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Mathematical Problems of Computer Science 34, 2010. 

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Mathematical Problems of Spin-Glass Systems in One-
Dimension 

Ashot S Gevorkyan, 

Institute for Informatics and Automation Problems, NAS of Armenia 

Spin glass models are very useful for investigations of a number of important  and difficult 
applied problems of physics, chemistry, material science, biology, evolution, organization 
dynamics, hard-optimization, environmental and social structures, human logic systems, 
financial mathematics etc (see for example [1-10]). It is possible to divide the considered mean-
field models of spin-glasses into two types. The first consists of the true random-bond models, 
where the coupling between interacting spins are taken to be independent random variables [11-
13]. The solution of these models was obtained by n-replica trick [11-13] and required the 
invention of sophisticated schemes of replica-symmetry breaking [13-14]. In the models of 
second type, the bond-randomness is expressed in terms of some underlining hidden site-
randomness and is thus of a superficial nature. It has been pointed out in the works, however, 
that this feature retains an important physical aspect of true spin-glasses, viz. that they are 
random with respect to the positions of magnetic impurities. The main way of investigation of 
spin-glasses systems is a method of numerical simulation. But, numerical studies of spin-glass 
problem is difficult to accomplish and, in general, only for a small number of spins, the 
numerical simulation is effectively solvable.  For development of new ideas and creation of 
program packages for numerical simulation of spin-glass systems the investigation of 1D model 
is very important. The modified 1D model can be useful also for analysis of series of real 
problems. 
 
   



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Fig. 1.  The red line shows visualization of numerical data while the green line shows 
analytic approximation by Cauchy function. 

 
      The statistical properties of ensemble of disordered 1D steric spin-chains (SSC) of various 
lengths are investigated. Using 1D spin-glass type classical Hamiltonian, the recurrent 
transcendental equations for stationary points and corresponding conditions for the construction 
of stable 1D SSCs are found. The ideal ensemble of spin-chains (non-interacting spin-chains) is 
analyzed and the latent interconnections between random angles and interaction constants for 
each set of three nearest-neighboring spins are found.  It is analytically proved and by numerical 
calculation is shown that the distribution of interaction constant satisfy Levy's alpha-stable 
distribution law [15] (see Fig. 1 and reference [16]). Energy distribution in ensemble is 
calculated depending on different conditions of possible polarization of spin-chains. It is 
specifically shown that the dimensional effects in the form of set of local maximums in the 
energy distribution arise when the number of spin-chains M<<Nx2 (where Nx is number of spins 
in a chain) while in the case when M ~ M<<Nx2 the energy distribution has one global maximum 
and ensemble of spin-chains satisfies the Birkhoff's ergodic theorem. An effective algorithm for 
parallel simulation of the problem which includes calculation of different statistic parameters of 
1D SSCs ensemble has been elaborated. 
      Finally, it is important to note that the developed scheme of solution of 1D SSCs problem 
can be used for analysis of different applied problems, also it can be useful for creation of 
effective program package for simulation of  3D spin-glass problem. 
 
 
[1]   K. Binder  and  A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and 
open questions. Rev. Mod. Physics, 58(4), 801-976 (1986). 
[2]  M. Me’zard, G. Parisi, M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, 
Singapore, 1987) 
[3] A. P. Young (ed.), Spin Glasses and Random Fields (World Scientific, Singapore, 1998) 
[4] R. Fisch and A. B. Harris, Phys. Rev. Let., 47, 620 (1981). 



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[5] C. Ancona-Torres, et al., Phys. Rev. Let.,  101, 057201 (2008). 
[6] A. Bovier, Statistical Mechanics of Disordered Systems: A Mathematical Perspective, 
Cambridge Series in Statistical and Probabilistic Mathematics, p 308 (2006). 
[7] Y. Tu, J. Tersoff and G. Grinstein, Phys. Rev. Lett., 81, 4899 (1998). 
[8] K. V. R. Chary, G. Govil, NMR in Biological Systems: From Molecules to Human (Focus on 
Structural Biology 6), Springer, p. 511, (2008). 
[9] E. Baake, M. Baake and H. Wagner, Phys. Rev. Let., 78(3), 559-562 (1997.) 
[10] A. S. Gevorkyan et al., New Mathematical Conception and Computation Algorithm for 
Study of Quantum 3D Disordered Spin System Under  the Influence of ExternalField, Trans. On 
Comput. Sci., VII, LNCS 132-153, Spinger-Verlage, 10.1007/978-3-642-11389-58 
[11] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett., 35, 1972 (1975). 
[12] B. Derrida, Phys. Rev., B 26, 2613 (1981). 
[13] G. Parisi, Phys. Rev. Lett. 43, 1754 (1979). 
[14]  A. J. Bray and M. A. Moore, Phys. Rev. Lett., 41, 1068 (1978). 
[15] J. P. Nolan, Stable Distributions: Models for Heavy Tailed Data (2009-02-21). 
en.wikipedia.org/Stable_/distribution. 
[15] A. S. Gevorkyan, H. G.  Abajyan and H. S. Sukiasyan,  Ideal Ensemble of Disordered 1D 
Steric Spin-Chains in External Field, J. Computational Science (in press).