Relaxation in spin-glasses far above the transition point

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Relaxation in spin-glasses far above the transition point

M.V. Feigel’Man, L.B. Ioffe

To cite this version:

M.V. Feigel’Man, L.B. Ioffe. Relaxation in spin-glasses far above the transition point. Journal de

Physique, 1986, 47 (3), pp.363-366. �10.1051/jphys:01986004703036300�. �jpa-00210213�

363

RELAXATION IN SPIN-GLASSES FAR ABOVE THE TRANSITION POINT

M.V. FEIGEL’MAN and L.B. IOFFE

Landau Institute for Theoretical Physics, Moscow, U.S.S.R.

(Reçu ^{Ze 12} juillet 1985, accepté ^{le 20 dgcembre} 1985)

Résumé.- Nous considérons deux modèles réa- listes de verres de spin : ^{le modèle} d’Edwards-Anderson avec interactions à lon- gue portée ^{et le} système ^de spins ^classi-

ques ^avec l’interaction RKKY. Nous montrons que, dans les deux modèles, une relaxation non-exponentielle ^{existe bien} au-dessus du

point ^de transition et même dans la région critique. ^{Nous obtenons la forme de la re-} laxation q(t) = S1 (0) S1 (t)> ^{pour les}

temps longs, ^{dans ces} deux modèles.

Abstract.- We consider two realistic models of spin glasses : ^the Edwards-Anderson ^mo- del with long-range interaction and the system ^{of classical vector} spins ^{with RKKY} interaction, and show that in the both ^mo- dels non-exponential relaxation occurs far above the transition point ^{and even in the} critical region. ^{We obtain the form of the} relaxation q(t) = S1 (0) S1 (t) > ^{for long}

times, in these models.

J. Physique ⁴⁷ (1986) ^363-366 MARS 1986,

Classification

Physics ^Abstracts

75.40

Recently Randeria, Sethna and Palmer [1] have published ^a very interest- ing paper on low-frequency relaxation in Ising spin glasses. They ^have shown that the Gaussian distribution of bond strength (J) ^{in the} Edwards-Anderson model of spin glasses ^{leads to} non-exponential ^relaxation of the magnetic moment at any temperature.

They have also shown that a more realistic distribution of J ^(such that J ^{cannot ex-}

ceed Jazz) ^{) leads to} non-exponential ^relaxa- tion in a wide temperature range above the transition point. Unfortunately the rough estimates of [1] produce only ^{a lower bound} for relaxation :

where d is the space dimension.

We have discovered that a slight modification of estimates [1] gives ^{an ex-}

act form of the q(t) relaxation at large times in at least two realistic models of

spin glasses. ^{First we consider a modified} Edwards-Anderson model with large - (but finite) - range interaction in the case of the Ising spins ^and formulate the analogous results for the vector case, then we consi- der ^a vector spin glass with RKKY-interac- tion.

The modified Edwards-Anderson model consists of variables U ^{(Ising or n-com-}

ponent vector) ^{on the sites} of a lattice, which interact with each other according to the Hamiltonian

where K(r) obeys ^{K(r) d3r = 1 this norm-}

alization provides ^{a critical} temperature

Tc ~ _the ^n-1 _number ^and _of Z - interacting neighbours ^K(r) rZd3r 3/2 >> ^{1 i.e. is}

large. ^{In the} vicinity ^{of the transition} point ^{the model} (2) has been studied in [2-4]. ^We consider only the pure relaxatio- nal dynamics ^of Ut’ governed by ^the Lange- vin equations for the vector spin :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703036300

364

and by ^{the anomalous Glauber} equations ^for

the dynamics ^{of the} Ising spin :

where pfal ^{is the} probability ^{of a} Icr ii

configuration.

The averaging ^{over fast} t - F-’ ^fluctua-

tions of a i yields ^a Langevin equation ^of motion for the average magnetization

where the effective energy HTAP ^coincides

[2-4] ^with the well-known result [5] :

The variables mi ^{can be regarded as}

a sum over eigenfunctions i ^{of the 2 is}

matrix :

Far above the transition tempera- ture the interaction between different ax

can be neglected therefore their dynamics is governed by ^the independent equations

(below we consider the Ising case) :

The states with Ea ^{T2 have a}

small Fl- 0 I -r- 2) relaxation time, ^whereas only ^states with E ;k > -U2 can relax exponent-

tially slowly, ^{but their number is} small, since the density of states of the J1j-ma-

trix decreases exponentially ^with energy :

p (E) N exp (-a(f-/,Eo ) 3 / 4 ) , Eo = Z - 4 I 3, ^{a - 1}

These are states replace ^{in our work the}

rare highly correlated ferromagnetic ^re-- gions ^{of ref} [1].

In the case of the Ising spins ^the

free energy H (a,,) ^{has two minima separated}

by the energy barrier ð E = 2 ) 2/8T

I(W§) , ^{hence the} relaxation rate of a

i

given ^{state N (with} e ;k > -V2 can be estima-- ted with exponential accuracy :

eigenstates ^N (e > Z-4 13 can be determi- ned from a saddle point approximation analogously to p (e) [4) ] ^: £(W(A) 4 ~

i

i

z- 1 al x 1 4. ^An eigenstate ^N gives ^{a contribu-} tion to the relaxation of q(t) proportion- tional to exp(-t/t,) ^{which produces an es-}

timate of q(t) :

The integral ⁱⁿ equation [9] ^{can be} evaluated by ^{means of the saddle} point asp-

proximation yielding :

Formula (10) is valid at times t such that

J1J

k(r)). Note, that the relaxation (10) is slower than the relaxation governed by the power law (q(t) - t’ °‘ ) .

We formulate the analogous ^result

for the case of the vector spins :

for

Now we consider the problem ^of RKKY-interaction of vector spins, namely, ^a system ^of randomly distributed classical vector spins cri (Iul 1=1) interacting ^with

each other according ^{to the} potential

Vij - V(r -r ) ^} aicr il ^{V(r) =} Vo cosp0rxr - 3

It is well-known (see e.g. [6]) that the transition temperature ^{of this} system ^is proportional to the concentration (c) ^of spins :

We remark here that our results hold irrespective ^{of the} fact whether T c ^is

a point ^{of a} phase transition or only ^a point ^{of crucial} slowing ^{down of} relaxation.

Local fluctuations of the concen-

tration c result in fluctuations of local

Tc. Therefore at any temperature ^{T above} T c

(as long ^{as there exists a} concentration c

such that spin-glasses with this concentra- tion have T. Z ^{T) there are regions of the}

systems ^which have a local Tc>T. ^{These re-}

gions ^{are clusters of N} highly ^correlated spins, relaxing only due to rotation of the cluster as a whole. The relaxation rate w

of the rigid system ^{of N} spins ^{can be ob-} tained straightforwardly ^{from the} Langevin equations (3) :

where wo ^is the rate of free spin ^relaxa- tion. The system ^{of N} spins ^{can be conside-}

red as rigid ^{if its} T ^{» T ; if, on the}

contrary, T - ^T « T ^{then w} increases :

where T is the effective reduced tempera- ture of the cluster T

(T -

T)/T .

^{In the}

framework of the scaling theory ^we get f«) - ° if T > T* so that correlation vo-

lume 13 (T) ~ T-3Y « V (V is the volume of the cluster). Below we show that at T - T

c

f (TT) , (-r) -

A probability ^{to find the cluster}

of N spins localized in the volume V (N >

cV) is exponentially ^{small :}

Its contribution to the relaxation of q(t) is :

where f(T) - qEA(T) - S>- is the Edwards- Anderson order parameter in the infinite

~

system ^{at the reduced} temperature T. (It

seems likely ^that f (T) - ^{f (T) at T} > T*,

but we shall not use it). Multiplying (16) by (17) ^we get the value of q(t) ^at large

(t >> w0-1 ) ^{times :}

where averaging is carried out over differ- ent volumes V. The integral (and the result of averaging) ⁱⁿ (18) ^can be evaluated with

an exponential accuracy from the saddle point approximation ^{over two} variables

(V,N) :

where

To ^{is the solution of the equation :}

At very ^small effective temperatures (T >>

T) f (T) - qE A (r) thus - (af’ 1 /aT) f (T) -

a - 1 so the value of -co ^{can be obtained}

from the equation :

Again ^we note that for our purpose

we do not need the value of qEA (T) ^{in the}

366

thermodynamic ^sense to = Si >2 , ^{E A -1 z}

^{but ra-}

ther its value at time larger ^than w-1 : :

qE A = S1 ⁽⁰⁾ Si ( t ) > t >> w- ^I that surely exists at low effective temperatures ^and is likely ^{to be} governed by ^{the mean field} solution.

In conclusion we have considered the Edwards-Anderson model of spin glass with long range interaction and shown that in the case of the Ising spins ^the large- time relaxation is governed by the law :

g(t) - exp(-ln 3 1 5 t) = Z- 4 / 3 (cf (10)) in the large ^time region (see (11)). ^We hope

that the Y 1 - x Er spin glass (x - 0.01) can

x

provide ^an example ^{of a} spin glass ^with long-range interaction [3-4] ^{and the exam-} ple ^of non-exponential relaxation above the real transition point ^(see [7] ^{for the} the view of experimental properties ^of Y 1 - x Er ^{). We} have also considered the problem

x

of the RKKY spin-glass and shown that the non-exponential relaxation persists ^far above the transition point, ^{in fact it} persists up to the maximum transition tem- perature of the spin glasses ^{of the same} chemical composition (but different stoe-

chiometry). ^{For the case of the classical}

vector spins ^the relaxation is governed by the exp - two - ^{law (cf.(19)) far above}

the transition point T >> Tc. ^{It seems like-}

ly ^that this estimate holds also for a sys- tem of quantum Heisenberg spins. Certainly, non-exponential relaxation occurs also at lower (T Z Tc) temperatures, ^but the power of (two) ^{in the exponent} decreases if T de- creases into the critical region. ^{The dif-}

ference between the exponential ^and

exp two - relaxation at T far above Tic

can be observed in experiments ^with large frequencies w - 10’ o .

Thus it seems likely ^{that this ef-}

fect can be observed most easily by ^neutron spin ^{echo (NSE)} technique. ^The existing ^NSE experiments [8] show that relaxation slows down and becomes non-exponential ^{far above}

To ^{(up to T -} 3T ). ^The relaxation of q(t)

at largest ^times may be governed by

exp (-lnat) law due to the presence of ani- sotropy.

We underline that the phenomenon considered here of the non-exponential ^re- laxation disappears in the limit ^{Z + ao} (in the EA model) ; thus our previous ^results [3] describing ^{the critical} slowing ^down hold since Ze f f --> oo at T--+Tc.

References

[1] ^{Randeria M., Sethna} J.P., Palmer R.G.

Phys. ^{Rev. Lett.} 54 (1985) 1321.

[2] Feigelman M.V., Ioffe L.B. J. de Physi-

que (Lettres) 45 (1984) L-475 [3] Feigelman M.V., Ioffe L.B. J. de Physi-

que (Lettres) 46 (1985) L-695 [4] Feigelman M.V., Ioffe L.B. ZhETF 89

(1985) 654

[5] Thouless D.I., Anderson P.W., Palmer R.G., Philos. Mag. 35 (1977) 593 ; Bray A.I., ^Moore M.A., ^J. Phys. C14

(1981) 2629.

[6] ^Larkin A.I., Khmelnistsky ^{P.E. ZhETF 58} (1970) 1789.

[7] ^Bouchiat H., Mailly ^D. preprint, Orsay 1984. Baberschke K., Pureur P., Fert A. et al. Phys. ^{Rev. B29}

(1984) 4999.

[8] Mezei F., J. Appl. Phys. 53 (1982) 7654

J. Magn. Magn. ^{MAt. 31-34} (1983)

1327.