A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field

ABSTRACT

We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequal-ity conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equa-tions for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribu-tion satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time peri-ods are obtained. Similarly, Edwards-Anderson’s ordering parameter depending on the external field ampli-tude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.

We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequal-ity conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equa-tions for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribu-tion satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time peri-ods are obtained. Similarly, Edwards-Anderson’s ordering parameter depending on the external field ampli-tude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.

KEYWORDS

Spin-Glass Hamiltonian, Birkhoff Ergodic Hypothesis, Statistic Distributions, Frustration, Fractal, Parallel Algorithm, Numerical Simulation

Spin-Glass Hamiltonian, Birkhoff Ergodic Hypothesis, Statistic Distributions, Frustration, Fractal, Parallel Algorithm, Numerical Simulation

Cite this paper

nullA. Gevorkyan, H. Abajyan and H. Sukiasyan, "A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field,"*Journal of Modern Physics*, Vol. 2 No. 6, 2011, pp. 488-497. doi: 10.4236/jmp.2011.26059.

nullA. Gevorkyan, H. Abajyan and H. Sukiasyan, "A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field,"

References

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[2] M. Mézard, G. Parisi, M. A. Virasoro, “Spin Glass Theory and Beyond”, World Scientific, Vol. 9, 1987.

[3] A. P. Young (ed.), “Spin Glasses and Random Fields”, World Scientific, 1998.

[4] R. Fisch and A. B. Harris, “Spin-glass model in continuous dimensionality”, Physical Review Letters, Vol. 47, No. 8, 1981, p.620.

[5] C. Ancona-Torres, D. M. Silevitch, G. Aeppli, and T. F. Rosenbaum, “Quantum and Classical Glass Transitions in LiHoxY1-xF4”, Physical Review Letters, Vol. 101, No. 5, 2008, pp. 057201.1-057201.4.

[6] A. Bovier, “Statistical Mechanics of Disordered Systems: A Mathematical Perspective”,Cambridge Series in Statistical and Probabilistic Mathematics, No. 18, 2006, p 308.

[7] Y. Tu, J. Tersoff and G. Grinstein, “Properties of a Continuous-Random-Network Model for Amorphous Systems”, Physical Review Letters, Vol. 81, No. 22, 1998, pp. 4899-4902.

[8] K. V. R. Chary, G. Govil, “NMR in Biological Systems: From Molecules to Human “, Springer, Vol. 6, 2008, p. 511.

[9] E. Baake, M. Baake and H. Wagner, “Ising Quantum Chain is a Equivalent to a Model of Biological Evolution”, Physical Review Letters, Vol. 78, No. 3, 1997, pp. 559-562.

[10] D. Sherrington and S. Kirkpatrick, “A solvable model of a spin-glass “, Physical Review Letters, Vol. 35, No. 26, 1975, pp. 1792-1796.

[11] B. Derrida, “Random-energy model: An exactly solvable model of disordered systems”, Physical Review B, Vol. 24, No. 5, 1981, pp. 2613-2626.

[12] G. Parisi, “Infinite Number of Order Parameters for Spin-Glasses”, Physical Review Letters, Vol. 43, No. 23, 1979, pp. 1754-1756.

[13] A. J. Bray and M. A. Moore, “Replica-Symmetry Breaking in Spin-Glass Theories”, Physical Review Letters, Vol.41, No. 15, 1978, pp. 1068-1072.

[14] J. F. Fernandez and D. Sherrington, “Randomly located spins with oscillatory interactions”, Physical Review B, Vol. 18, No. 11, 1978, pp. 6270-6274.

[15] F. Benamira, J. P. Provost and G. J. Vallée, “Separable and non-separable spin glass models”, Journal de Physique, Vol. 46, No. 8, 1985, pp. 1269-1275.

[16] D. Grensing and R. Kühn, “On classical spin-glass models”, Journal de Physique, Vol. 48, No. 5, 1987, pp. 713-721.

[17] A. S. Gevorkyan et al., “New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System Under the Influence of External Field”, Transactions on computational science VII, 2010, pp. 132-153.

[18] S. F. Edwards and P. W. Anderson, “Theory of spin glasses”, Journal of Physics F, Vol. 5, 1975, pp. 965-974.

[19] A. S. Gevorkyan, H. G. Abajyan and H. S. Sukiasyan. ArXiv: cond-mat.dis-nn 1010.1623v1.

[20] I. Ibragimov and Yu. Linnik, “Independent and Stationary Sequences of Random Variebles”, Mathematical Reviews, Vol. 48, 1971, pp. 1287-1730.

[1] [1] K. Binder and A. P. Young, “Spin glasses: Experimental facts, theoretical concepts and open Questions”, Reviews of Modern Physics, Vol. 58, No.4, 1986, pp. 801-976.

[2] M. Mézard, G. Parisi, M. A. Virasoro, “Spin Glass Theory and Beyond”, World Scientific, Vol. 9, 1987.

[3] A. P. Young (ed.), “Spin Glasses and Random Fields”, World Scientific, 1998.

[4] R. Fisch and A. B. Harris, “Spin-glass model in continuous dimensionality”, Physical Review Letters, Vol. 47, No. 8, 1981, p.620.

[5] C. Ancona-Torres, D. M. Silevitch, G. Aeppli, and T. F. Rosenbaum, “Quantum and Classical Glass Transitions in LiHoxY1-xF4”, Physical Review Letters, Vol. 101, No. 5, 2008, pp. 057201.1-057201.4.

[6] A. Bovier, “Statistical Mechanics of Disordered Systems: A Mathematical Perspective”,Cambridge Series in Statistical and Probabilistic Mathematics, No. 18, 2006, p 308.

[7] Y. Tu, J. Tersoff and G. Grinstein, “Properties of a Continuous-Random-Network Model for Amorphous Systems”, Physical Review Letters, Vol. 81, No. 22, 1998, pp. 4899-4902.

[8] K. V. R. Chary, G. Govil, “NMR in Biological Systems: From Molecules to Human “, Springer, Vol. 6, 2008, p. 511.

[9] E. Baake, M. Baake and H. Wagner, “Ising Quantum Chain is a Equivalent to a Model of Biological Evolution”, Physical Review Letters, Vol. 78, No. 3, 1997, pp. 559-562.

[10] D. Sherrington and S. Kirkpatrick, “A solvable model of a spin-glass “, Physical Review Letters, Vol. 35, No. 26, 1975, pp. 1792-1796.

[11] B. Derrida, “Random-energy model: An exactly solvable model of disordered systems”, Physical Review B, Vol. 24, No. 5, 1981, pp. 2613-2626.

[12] G. Parisi, “Infinite Number of Order Parameters for Spin-Glasses”, Physical Review Letters, Vol. 43, No. 23, 1979, pp. 1754-1756.

[13] A. J. Bray and M. A. Moore, “Replica-Symmetry Breaking in Spin-Glass Theories”, Physical Review Letters, Vol.41, No. 15, 1978, pp. 1068-1072.

[14] J. F. Fernandez and D. Sherrington, “Randomly located spins with oscillatory interactions”, Physical Review B, Vol. 18, No. 11, 1978, pp. 6270-6274.

[15] F. Benamira, J. P. Provost and G. J. Vallée, “Separable and non-separable spin glass models”, Journal de Physique, Vol. 46, No. 8, 1985, pp. 1269-1275.

[16] D. Grensing and R. Kühn, “On classical spin-glass models”, Journal de Physique, Vol. 48, No. 5, 1987, pp. 713-721.

[17] A. S. Gevorkyan et al., “New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System Under the Influence of External Field”, Transactions on computational science VII, 2010, pp. 132-153.

[18] S. F. Edwards and P. W. Anderson, “Theory of spin glasses”, Journal of Physics F, Vol. 5, 1975, pp. 965-974.

[19] A. S. Gevorkyan, H. G. Abajyan and H. S. Sukiasyan. ArXiv: cond-mat.dis-nn 1010.1623v1.

[20] I. Ibragimov and Yu. Linnik, “Independent and Stationary Sequences of Random Variebles”, Mathematical Reviews, Vol. 48, 1971, pp. 1287-1730.