Relaxation in spin glasses

Journal of Magnetism and Magnetic Materials 7 (1978) 265-266 0 North-Holland Publishing Company

RELAXATIONINSPINGLASSES I.RIESS, M. FIBICH and A. RON Physics Department,

Technion-Israel Institute of Technology,

Haifa, Israel

We discuss the relaxation of the magnetization, M, and the order parameter q, in spin glasses with arbitrary spin S, within the mean field approximation. It is shown that the rate of the relaxation of the order parameter is slowed down as the spin glass freezing temperature is approached from either above or below. The relaxation of M for T > Tf and the fast relaxation of M for T < Tf show no critical behavior near Tf.

Eq. (3) becomes

The relaxation of the order parameter in spin glasses, q, was recently considered by Edwards and Anderson [ 1] for a classical two-dimensional spin system solving the Fokker-Planck equation. Fischer [2] has used the Glauber rate equation [3] for the Ising S = $ model with the mean field approximation (MFA) [4]. We have extended the Glauber rate equation within the MFA to spin S > i and used it to calculate the relaxation of the spin glass order parameter 4 and the magnetization M. In the MFA the spin glass is given by the distribution P(B) of the (temporal) local mean fields. Let Bi(t) be the local field acting on the jth spin at the moment t. It defines a momentary quasi-equilibrium value of the local magnetization, (&s;>qe= SCM&(t)/z-)

)

dM/dt = -M/r ,

and the magnetization exhibits no critical behavior near the spin glass transition temperature Tf. Eqs. (2) and (4) do not hold at low temperature T < Tf when the system is close to one of its low energy levels. The spin glass order parameter is defined similar to Fischer’s definition [5]

The rate equation for 4 is [using eqs. (l), (2) and (5) and 6;) + cpi”,” z 2h;)q”)9e]

(1)

where gs(x) is the Brillouin function. It can be argued that within the MFA the local magnetization relaxes at any given moment t, to the quasi-equilibrium value given by eq. (l), d$) --_ dt

_

dq -_=_-+-

q

dt

T

s*

-

r_- s

We assume that the width of the distribution P(B) is proportional to 4 ‘I2 [6]. Using P(B) dB + P(B/&) dB/& and x = BJ&, eq. (7) becomes

Gsi”)- +qe 7

(2)

’

dM

s

-= _!?!,s r

r__

P(B) qs

; 0

dB

s

_=-4+s1-

dq

where r-’ is the relaxation rate. The rate equation the macroscopic magnetization M = X&S;) is then

dt

(4)

dt

for

T

r_-

(7)

This differential equation can be solved for T > Tf since then 4 + 0 and for T < Tf as q approaches its equilibrium value q(t + 00) since then 4 - q(m) + 0 and it is possible to expand the cx3~(xfi/7’) function about the equilibrium value. Using this and the relations between Tf and q(m) and the second and fourth moments of the distribution function P(x), as shown

(3)

P(B) is symmetric with respect to B -+ -B (for any given M, as long as the temperature is not too low and the system is not close to one of its low energy levels). 265

I. Riess et al. /Relaxation

266

References

by Riess and Klein [7], one obtains

T > Tf, (84

40) = T
in spin glasses

(8b)

This shows that the spin glass order parameter exhibits a critical slowing down as the critical temperature Tf is approached from both sides.

[l] [2] [3] [4] [S] [6] [7]

S.F. Edwards and P.W. Anderson, J. Phys. F6 (1976) 1927. K.H. Fischer, Physica 86-88B (1977) 813. R.J. Glauber, J. Math. Phys. 4 (1963) 294. R. Kubo and M. Suzuki, J. Phys. Sot. Jap. 24 (1968) 51 K.H. Fischer, Phys. Rev. Lett. 34 (1975) 1438. M.W. Klein, Phys. Rev. 14 (1976) 5008. 1. Riess and M.W. Klein, Phys. Rev. B15 (1977) 6001.