Advertisement

Physics of Particles and Nuclei Letters

, Volume 9, Issue 6–7, pp 530–540 | Cite as

A new parallel algorithm for simulation of spin glasses on scales of space-time periods of external fields with consideration of relaxation effects

  • A. S. Gevorkyan
  • H. G. Abajyan
Physics of Solid State and Condensed Matter

Abstract

We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.

Keywords

External Field Parallel Algorithm Spin Chain Spin Glass Nucleus Letter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Binder and A. P. Young, “Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions,” Rev. Mod. Phys. 58, 801–976 (1986).ADSCrossRefGoogle Scholar
  2. 2.
    M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Sci., Singapore, 1987), Vol. 9.MATHGoogle Scholar
  3. 3.
    Spin-Glasses and Random Fields, Ed. by A. P. Young (World Sci., Singapore, 1998).Google Scholar
  4. 4.
    S. F. Edwards and P. W. Anderson, “Theory of Spin Glasses,” J. Phys. F: Met. Phys. 9, 965 (1975).ADSCrossRefGoogle Scholar
  5. 5.
    R. Fisch and A. B. Harris, “Spin-Glass Model in Continuous Dimensionality,” Phys. Rev. Lett. 47, 620 (1981).ADSCrossRefGoogle Scholar
  6. 6.
    C. Ancona-Torres, D. M. Silevitch, G. Aeppli, and T. F. Rosenbaum, “Quantum and Classical Glass Transitions in LiHOxY(1 − x)F4,” Phys. Rev. Lett. 101, 057201 (2008).ADSCrossRefGoogle Scholar
  7. 7.
    A. Bovier, Statistical Mechanics of Disordered Systems: A Mathematical Perspective in Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge Univ. Press, Cambridge, 2006), p. 308.Google Scholar
  8. 8.
    Y. Tu, J. Tersoff, and G. Grinstein, Phys. Rev. Lett. 81, 4899 (1998).ADSCrossRefGoogle Scholar
  9. 9.
    K. V. R. Chary and G. Govil, NMR in Biological Systems: From Molecules to Human (Focus on Structural Biology 6) (Springer, Berlin, 2008), p. 511.Google Scholar
  10. 10.
    E. Baake, M. Baake, and H. Wagner, “Ising Quantum Chain is an Equivalent to Model of Biological Evolution,” Phys. Rev. Lett. 78, 559–562 (1997).ADSCrossRefGoogle Scholar
  11. 11.
    D. Sherrington and S. Kirkpatrick, “A Solvable Model of a Spin-Glass,” Phys. Rev. Lett. 35, 1972 (1975).CrossRefGoogle Scholar
  12. 12.
    B. Derrida, “Random-Energy Model: An Exactly Solvable Model of Disordered Systems,” Phys. Rev. B: Condens. Matter Mater. Phys. 24, 2613 (1981).MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    G. Parisi, “Infinite Number of Order Parameters for Spin Glasses,” Phys. Rev. Lett. 43, 1754 (1979).ADSCrossRefGoogle Scholar
  14. 14.
    A. J. Bray and M. A. Moore, “Replica-Symmetry Breaking in Spin-Glass Theories,” Phys. Rev. Lett. 41, 1068 (1978).ADSCrossRefGoogle Scholar
  15. 15.
    J. F. Fernandez and D. Sherrington, “Randomly Located Spins with Oscillatory Interactions,” Phys. Rev. B: Condens. Matter Mater. Phys. 18, 6271 (1978).ADSCrossRefGoogle Scholar
  16. 16.
    F. Benamira, J. P. Provost, and G. J. Vallee, “Separable and Nonseparable Spin-Glass Models,” J. Phys. 46, 1269 (1985).MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Grensing and R. Kuhn, “On Classical Spin-Glass Models,” J. Phys. 48, 713 (1987).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. S. Gevorkyan and H. Chin-Kun, “On a Mathematical Approach for the Investigation of Some Statistical Phenomena of a Disordered 3D Spin System in the External Field,” in Proceedings of the ISAAC Conference on Analysis, Yerevan, Armenia, 2002, Ed. by G. A. Barsegian (Natl. Acad. Sci. Armenia, Yerevan, 2004), pp. 164–178.Google Scholar
  19. 19.
    A. S. Gevorkyan, C.-K. Hu, and S. Flach, “New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System under the Influence of External Field,” Trans. Comp. Sci. 7, 132–153 (2010).MathSciNetGoogle Scholar
  20. 20.
    A. S. Gevorkyan, H. G. Abajyan, and H. S. Sukiasyan, “A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field,” Int. J. Mod. Phys. A 2, 488–497 (2011).Google Scholar
  21. 21.
    A. V. Bogdanov, A. S. Gevorkyan, and G. V. Dubrovskiy, “On Mechanisms of Proton-Hydrogen Resonance Recharge at Moderate Energies,” Sov. Phys. Tech. Lett. 9, 343–348 (1983).Google Scholar
  22. 22.
    A. S. Gevorkyan, H. G. Abajyan, and E. A. Ayryan, “On Modeling of Statistical Properties of Classical 3D Spin Glasses” (2011). arXiv:1107.2125v1Google Scholar
  23. 23.
    C. M. Bender, J.-H. Chen, D. W Darg, and K. A Milton, “Classical Trajectories for Complex Hamiltonians,” J. Phys. A: Math. Gen. 39, 4219 (2006).MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    C. M. Bender and D. W. Darg, “Spontaneous Breaking of Classical PT Symmetry,” J. Math. Phys. 48, 2703 (2007).MathSciNetGoogle Scholar
  25. 25.
    C. M. Bender and D. W. Hook, “Exact Isospectral Pairs of PT Symmetric Hamiltonians,” J. Phys. A: Math. Theor. 41, 1751–8113 (2008).Google Scholar
  26. 26.
    A. S. Gevorkyan, A. V. Bogdanov, and G. Nyman “Regular and Chaotic Quantum Dynamic in Atom-Diatom Reactive Collisions,” Phys. At. Nucl. 71, 876–883 (2008).CrossRefGoogle Scholar
  27. 27.
    A. V. J. Smilga, “Cryptogauge Symmetry and Cryptoghosts for Crypto-Hermitian Hamiltonians,” Phys. A: Math. Theor. 41, 4026 (2008).MathSciNetGoogle Scholar
  28. 28.
    Ch. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976), p. 599Google Scholar
  29. 29.
    D. J. Griffith, Introduction to Electrodynamics (Prentice Hall, New Jersey, 1989), p. 192.Google Scholar
  30. 30.
    R. Becker, Electromagnetic Fields and Interactions (Dover, New York, 1982).Google Scholar
  31. 31.
    C. Itzykson and J. M. Drouffe, Statistical Field Theory: From Brownian Motion to Renormalization and Lattice Gauge Theory (Cambridge Univ. Press, Cambridge, 1991), Vol. 2, p. 428.Google Scholar
  32. 32.
    A. S. Gevorkyan, H. G. Abajyan, and H. S. Sukiasyan, “Statistical Properties of Ideal Ensemble of Disordered 1D Steric Spin Chains” (2010). arXiv:cond-mat.disnn1010.1623v1Google Scholar
  33. 33.
    I. Ibragimov and Yu. Linnik, Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, Groningen, 1971), Vol. 48, pp. 1287–1730.MATHGoogle Scholar
  34. 34.
    C. J. Thompson, Phase Transitions and Critical Phenomena (Academic, New York, 1972) Vol. 1, pp. 177–226.Google Scholar
  35. 35.
    Spin Glasses, Ed. by E. Bolthausen and A. Bovier (Springer, Berlin, 2007), Vol. 163, pp. 1900–2075.MATHGoogle Scholar
  36. 36.
    G. Wannier, Statistical Physics (Dover, New York, 1987), p. 532.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institute for Informatics and Automation ProblemsNAS of ArmeniaYerevanArmenia
  2. 2.Joint Institute for Nuclear ResearchDubnaRussia

Personalised recommendations