Size dependence of relaxation time in the fuzzy spin model

Journal of Magnetism and Magnetic Materials 104-1117 (1992) 1595-1596 North-Hoihmd i.

Size dependence of relaxation time in the fuzzy spin model T. Kawasaki and S. Miyashita Physics Department, College of Liberal Arts and Sciences, I@oto Unicersia.,, Kyoto 606, Japan Distribution of relaxation time in the fuzzy spin model (FSM) is studied when the model is quenched from an infinite temperature to a temperature below the ordering point. It is found that the time becomes longer proportionally to an exoonential of the linear size of the system L, namely rot exp(aL). Spin-glasses show extremely slow relaxation near and below the phase transition temperature, owing to nearly degenerate metastable states. This muitivalley structure of the energy is attributed to frustrated situation of interactions between spins. It is, however, demonstrated [1,2] that the nearly degenerate rectastable states are not unique in the spin-glasses, but that a random system without frustrations also can have a similar feature. T h e random system has fairly stable metastable states, showing slow relaxation. Therefore the aim of this p a p e r is to report essential and quantatire character of the relaxation time in the model only with randomness. The new r a n d o m spin model, n a m e d the fuzzy spin model, is defined by the Hamiltonian [1]

;;U = - J Y] SiS j ,

where each S i is an lsing spin, S i = 4- I Si[ , with random magnitudes between 0 and S, J is a positivc valuc, ~,"d the summation goes aq over the nearest neighbor pairs. This modcl ~s essentially fcrromag~ct!c and therefore has a unique ferromagnetic ground state. The model has no frustrations in itself, but has many metastable states because of the r a n d o m distribution of spin magnitudes. So far the static critical properties have been studied and found to be the same as those of the standard Ising model [1]. It however turned out that it takes a very long time to reach the equilibrium state when the system is quenched from an infinite to a certain temperature below the critical point. For example, more than several 10 million MCS arc often rcquired for the above relaxation even in a rather smaller square lattice with 30 × 30. Typical relaxations ol magnetization and energy are shown in fig. 1, where stcpwise relaxations of magnetization arc c~earty seen. , ncy correspond to cooperative overturns of certain magnetic domains. It should be attributed to the existcncc of raany metastable states inherent in the present random model. The clustering of spin directions associated with spins having larger length causes spinflip unflexible in the relaxation process [1]. Here we note that the energy relaxation is rather insensitive to show the stepwise relaxation. . s

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In order to study these characteristics, the relaxation time "r is intensively studied for various lattice sizes and the size-dependence is investigated. We used systems with sizes of L = 8 to 30, where L is the linear dimension of the square lattice size. For each lattice several hundreds Monte Carlo simulations are carried out with different ram.ore number sequenccs and wc studied distribution of the relaxation time r. The results are shown in fig. 2, where the relaxation time (RT) is scaled in logarithm and lattice size in normal scale. In measuring R T we regard states as equilibrium ones when the magnetization of the system rcachs to a value which is 98% of the simulated equilibrium value d e t e r m i n e d in advance [1]. Averaged values of R T for the same initial configuration are also plotted on the axis of each lattice size in fig. 2. Different symbols indicate the data from the different initial configuration. Though the deviations are rathcr large, wc now guess from fig. 2 that the relaxation time becomes longer proportionally to an exponential of the linear size of the system L: rcx e x p ( a L )

(a > 0).

O n the other hand the relaxation time in the normal Ising system depends on L as L 3 in two dimension [3], which is much faster than in the present system. The exponential d e p e n d e n c e however is also found for

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Pig. I. Relaxation behavior of magnetization and energy. Stepwise relaxations are clearly seen. where domains are turned over suddeniy. Energy relaxation is rather insensitive to the domain overturns.

0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights rese~'ed

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a process that the magnetization jumps from a stable value M to another one - M below the critical point. Thus the exponential divergence of the relaxation time in the present system could be considered to be associated with the symmetry breaking phenomena. We then have come to a question that "Does the system can rcach an equilibrium in the thermodynamic limit'?". Wc may doubt the existence of the equilibrium state fi)r the system, although the ferromagnetic state is a trivial equilibrium state in a common scnsc. The stable clustcr,: of spins consist of large spins surrounded by small spins. Thus cach cluster behavcs as an indcpcndcnt clus~cn'. The sizc of cluster increases with the size of the l,tticc. This may give the above size dependence of the relaxation. When the cluster overturns, the magnetization changes largely, but it does not cause large energy changes as we see in fig. 1. This is assured by examination of the snapshots (fig. 3) of spin-flipping through the simulation. We have intended to reveal an essential aspect of general glass phenomena through the study of the slow relaxation phenomena using the fuzzy spin model, not with frustrations but only with randomness. We conelude that one of the main mechanisms of the slow relaxation proccss scen in the spin-glasscs is explained by the random nature inherent in the system. References [I] T, Kawasaki, P::,)g. T h c o r . Phys. 84 (1991,))213. [2] A.E. Jacob and C.M. C o r a m , Phys. Rev. B 36 (1987) 3844. [3] S. Miyashita and !t. T a k a n o , Prog. T h e o r . Phys. 73 (191,;5) II,,.

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