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The ±J spin-glass model: a new exponent
Journal of Magnetism and Magnetic Materials 104-107 (1992) 1647-1648 North-Holland
mtma i
The _+J spin-glass model" a new exponent J. P o u l t e r ~ and J.A. B l a c k m a n b " FTS, Physics Department, Chulalongkorn University, Bangkok 10330, Thaihmd
t, Department of Physics, Unil'ersityof Reading, Reading RG6 2AF, UK A variant of the Pfaffian method is used to study the two-dimensional + J ising model (with concentration p of negative bonds). The method is particularly well suited to this type of problem because the frustration manifests itself as a gauge invariant feature of the formalism which can be factored out as a separable part of the theory. The disappearance of ferromagnetic order at a critical concentration, Pc, can be related to a sort of Anderson transition. A new exponent is introduced which describes the distribution of extended states, and this is then related to the exponent ,7 of the spin-spin correlation functions. The + J Ising model is one the canonical systems used to study the effects of frustration. If p represent the concentration of negative bonds, then it is well known [1] that, in 2 dimensions, ferromagnetic order disappears at some critical concentration Pc (12-15%); for p = 50%, one has a zero temperature spin glass whose spin correlations show a power law decay with distance.
[ < s0st~ >z] ,,v = C/R".
(1)
For the 2 d Ising system one can, of course, write down a formally exact solution for the zero field thermodynamic properties [2]. For example, onc may cxpress the partition function for an N site lattice at temperature T as
Z = 2N{ FIcosh( J i J k T ) } (det D) l/z,
(2)
where Ju is the coupling between neighbouring spins at sites i and j, and D is an Hermitian matrix (it is more often expnSssed in terms of a Pfaffian). The matrix elements of D are functions of the Ju and T (see refs. [3,4] for the explicit values). It is instructive to consider the eigenvalues of D. Clearly they relate to physical observables; the free energy is proportional to the sum of their logarithms, for example. They do, however, have a d e e p e r significance in the frustration problem. A subset of "he eigcnvalues, equal in number to the frustrated plaquettes of the system, approach the value zero as T---,0. They can bc written generally as + ( X / 2 ) exp(-2rJ/kT), and it can bc shown [3,4] that thc set of r and X of these cigcnstatcs relate to the ground state energy and entropy oI the lsing system. Also the corresponding eigenvectors are localized spatially on the frustrated plaquettes. The association with the frustration is u n a m b i g u o u s and so we refer to them as frustration eigenstates. Although these eigenstates are localized on the
frustrated plaquettes, they can bc extended over several of them. It was observed earlier [3] that the dcgrec of the extension depends on p. The purpose of thc present work is to investigate this behaviour quantitatively and to offer an interpretation of its physical significance. The eigenstates of D occur in pairs, +e, and the corresponding eigenvectors are complex conjugates of each other I p> + i I q>. A convenient definition of thc spatial extent of these states is the (Manhattan) distancc between thc 'ccntrcs of mass' of [ p ) and I q). The frustrated cigcnstates have e = 0 at T = 0, but it is necessary to examine them as T---, 0 rather than precisely at the limit it:~clf. The cigcnstatcs (as a degenerate set at T = 0) can bc obtained by inspection for a sample containing a random configuration of negative bonds. "Ihc T --, 0 bchaviour can b.." determined numerically for quite large samples using perturbation theory with T as the small parameter. A range of c~mfigurations and sample sizes have bccn stud:cd, the most useful information coming from the larger samples (128 x 128, 256 × 64, 512 × 32). To facilitate the discussion of the extension of the frustration eigenstates, it is convenient to introduce a distribution function N t. This quantity is defined to bc the n u m b e r of states whose size is larger than 1 in an L~ × L 2 system. We might expect, in the large ! limit, to be able to fit the distribution function to some analytic expression. The following is found to provide the best fit
N I =AilL21 -~'.
(3)
This is illustrated in fig. 1 for three values of p. where the distributions have bccn averaged over 5(I samples (deviations at high l are due to finite sample sizes). It has not been possible to perform numerical calculations on larger samples but, even with the rather restricted range of l that one can explore, eq. (3) provides a much better fit than, say, the obvious alternative function, exp( - l/().
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1648
J. Poulter, J.A. Bhwkman / The +_-Jspin-glass model
It was shown qualitatively [3] and has bccn demonstratcd quantitatively here that the occurrence of extended states is associated with the destruction of long-range order. Let us generalize that idea somewhat. The much studied two dimensional +J spin glass ( p = 5 0 ' % ) is believed to have its spin correlations given by eq. (1). We can relate ,/ to the new exponent p as follows. The spin glass susceptibility is defined as
10' -
10' 10 Z 1 0 ~' -
Xs¢; = N - t
10 '-
(S i U
10 "
I
I
I
I
I
4
7
10
20
40
I
I
70 100
I
200
Fig. I. Number N t of 'states' (no,'malized to an area of 128-" sites) thai are more extended than i (Manhattan distance in lattice spacings) for p = 5£+ (triangle,;), p = 8% (octagons)
and p = 50% (squares). The exponent p has the values 4.73 + 0.22 at p = 5%, 3.12 at p = 8% and 1.60_+ 0.08 at p = 50%. The expression (3) suggests that, for a square sample ( L × L), the number of states whose size is of the order of L varies at L"-". It is convenient to define extended states as those whose size is O(L). If O > 2, then the n u m b e r of extended states approaches zero as L ~ oo while, if p < 2, the n u m b e r approaches oo. Wc can see from the values of p obtained numerically that concentrations p of 5% and 8% clearly iic in the first rcgimc and the states are localized. At p = 50%, however, one is clearly in the second regime. We propose the schematic dependence of p on p that is shown in fig. 2 with p going through the critical value of 2 at p,:. 5,0--
4.54.03.53.02.5-
O
1.5-
From eq. (1), this leads to the finite size scaling form for a system of linear dimension L Xso(L) ~ L "-n.
(5)
Now consider the frustration states of the current development. For a sample of size L the n u m b e r of states whose size is of the order of L scales, from eq. (3), as Nt, ~ L " " . We have stated that the extended states destroy correlations and so one might infer that spins will be correlated within an area whose size scales as L 2 / N t ; that is, as L p. Thus in the double summation in (4), there are on average ~ L p sites j within the region of correlation of j. This leads to Xs~; ~ L ° which, comparing eq. (5), gives r/=2-p.
(6)
From the p calculated numerically at p = 50%, we obtain the value 77 = (I.40 + 0.08. Interestingly this agrees with the earlier direct calculation of r/ by Morgenstern and Horncr [5] rather than the more recently obtained [6,7] values around (I.2. A feature peculiar to the current calculation is that it is done at T = 11 rather than being an extrapolation from finite temperature; also the sample sizes studied were larger than those generally used. We can see from fig. 1 that, at p = 50%, one needs to use length scales larger than about 40 lattice spacings to be in the asymptotic regime. The need for very large samples may account bar the discrepancy between different calculations.
[1] K. Binder and A.P. Young, Rev. Mod. Phys, 58 (1986) 801. [2] H.S. Green and C.A. Hurst, Order-Disorder Phenomena ~IIIL%-ICI%.,I~II&.,%,,q
0
(4)
References
2.0-
1.0
. .Ia v
i-
i
1o
20
i
3o
i0
4
T
50
P
Fig. 2. Exponent p in eq. (3) as a function of concentration p (as a percentage) of negative bonds. The ful[ line is schematic. Circles are calculated data points and square is postulated value at Pc,
L.,,Jr|U%Y|Iq
I 7%P"T~',
[3] J. Poulter and J.A. Blackman, J. Phys. C 19 (1986) 569. [4] J.A. Blackman and J. Pouiter, Ph3s. Rev. B, m press. [5] I. Morgenstcrn arid 1t. ltorncr, Phys. Rev. B 25 (1982) 5O4. [b] J.-S. Wang and R,H.S. Swendsen, Phys. Rev. B 37 (1988) 7745. [7] R.N. Bhatt and A.P. Young, Phys. Rev. B 37 (1988) 5606.
mtma i
The _+J spin-glass model" a new exponent J. P o u l t e r ~ and J.A. B l a c k m a n b " FTS, Physics Department, Chulalongkorn University, Bangkok 10330, Thaihmd
t, Department of Physics, Unil'ersityof Reading, Reading RG6 2AF, UK A variant of the Pfaffian method is used to study the two-dimensional + J ising model (with concentration p of negative bonds). The method is particularly well suited to this type of problem because the frustration manifests itself as a gauge invariant feature of the formalism which can be factored out as a separable part of the theory. The disappearance of ferromagnetic order at a critical concentration, Pc, can be related to a sort of Anderson transition. A new exponent is introduced which describes the distribution of extended states, and this is then related to the exponent ,7 of the spin-spin correlation functions. The + J Ising model is one the canonical systems used to study the effects of frustration. If p represent the concentration of negative bonds, then it is well known [1] that, in 2 dimensions, ferromagnetic order disappears at some critical concentration Pc (12-15%); for p = 50%, one has a zero temperature spin glass whose spin correlations show a power law decay with distance.
[ < s0st~ >z] ,,v = C/R".
(1)
For the 2 d Ising system one can, of course, write down a formally exact solution for the zero field thermodynamic properties [2]. For example, onc may cxpress the partition function for an N site lattice at temperature T as
Z = 2N{ FIcosh( J i J k T ) } (det D) l/z,
(2)
where Ju is the coupling between neighbouring spins at sites i and j, and D is an Hermitian matrix (it is more often expnSssed in terms of a Pfaffian). The matrix elements of D are functions of the Ju and T (see refs. [3,4] for the explicit values). It is instructive to consider the eigenvalues of D. Clearly they relate to physical observables; the free energy is proportional to the sum of their logarithms, for example. They do, however, have a d e e p e r significance in the frustration problem. A subset of "he eigcnvalues, equal in number to the frustrated plaquettes of the system, approach the value zero as T---,0. They can bc written generally as + ( X / 2 ) exp(-2rJ/kT), and it can bc shown [3,4] that thc set of r and X of these cigcnstatcs relate to the ground state energy and entropy oI the lsing system. Also the corresponding eigenvectors are localized spatially on the frustrated plaquettes. The association with the frustration is u n a m b i g u o u s and so we refer to them as frustration eigenstates. Although these eigenstates are localized on the
frustrated plaquettes, they can bc extended over several of them. It was observed earlier [3] that the dcgrec of the extension depends on p. The purpose of thc present work is to investigate this behaviour quantitatively and to offer an interpretation of its physical significance. The eigenstates of D occur in pairs, +e, and the corresponding eigenvectors are complex conjugates of each other I p> + i I q>. A convenient definition of thc spatial extent of these states is the (Manhattan) distancc between thc 'ccntrcs of mass' of [ p ) and I q). The frustrated cigcnstates have e = 0 at T = 0, but it is necessary to examine them as T---, 0 rather than precisely at the limit it:~clf. The cigcnstatcs (as a degenerate set at T = 0) can bc obtained by inspection for a sample containing a random configuration of negative bonds. "Ihc T --, 0 bchaviour can b.." determined numerically for quite large samples using perturbation theory with T as the small parameter. A range of c~mfigurations and sample sizes have bccn stud:cd, the most useful information coming from the larger samples (128 x 128, 256 × 64, 512 × 32). To facilitate the discussion of the extension of the frustration eigenstates, it is convenient to introduce a distribution function N t. This quantity is defined to bc the n u m b e r of states whose size is larger than 1 in an L~ × L 2 system. We might expect, in the large ! limit, to be able to fit the distribution function to some analytic expression. The following is found to provide the best fit
N I =AilL21 -~'.
(3)
This is illustrated in fig. 1 for three values of p. where the distributions have bccn averaged over 5(I samples (deviations at high l are due to finite sample sizes). It has not been possible to perform numerical calculations on larger samples but, even with the rather restricted range of l that one can explore, eq. (3) provides a much better fit than, say, the obvious alternative function, exp( - l/().
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1648
J. Poulter, J.A. Bhwkman / The +_-Jspin-glass model
It was shown qualitatively [3] and has bccn demonstratcd quantitatively here that the occurrence of extended states is associated with the destruction of long-range order. Let us generalize that idea somewhat. The much studied two dimensional +J spin glass ( p = 5 0 ' % ) is believed to have its spin correlations given by eq. (1). We can relate ,/ to the new exponent p as follows. The spin glass susceptibility is defined as
10' -
10' 10 Z 1 0 ~' -
Xs¢; = N - t
10 '-
(S i U
10 "
I
I
I
I
I
4
7
10
20
40
I
I
70 100
I
200
Fig. I. Number N t of 'states' (no,'malized to an area of 128-" sites) thai are more extended than i (Manhattan distance in lattice spacings) for p = 5£+ (triangle,;), p = 8% (octagons)
and p = 50% (squares). The exponent p has the values 4.73 + 0.22 at p = 5%, 3.12 at p = 8% and 1.60_+ 0.08 at p = 50%. The expression (3) suggests that, for a square sample ( L × L), the number of states whose size is of the order of L varies at L"-". It is convenient to define extended states as those whose size is O(L). If O > 2, then the n u m b e r of extended states approaches zero as L ~ oo while, if p < 2, the n u m b e r approaches oo. Wc can see from the values of p obtained numerically that concentrations p of 5% and 8% clearly iic in the first rcgimc and the states are localized. At p = 50%, however, one is clearly in the second regime. We propose the schematic dependence of p on p that is shown in fig. 2 with p going through the critical value of 2 at p,:. 5,0--
4.54.03.53.02.5-
O
1.5-
From eq. (1), this leads to the finite size scaling form for a system of linear dimension L Xso(L) ~ L "-n.
(5)
Now consider the frustration states of the current development. For a sample of size L the n u m b e r of states whose size is of the order of L scales, from eq. (3), as Nt, ~ L " " . We have stated that the extended states destroy correlations and so one might infer that spins will be correlated within an area whose size scales as L 2 / N t ; that is, as L p. Thus in the double summation in (4), there are on average ~ L p sites j within the region of correlation of j. This leads to Xs~; ~ L ° which, comparing eq. (5), gives r/=2-p.
(6)
From the p calculated numerically at p = 50%, we obtain the value 77 = (I.40 + 0.08. Interestingly this agrees with the earlier direct calculation of r/ by Morgenstern and Horncr [5] rather than the more recently obtained [6,7] values around (I.2. A feature peculiar to the current calculation is that it is done at T = 11 rather than being an extrapolation from finite temperature; also the sample sizes studied were larger than those generally used. We can see from fig. 1 that, at p = 50%, one needs to use length scales larger than about 40 lattice spacings to be in the asymptotic regime. The need for very large samples may account bar the discrepancy between different calculations.
[1] K. Binder and A.P. Young, Rev. Mod. Phys, 58 (1986) 801. [2] H.S. Green and C.A. Hurst, Order-Disorder Phenomena ~IIIL%-ICI%.,I~II&.,%,,q
0
(4)
References
2.0-
1.0
. .Ia v
i-
i
1o
20
i
3o
i0
4
T
50
P
Fig. 2. Exponent p in eq. (3) as a function of concentration p (as a percentage) of negative bonds. The ful[ line is schematic. Circles are calculated data points and square is postulated value at Pc,
L.,,Jr|U%Y|Iq
I 7%P"T~',
[3] J. Poulter and J.A. Blackman, J. Phys. C 19 (1986) 569. [4] J.A. Blackman and J. Pouiter, Ph3s. Rev. B, m press. [5] I. Morgenstcrn arid 1t. ltorncr, Phys. Rev. B 25 (1982) 5O4. [b] J.-S. Wang and R,H.S. Swendsen, Phys. Rev. B 37 (1988) 7745. [7] R.N. Bhatt and A.P. Young, Phys. Rev. B 37 (1988) 5606.