described by the Landau-Lifshitz equations with the Landau damping term 100 classical

spins

as; - = S, x Bef, at

of constant

length.

;S;x (S, x Be,;)

(1)

and a Hamiltonian

-1

N-l Z=

-J

c

i=

S,S,+, 1

These were (i) chaos via

(a) 0

(2)

+i

2n

1=1

i=l

where Bef, = - tH/dSi is an effective magnetic field, A the damping coefficient, and J and K are ferromagnetic exchange and anisotropy constants, respectively. We impose periodic boundary conditions. We transformed eqs. (1) and (2) to the spherical coordinate

1)

-1 1

I

i (site)

(b) 100

Fig. 2. h = 1.96; (a) Poincart section of the two-dimensional torus 2T2, (b) spatial structure: profile cos 0, at one moment.

i (site)

1

100

11

0

(b) /

%

2n

Fig. 1. h = 0.68; (a) spatial structure: 50 profiles cos 8, overlayed (for SO periods), (b) PoincarC section of the two-dimensional torus T*. 0312-8853/92/$05.00

0 1992 - Elsevier

Science

Publishers

system {ei, @J, i = 1,. . , 100 and we introduced new dimensionless parameters E = J/K and h = B,,/K,. To observe the system’s dynamics we chose the method of PoincarC section: we observed the system stroboscopically with a time step equal to the period of the external field oscillations T = 27r/w,,. The numerical calculations were performed using the predictor-corrector method. To begin we mention only the dynamics of a single spin under the same conditions as we consider our system in this paper. It looks very simple. For 1.4 < h < 1.8, spin evolves quasiperiodically on a two-dimensional torus T* and for the field weaker or stronger it evolves periodically. Chaotic motion does not appear. For a chain of N = 100 spins the picture looks completely different: the complex system of attractors of different types is found. We observe three routes of the transition of the system from regular to chaotic behaviour as we change the value of the external field amplitude h.

B.V. All rights reserved

868

A. Jaroszewicz,

A. Sukiennicki

/ 3 routes to chaos in Heisenherg spin chuins

1

a) 0

1

+i

i

(site)

I

2rI

100

Fig. 3. /z = 1.95; (a) Poincark section of the three-dimensional torus T’, (b) spatial structure: profile cos 0, at one moment.

When we start with a chain magnetized homogeneously along the easy axis z (with a small noise added) we observe periodic motion as far as an amplitude h is small enough: an attractor of homogeneous spatial structure is found for h < 0.935 and the second one of modulated spatial structure for 0.935 < h < 0.97. For 0.97
appears and the system evolves periodically. For 0.68 1.97 a periodic attractor (limit cycle) of homogeneous spatial structure is observed. For 1.963 < h < 1.97 double-limit cycle and modulated spatial structure appears. For lower fields 1.955 < h < 1.963 we observe quasi-periodic motion on a two-dimensional double torus 2T2 (fig. 2a). WC find two incommensurate fundamental frequencies in a power spectrum wg and W, = O.O21w,,, while subharmonies wo/2 also appear. Fig. 2b presents the spatial structure of this attractor. For 1.94 < h < 1.955 we cnter another quasi-periodic regime: The system evolves on a three-dimensional torus T” (fig. 3a). The power spectrum consists of three incommensurate frequencies: wg, W, = O.OOSw,, and w2 = OSO7w,,. Fig. 3b prcsents modulated spatial structure of the attractor. For h < 1.94 the system evolves chaotically. The complex behaviour of the system described above is a result of its spatially extended structure, which can be seen when compared with the dynamics of a single spin.