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

## 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## Preview

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## References

- 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.M. Mézard, G. Parisi, and M. A. Virasoro,
*Spin Glass Theory and Beyond*(World Sci., Singapore, 1987), Vol. 9.MATHGoogle Scholar - 3.
*Spin-Glasses and Random Fields*, Ed. by A. P. Young (World Sci., Singapore, 1998).Google Scholar - 4.S. F. Edwards and P. W. Anderson, “Theory of Spin Glasses,” J. Phys. F: Met. Phys.
**9**, 965 (1975).ADSCrossRefGoogle Scholar - 5.R. Fisch and A. B. Harris, “Spin-Glass Model in Continuous Dimensionality,” Phys. Rev. Lett.
**47**, 620 (1981).ADSCrossRefGoogle Scholar - 6.C. Ancona-Torres, D. M. Silevitch, G. Aeppli, and T. F. Rosenbaum, “Quantum and Classical Glass Transitions in LiHO
_{x}Y(1 −*x*)F_{4},” Phys. Rev. Lett.**101**, 057201 (2008).ADSCrossRefGoogle Scholar - 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.Y. Tu, J. Tersoff, and G. Grinstein, Phys. Rev. Lett.
**81**, 4899 (1998).ADSCrossRefGoogle Scholar - 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.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.D. Sherrington and S. Kirkpatrick, “A Solvable Model of a Spin-Glass,” Phys. Rev. Lett.
**35**, 1972 (1975).CrossRefGoogle Scholar - 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.G. Parisi, “Infinite Number of Order Parameters for Spin Glasses,” Phys. Rev. Lett.
**43**, 1754 (1979).ADSCrossRefGoogle Scholar - 14.A. J. Bray and M. A. Moore, “Replica-Symmetry Breaking in Spin-Glass Theories,” Phys. Rev. Lett.
**41**, 1068 (1978).ADSCrossRefGoogle Scholar - 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.F. Benamira, J. P. Provost, and G. J. Vallee, “Separable and Nonseparable Spin-Glass Models,” J. Phys.
**46**, 1269 (1985).MathSciNetCrossRefGoogle Scholar - 17.D. Grensing and R. Kuhn, “On Classical Spin-Glass Models,” J. Phys.
**48**, 713 (1987).MathSciNetCrossRefGoogle Scholar - 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.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.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.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.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.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.C. M. Bender and D. W. Darg, “Spontaneous Breaking of Classical PT Symmetry,” J. Math. Phys.
**48**, 2703 (2007).MathSciNetGoogle Scholar - 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.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.A. V. J. Smilga, “Cryptogauge Symmetry and Cryptoghosts for Crypto-Hermitian Hamiltonians,” Phys. A: Math. Theor.
**41**, 4026 (2008).MathSciNetGoogle Scholar - 28.Ch. Kittel,
*Introduction to Solid State Physics*(Wiley, New York, 1976), p. 599Google Scholar - 29.D. J. Griffith,
*Introduction to Electrodynamics*(Prentice Hall, New Jersey, 1989), p. 192.Google Scholar - 30.R. Becker,
*Electromagnetic Fields and Interactions*(Dover, New York, 1982).Google Scholar - 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.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.I. Ibragimov and Yu. Linnik,
*Independent and Stationary Sequences of Random Variables*(Wolters-Noordhoff, Groningen, 1971), Vol. 48, pp. 1287–1730.MATHGoogle Scholar - 34.C. J. Thompson,
*Phase Transitions and Critical Phenomena*(Academic, New York, 1972) Vol. 1, pp. 177–226.Google Scholar - 35.
*Spin Glasses*, Ed. by E. Bolthausen and A. Bovier (Springer, Berlin, 2007), Vol. 163, pp. 1900–2075.MATHGoogle Scholar - 36.G. Wannier,
*Statistical Physics*(Dover, New York, 1987), p. 532.MATHGoogle Scholar