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Critical exponents and fractal dimension of realistic spin glasses
Volume 104A, number 2
PHYSICS LETTERS
13 August 1984
CRITICAL EXPONENTS AND FRACTAL DIMENSION OF REALISTIC SPIN GLASSES Debashish CHOWDHURY and Jayanta K. BHATTACHARJEE Department o f Physics, Indian Institute o f Technology, Kanpur-208016, India
Received 4 April 1984 Revised manuscript received 8 June 1984
We estimate the various critical exponents for the spin glass transition in real three-dimensional spin glasses from available experimental data and check their consistency in several ways. The fractal dimension of these systems turns out to be 2. This is in agreement with a recent result that systems with fractal dimension 2 exhibit 1/f noise.
The most widely studied and best understood model of spin glasses (SGs) is the Sherrington-Kirkpatrick (SK) model [1]. The latter model is based on infinitely weak and infinitely long.ranged exchange interaction. On the other hand Smith [2] developed a percolation model for realistic SGs. Some of the drawbacks of this model have been rectified by incorporating frustration and finite mean free path effects [3,4]. The latter model has a very close connection with some earlier models [5]. It can also explain logarithmic relaxation of magnetization [6] and thus leads to a 1/f noise spectrum [7]. In the percolation model the SG transition is described as a percolation of clusters of dynamically correlated spins. One draws a sphere of correlation around each spin; the volume of the sphere increases with decreasing temperature [4]. The problem of SG transition is thereby geometrized and reduces to a problem of percolation of overlapping spheres [8]. This type of percolation problem is quite different from usual site-bond percolation problems where one assumes only nearest neighbour bonds. Even for damped RKKY interaction the interaction is quite long.ranged, although not infinitely ranged. Therefore there is no a priori reason to believe that the critical exponents for the nearest-neighbour sitebond percolation should also hold in the case of sphere percolation. The aim of this paper is to estimate the critical exponents for the sphere percolation problem assuming that a SG transition in real SGs can, indeed, be described as a percolation process. No new 100
effort will be made in this paper to prove the validity of the latter assumption. We shall compute the critical exponents from available experimental data and check their consistency in several ways. The fractal dimension turns out to be 2. The critical exponents for percolation on a continuum are defined as follows: S= Ip-Pcl -~ ,
(la)
o: Ip - Pcl - v ,
(lb)
P~ IP-Pcl # ,
(lc)
where p is the volume fraction, S is the size of percolating clusters, ~ is the correlation length and P is the percolation probability. In the case of the SG-percolation problem the volume of a sphere is inversely proportional to the temperature [4]. Therefore, ( l a ) - ( l c ) get modified as So~ I T - Tgl - ~ ,
(2a)
= I T - Tg I - " ,
(2b)
pcc I T - Tgl ~ •
(2c)
One must remember that ~ is the correlation length associated with the correlation function C(r - r', p ) of the percolation problem. The latter is def'med as follows [4]: C(r - r', p) is the joint probability of (a) the point r being occupied by a sphere centre 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 104A, number 2
PHYSICS LETTERS
which belongs to a finite cluster, and (b) the poInt r' also being occupied by a sphere centre which belongs to the same cluster, p(T) is the volume fraction. Let us begin with an estimation of the critical exponent 3'. Assuming that the whole cluster turns over together, it was shown earlier [4] that the relaxation time r(S) o f a cluster o f size S is given by r(S) = r 0 exp(aS),
(3)
where a is a constant. Substituting (2a) into (3) we get
r(S) : r 0 exp[C1/(T- Tg)~/],
(4)
where C 1 is another constant. If the measurement time r m ~> r(S) a cluster of size S will appear frozen. So we can recast (4) as rm = 1"0 exp{C1/[Tg(6O) - Tg(0)]'r) ,
(5)
where Tg(~) is the freezing temperature corresponding to the observation frequency co = rm 1. From (5), we get
13 August 1984
[ln(rm/ro)]-l/7 = const X [Tg(6O) - Tg(0)].
(6)
Taking the experimental data of the 1.h.s. of eq. (6) from ref. [9] we get very good fit with eq. (6) provided 3' = 0.8 -+ 0.1 and Tg(0) = 26 K as shown in fig. 1. This seems to be physically consistent. Next let us compute v. We know that neutron scattering experiment probes a length scale ~ ~ 1[q. There. fore, (2b) can be recast as In q = const + v ln[Tg(0) - Tg(q)], where Tg(q) corresponds to the maximum in ×(q). The gradient of the curve of In q versus In [Tg(0) Tg(q)] taken from experimental data [10] leads to v = 1.2 (see fig. 2). ' The percolation probability P(T) has been calculated by Chowdhury and Mookerjee [11] comparing experimental data on susceptibility with theoretical predictions o f the percolation model [4]. Besides,
from (2c) In P = const + 3 In(Tg - T ) .
/ 2.s
~,u~t (10 at. 'Y,)
0.020
0.01
2.C
[
--~
~~o
t
~E 0.012
I
C._u_uMn ( 4.6 cat.%)
Zo = 1:'3coos ~'= 0.8 Tg(o) = 26K ls
1.0 "e
~
026
~27
O.S / 0.?
28
I 1.2
29
I I.? ,.
I,, 2. 9
J-T~co)- T~ cq)-I
Tg(,~)
Fig. 2. In q as a function of ln[Tz(0) - Tg(q)] for AuFe (10 Fig. 1. [ln(rm/rO)] -10' as a function of Tg(¢O) for C__uuMn(4.6 at%) alloy. Tg(¢o) is in units of K. Data are taken from ref. [9].
at%)alloy,q is in units of 10-3A -1 and Tg in unit~oof K. Data are taken from ref. [10].
101
PHYSICS LETTERS
Volume 104A, number 2
-0.6
-0.8 J3 - 1 . 3 7
-1.2
0,5
~l
0.75
I I i.25 tn (Tg-T) 1.0
I.Ii
Fig. 3. In P(T) as a function of ln(Tg - T) for the amorphous spin glass alloy (Feo.64Mno.ad)TsP16B6A13. T and Tg are in units of K. Data are taken from ref. [ 11 ]. Plotting In P(T) against ln(Tg - T) we get fl = 1.37 (see fig. 3). Thus, comparison o f the percolation theory o f SG transition with experiments yield 7 = 0.8, v = 1.2 and fl = 1.37; being drastically different from the corresponding values for site-bond percolation problems. Now let us check the consistency of the values of the exponents calculated above. We know the scaling relation that relates fl, v and the euclidean dimension d with the fractal dimension Df [12]: Of = d -
~lv .
(7)
Substituting fl = 1.37 and v - 1.2 for d = 3 we get Df ~2. On the other hand, Hertz [13] suggested that r = r0
exp[Ca/(r- r g ) ~ ] ,
where
= W f - 1)v,
(8)
because a cluster can disorder by propagating a domain wall through it [13]. Substituting 7 = 0.8, v = 1.2 in (8) we again get Df ~- 2 and hence the consistency o f the values of the exponents. Alexander and Orbach [14] have proved that the fracton dimensionality D F = 2Df/(2 + 0) = 4 / 3 ,
(9)
almost independently o f space dimensionality, where 102
13 August 1984
the diffusion constant Deft(r) ~ r -°. At present no numerical value of 0 for the sphere percolation problem is available. However, if we assume that the value of 0 is the same for usual site-bond percolation and sphere percolation, then eq. (9) leaves no choice but that Df is the same as in the ordinary site-bond percolation problem (Df ~ 2.5) [ 15]. But this is inconsistent with eq. (7) and the values of v and fl we found above. However, it seems that there is a way out * 1 Suppose it is the backbone, not the entire percolation cluster, that is relevant to the freezing in the spin glass. Then instead of Df ~ 2.5 we should use the backbone fractal dimensionality, which is about 2 [16], again in agreement with our estimation above. From the above discussion we conclude that Df = 2 for real SGs in three dimensions. This is in agreement with the recent observation [17] that 1If noise is the characteristic of systems with fractal dimensionality Df = 2 because SGs exhibit l / f noise [7] which is a consequence of the logarithmic relaxation o f magnetization. ,1 We thank a referee for suggesting this argument.
References [1] D. Shcrrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [2] D.A. Smith, J. Phys. F5 (1975) 2148. [3] A.A. Abrikosov, J. Low Temp. Phys. 33 (1978) 505; in: Lecture notes in physics, Vol. 115 (Springer, Berlin, 1980) p. 251. [4] A. Mookerjee and D. Chowdhury, J. Phys. F13 (1983) 431. [5] D. Chowdhury and A. Mookerjee, J. Phys, F13 (1983) L19. [6] D. Chowdhury and A. Mookerjee, J. Phys. F14 (1984) 245; Phys. Lett. 99A (1983) 111. [7] D. Chowdhury and A. Mookerjee, Solid State Commun. 48 (1983) 887. [8] J. Kertesz, J. Phys. (Paris) Lett. 42 (1981) L393. [9] J.L. Tholence, Solid State Commun. 35 (1980) 113. [10] A.P. Murani, Phys. Rev. Lett. 37 (1976) 450. [ 11 ] D. Chowdhury and A. Mookerjee, Physica B, to be published. [121 D. Stauffer, Phys. Rep. 54 (1979) 1. [13] J.A. Hertz, Order and disorder in spin glasses, NORDITA Preprint No. 83[12. [14] S. Alexander and R. Orbach, J. Phys. (Paris) Lett. 43 (1982) L625. [15] R.B. Pandey and D. Stauffer, J. Phys. A16 (1983) L511. [16] Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kkkpatriek, Phys. Rev. Lett. 47 (1981) 1771. [17] R. Benzi, L. Peliti and A. Vulpiani, Lett. Nuovo Cimento 36 (1983) 471.
PHYSICS LETTERS
13 August 1984
CRITICAL EXPONENTS AND FRACTAL DIMENSION OF REALISTIC SPIN GLASSES Debashish CHOWDHURY and Jayanta K. BHATTACHARJEE Department o f Physics, Indian Institute o f Technology, Kanpur-208016, India
Received 4 April 1984 Revised manuscript received 8 June 1984
We estimate the various critical exponents for the spin glass transition in real three-dimensional spin glasses from available experimental data and check their consistency in several ways. The fractal dimension of these systems turns out to be 2. This is in agreement with a recent result that systems with fractal dimension 2 exhibit 1/f noise.
The most widely studied and best understood model of spin glasses (SGs) is the Sherrington-Kirkpatrick (SK) model [1]. The latter model is based on infinitely weak and infinitely long.ranged exchange interaction. On the other hand Smith [2] developed a percolation model for realistic SGs. Some of the drawbacks of this model have been rectified by incorporating frustration and finite mean free path effects [3,4]. The latter model has a very close connection with some earlier models [5]. It can also explain logarithmic relaxation of magnetization [6] and thus leads to a 1/f noise spectrum [7]. In the percolation model the SG transition is described as a percolation of clusters of dynamically correlated spins. One draws a sphere of correlation around each spin; the volume of the sphere increases with decreasing temperature [4]. The problem of SG transition is thereby geometrized and reduces to a problem of percolation of overlapping spheres [8]. This type of percolation problem is quite different from usual site-bond percolation problems where one assumes only nearest neighbour bonds. Even for damped RKKY interaction the interaction is quite long.ranged, although not infinitely ranged. Therefore there is no a priori reason to believe that the critical exponents for the nearest-neighbour sitebond percolation should also hold in the case of sphere percolation. The aim of this paper is to estimate the critical exponents for the sphere percolation problem assuming that a SG transition in real SGs can, indeed, be described as a percolation process. No new 100
effort will be made in this paper to prove the validity of the latter assumption. We shall compute the critical exponents from available experimental data and check their consistency in several ways. The fractal dimension turns out to be 2. The critical exponents for percolation on a continuum are defined as follows: S= Ip-Pcl -~ ,
(la)
o: Ip - Pcl - v ,
(lb)
P~ IP-Pcl # ,
(lc)
where p is the volume fraction, S is the size of percolating clusters, ~ is the correlation length and P is the percolation probability. In the case of the SG-percolation problem the volume of a sphere is inversely proportional to the temperature [4]. Therefore, ( l a ) - ( l c ) get modified as So~ I T - Tgl - ~ ,
(2a)
= I T - Tg I - " ,
(2b)
pcc I T - Tgl ~ •
(2c)
One must remember that ~ is the correlation length associated with the correlation function C(r - r', p ) of the percolation problem. The latter is def'med as follows [4]: C(r - r', p) is the joint probability of (a) the point r being occupied by a sphere centre 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 104A, number 2
PHYSICS LETTERS
which belongs to a finite cluster, and (b) the poInt r' also being occupied by a sphere centre which belongs to the same cluster, p(T) is the volume fraction. Let us begin with an estimation of the critical exponent 3'. Assuming that the whole cluster turns over together, it was shown earlier [4] that the relaxation time r(S) o f a cluster o f size S is given by r(S) = r 0 exp(aS),
(3)
where a is a constant. Substituting (2a) into (3) we get
r(S) : r 0 exp[C1/(T- Tg)~/],
(4)
where C 1 is another constant. If the measurement time r m ~> r(S) a cluster of size S will appear frozen. So we can recast (4) as rm = 1"0 exp{C1/[Tg(6O) - Tg(0)]'r) ,
(5)
where Tg(~) is the freezing temperature corresponding to the observation frequency co = rm 1. From (5), we get
13 August 1984
[ln(rm/ro)]-l/7 = const X [Tg(6O) - Tg(0)].
(6)
Taking the experimental data of the 1.h.s. of eq. (6) from ref. [9] we get very good fit with eq. (6) provided 3' = 0.8 -+ 0.1 and Tg(0) = 26 K as shown in fig. 1. This seems to be physically consistent. Next let us compute v. We know that neutron scattering experiment probes a length scale ~ ~ 1[q. There. fore, (2b) can be recast as In q = const + v ln[Tg(0) - Tg(q)], where Tg(q) corresponds to the maximum in ×(q). The gradient of the curve of In q versus In [Tg(0) Tg(q)] taken from experimental data [10] leads to v = 1.2 (see fig. 2). ' The percolation probability P(T) has been calculated by Chowdhury and Mookerjee [11] comparing experimental data on susceptibility with theoretical predictions o f the percolation model [4]. Besides,
from (2c) In P = const + 3 In(Tg - T ) .
/ 2.s
~,u~t (10 at. 'Y,)
0.020
0.01
2.C
[
--~
~~o
t
~E 0.012
I
C._u_uMn ( 4.6 cat.%)
Zo = 1:'3coos ~'= 0.8 Tg(o) = 26K ls
1.0 "e
~
026
~27
O.S / 0.?
28
I 1.2
29
I I.? ,.
I,, 2. 9
J-T~co)- T~ cq)-I
Tg(,~)
Fig. 2. In q as a function of ln[Tz(0) - Tg(q)] for AuFe (10 Fig. 1. [ln(rm/rO)] -10' as a function of Tg(¢O) for C__uuMn(4.6 at%) alloy. Tg(¢o) is in units of K. Data are taken from ref. [9].
at%)alloy,q is in units of 10-3A -1 and Tg in unit~oof K. Data are taken from ref. [10].
101
PHYSICS LETTERS
Volume 104A, number 2
-0.6
-0.8 J3 - 1 . 3 7
-1.2
0,5
~l
0.75
I I i.25 tn (Tg-T) 1.0
I.Ii
Fig. 3. In P(T) as a function of ln(Tg - T) for the amorphous spin glass alloy (Feo.64Mno.ad)TsP16B6A13. T and Tg are in units of K. Data are taken from ref. [ 11 ]. Plotting In P(T) against ln(Tg - T) we get fl = 1.37 (see fig. 3). Thus, comparison o f the percolation theory o f SG transition with experiments yield 7 = 0.8, v = 1.2 and fl = 1.37; being drastically different from the corresponding values for site-bond percolation problems. Now let us check the consistency of the values of the exponents calculated above. We know the scaling relation that relates fl, v and the euclidean dimension d with the fractal dimension Df [12]: Of = d -
~lv .
(7)
Substituting fl = 1.37 and v - 1.2 for d = 3 we get Df ~2. On the other hand, Hertz [13] suggested that r = r0
exp[Ca/(r- r g ) ~ ] ,
where
= W f - 1)v,
(8)
because a cluster can disorder by propagating a domain wall through it [13]. Substituting 7 = 0.8, v = 1.2 in (8) we again get Df ~- 2 and hence the consistency o f the values of the exponents. Alexander and Orbach [14] have proved that the fracton dimensionality D F = 2Df/(2 + 0) = 4 / 3 ,
(9)
almost independently o f space dimensionality, where 102
13 August 1984
the diffusion constant Deft(r) ~ r -°. At present no numerical value of 0 for the sphere percolation problem is available. However, if we assume that the value of 0 is the same for usual site-bond percolation and sphere percolation, then eq. (9) leaves no choice but that Df is the same as in the ordinary site-bond percolation problem (Df ~ 2.5) [ 15]. But this is inconsistent with eq. (7) and the values of v and fl we found above. However, it seems that there is a way out * 1 Suppose it is the backbone, not the entire percolation cluster, that is relevant to the freezing in the spin glass. Then instead of Df ~ 2.5 we should use the backbone fractal dimensionality, which is about 2 [16], again in agreement with our estimation above. From the above discussion we conclude that Df = 2 for real SGs in three dimensions. This is in agreement with the recent observation [17] that 1If noise is the characteristic of systems with fractal dimensionality Df = 2 because SGs exhibit l / f noise [7] which is a consequence of the logarithmic relaxation o f magnetization. ,1 We thank a referee for suggesting this argument.
References [1] D. Shcrrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [2] D.A. Smith, J. Phys. F5 (1975) 2148. [3] A.A. Abrikosov, J. Low Temp. Phys. 33 (1978) 505; in: Lecture notes in physics, Vol. 115 (Springer, Berlin, 1980) p. 251. [4] A. Mookerjee and D. Chowdhury, J. Phys. F13 (1983) 431. [5] D. Chowdhury and A. Mookerjee, J. Phys, F13 (1983) L19. [6] D. Chowdhury and A. Mookerjee, J. Phys. F14 (1984) 245; Phys. Lett. 99A (1983) 111. [7] D. Chowdhury and A. Mookerjee, Solid State Commun. 48 (1983) 887. [8] J. Kertesz, J. Phys. (Paris) Lett. 42 (1981) L393. [9] J.L. Tholence, Solid State Commun. 35 (1980) 113. [10] A.P. Murani, Phys. Rev. Lett. 37 (1976) 450. [ 11 ] D. Chowdhury and A. Mookerjee, Physica B, to be published. [121 D. Stauffer, Phys. Rep. 54 (1979) 1. [13] J.A. Hertz, Order and disorder in spin glasses, NORDITA Preprint No. 83[12. [14] S. Alexander and R. Orbach, J. Phys. (Paris) Lett. 43 (1982) L625. [15] R.B. Pandey and D. Stauffer, J. Phys. A16 (1983) L511. [16] Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kkkpatriek, Phys. Rev. Lett. 47 (1981) 1771. [17] R. Benzi, L. Peliti and A. Vulpiani, Lett. Nuovo Cimento 36 (1983) 471.