Critical dynamics of the random spherical model

Volume 75A, number 5

PHYSICS LETFERS

4 February 1980

CRITICAL DYNAMICS OF THE RANDOM SPHERICAL MODEL Y. UENO Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan Received 28 June 1979

The spherical model with random infinite-range interactions which shows a spin-glass property is investigated rigorously by the kinetic model. The uniform magnetization as well as the order parameter shows a critical slowing down. The static nonlinear susceptibility is found to diverge at the freezing temperature.

Recently there has been a great deal of interest in the dynamical aspects of random magnets, the so-called spin glasses. However, there has not been so much theoretical work since Edwards and Anderson [1]. Kinzel and Fischer [2] studied the dynamical properties of the Ising spins with infinite-range interactions by the kinetic model following Suzuki and Kubo [3]. By the same method, Hamada [4] investigated those of the Bethe lattice with nearest-neighbor interactions taking into account the random ordered phase (ROP) [5]. The result of Kinzel and Fischer [2] has an inconsistent property, as pointed out by themselves, that the sum rule ~~(S~(t)YrNdoes not hold; it leads to some results which are not good. They diagonalized the random matrix of exchange interaction, which is not right for an Ising system having the stronger restriction of S~= 1. However, this treatment is rigorous in the spherical model. (In the note added in proof of their published paper they stated this point and obtained the result for this model. It seems their treatment is not justified because spins can take continuous values in this model although details are not clear.) In the present letter we use the kinetic model suitable for spherical spin systems [6] and derive the dynamical properties above the freezing temperature Tf and discuss the magnetic field dependence of the static susceptibility. Following Tomita [6] we introduce the ~master equation for a spin distribution t):

a ~F(s1

t)

=

—

EZ W(s1 -÷ s~+ i~)P(s1t) 1

±~

~ W(s~+

+ I

s~)P(s~ +~

.~ —~

(1)

t).

±~

Here W(s~ s~ ±~) is the probability with which one spin s~ jumps by ±~by the interaction with a heat bath: -~

W(s~-÷ s~+

=

~—

—i exp ~i—} [4I~(st +

—

2 cI({st}) =

E s~ fI ~J~1s~S1 !3Eh1s~, —

—

(3)

(zj)

where the J~1’sare random infinite-range interactions and h~is an external field. The parameter p is introduced to assure the spherical condition [7]: N~ ~(s?(t)) =

(4)

1,

where (K(S~))=

K(s~)F(s1,

...,

s~t).

Let us take’ the limit of ~ Then eq. (1) is reduced to a Fokker—Planck-type equation: a 1 a ‘a a‘ ~ +-~-—)~ (5) 5i 5i 5i -

T i

This equation gives the evolution equations for 383

Volume 75A, number 5 (Sk(t)>

and

PHYSICS LETTERS

(Sk(t)Sl(t)),

2r ~(Sk)

respectively:

—~~~k> + 13EJk/(S/) +

2r~(sksl)=

26k,l

—

4 February 1980

variable q0 for the maximum eigenvalue J~= 2~1 is the order parameter in the sense of the ROP [5]. The correlation function for = q0 is then 2 -i-(1 -l-f32J2)A(131)132h2]* = [(1 ~3J) Thus we have the freezing temperature Tf = (15) J and the critical index y = 2. These values agree with the static rigorous solution [8]. The

13hk,

(6)

2p
13E(Jk/~/5l>+ JlJ). (7) / Following Kosterlitz et al. [8], we take the diagonal representation for j 11, namely, j~1= X (iJX) (XI 1>, Si = ~~(iIX>q~ and h~= ~~(ilX)h~.Then we obtain from eqs. (6) and (7) +

—

Now we return to the dynamics. In so far as the system is not far from equilibrium we may put i4t) = ~ Then we obtain from eq. (8): lie 13J \ exp Using the_fluctuation—distJ, (16) 6q~= (q~)~. 2r / where sipation theorem [3] as usual and <~1S 1. (t)) cx c5~we obtam the dynamical susceptibility above Tf,

(

—

—

2r ~(q~)

=

1

—(p

—

—

~~)(q~)

+ 13h~,

(8)

(~—$J~)(q~> +13h~(q~>.

(9)

Using eqs. (4) and (7) we obtain the parameter I +N1EJ~(q~>+13l’fl1Eh~(q~). These equations give the equilibrium values:

(10)

I3J~)~

(11)

p(t)

=

l3hx/(lie

1I(pe I3Jx)+(qx>~q, is the solution of the equation

=

and =

~e

1~ 1 NX/ie

~ 131X

N

(j3h~)2

~

(lie

—

~)2

1 X(W,h)’{Pe 21372

(12)

4132J2] 1/2}

(17)

(14) when the field is uniform. For h = 0, x(w, 0) agrees with that obtained by Kinzel and Fischer [2] written in the note added in proofanditisshowninfig. 1. The relaxation time for the order parameter is obtained as

(13)

2)~”2exp[—4/2a2], F(J,1) = (2ira

2,for which the where a is the variance JN~’ eigenvalue density p(J) ais p(J) = (27r72)_1 X (472 —J2)1”2 [9]. Let us consider the uniform field h ~ J. If one is in any one of the eigenstates IX), one may see this field completely random, 2. Using namely, white. h~= h from eq. these results theTherefore parameterone ~e has is obtained (13) as

=

+132J2)[l +A(13J)132h2] ÷0(h4). Here A(flJ) is a monotonically decreasing function(14) of 131, which takes values from 1 at i~J= 0 to ~ at g3Jr 1.

f

2.

(q 0q0(t))dt/(q~2r/(1

—

TfIfl

(18)

0

As expected q 0 shows a critical slowing down with the indexmagnetization uniform = 2 = ‘y. However, (m) = N’~ it isZ~(s surprising that the 1)has the same behavior with the index ~m = 1 as shown below: ~

f (tnrn(t))dt/(in2> ~f(s~s1(t)>dt/(s~) =

Tm

=

=

0

=

384

2

[(lie — 2iwr) where ~e is given in eq. —

Now we consider a symmetric gaussian distribution for ~

lie(1

2iwr — —

awx(O, 0)/i13x(0, 0) = 2r/13(1 T~/T2). (19) Finally it is quite interesting to investigate the static susceptibility for h * 0, which is expressed for —

~

land T~Tfas

PHYSICS LETTERS

Volume 75A, number 5 Rex (a’) /x (0)

Imx (ac) 0.5

Ix (0)

O.3~ O.4[

_

~T1f

0.2

4 February 1980

T>>Tf

0.2

T=Tf

T>>

0.1

00

00 ~.fl.

(a)

(b)

Fig. 1. Real part (a) and imaginary part (b) of the dynamical susceptibility x(w, 0) for h

~0, h) —

2J2)[1 +A(W~)132h2]

+ 13

~{(l 2131

=

[(1

2J2)2

—

+

2(1

+ 132J2)2A(/3J)132h2] 1/2}~

0.

parameter weremodes, so, ~2 more wouldprecisely, be 4), butonit the depends also (if on itother density of states near = 21. Generally for the nth order susceptibility we have

13

(20) The variation of x(0,h) for some small values of h is shown in fig. 2. Expanding x(O, h) in J3h we obtain the nonlinear susceptibility x 2J2)/(1 2~ 2J2) (21) = —A J)13~(l+ j3 13 which diverges with a negative sign at T Tf [10—12]. The critical index for it is ~2 = 1. Very recently Miyako et al. [13] have observed this property in some magnetic dilute alloys. x 2 isthis alsoexpression expressedwe as 3 = —6N’E~
x (0, h) 1 a

b

b : h/Tf =0.05 a:h/Tf0.O c

~

01

h/Tf =0.1 =0.2

__________________________________ ‘ 2 T/Tf

Fig. 2. The static susceptibility x(O, h) versus temperature (T> Tf) for various fields.

~

~-‘(—1)’~(T—Tf)~~

forn~1.

The author express his sincere thanks to Professor T.would Oguchilike andtoDr Y. Hamada for their critical reading of the manuscript and useful discussions. References [1] S.F. Edwards and P.W. Anderson, J. Phys. F6 (1976) 1927. [2] W. Kinzel and K.H. Fischer, Solid State Commun. 23 (1977) 687. [3] M. Suzuki and R. Kubo, J. Phys. Soc. Japan 24 (1968) 51. [4] Y. Hamada, Prog. Theor. Phys. 60 (1978) 937. [5] Y. Ueno and T. Oguchi, J. Phys. Soc. Japan 40 (1976) 1513; T. Oguchi and Y. Ueno, J. Phys. Soc. Japan 43 (1977) 406,764. [6] H. Tomita,Prog. Theor. Phys. 59(1978)1391,1116. [7] M. Suzuki, Phys. Lett. 43A (1973) 245. [8] MJ. Kosterlitz, D.J. Thouless and R.C. Jones, Phys. Rev. Lett. 36 (1976) 1217. [9] M.L. Mehta, Random matrices and the statistical theory of energy levels (Academic Press, New York, 1967). [10] S. Katsura, Prog. Theor. Phys. 55 (1976) 1049. [11] M. Suzuki, Prog. Theor. Phys. 58 (1977) 1151. [12] H. Takayama, J. Phys. Soc. Japan 45 (1978) 382. [13] Y. Miyako, S. Chikazawa, T. Saito and Y.G. Yuochunas, preprint.

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