A spin-glass-like phase in complex nonmagnetic systems. Various kinds of replica symmetry breaking
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ESSION “STATISTICAL MECHANICS, KINETICS AND QUANTUM THEORY OF CONDENSED MATTER”
A SpinGlassLike Phase in Complex Nonmagnetic Systems. Various Kinds of Replica Symmetry Breaking1 E. E. Tareyeva, T. I. Schelkacheva, and N. M. Chtchelkatchev Institute for High Pressure Physics RAS, Troitsk, Russia Abstract—A brief review of the works of the authors on generalized spin glass models is given. The problem of the dependence of transition scenario on different factors is discussed. A classification of spin glasses behavior as depending on symmetry characteristics of systems is proposed. DOI: 10.1134/S1063779610070245 1
In this report dedicated to the memory of Nikolay Nikolaevich Bogolyubov, we are dealing with spin glass theory. We note that spin glass (SG) theory is an important field of modern statistical mechanics, where, just as in other fields, Bogolyubov’s concept of quasiaverages [1] is most important for understand ing the essence. While usually quasiaverages are intro duced using an external field, the quasiaverages describing replica symmetry breaking (RSB) in SG are defined through infinitesimal interaction between the 2
replicas.
Our aim now is twofold: first, we show that there exists a number of real complex nonmagnetic physical systems that have much in common with the tradi tional spin glasses and that can be described using the standard methods of SG theory; second, we use our results to clear some points in the classification of the different kinds of SG behavior. Extending the class of models permits considering the role of different fac tors in the scenarios for the appearance of SGtype nonergodic states. The theory of spin glasses appeared as an attempt to describe unordered equilibrium freezing of spins in actual dilute magnetic systems with disorder and frus tration. This problem was soon solved in principle by Sherrington and Kirkpatrick, Edwards and Anderson, and Parisi (see [2] for a review). The Sherrington– Kirkpatrick (SK) Hamiltonian is of the form 1 H = – J ij U i U j , 2i≠j
∑
(1)
1 The article is published in the original. 2 See, e.g. Section 12 in Moskalenko V.A. et all The SelfConsistent
Field Method in the Theory of Glassy States of Spin and Quadru pole Systems (Shtiintsa, Kishinev, 1990).
and describes Ising spins U located at the lattice sites i, j and the quenched interactions Jij are distributed with Gaussian probability 2
( J ij – J 0 ) 1 exp – P ( J ij ) = , 2 2πJ 2J
(2)
with J = J˜ / N, J0 = ˜J 0 /N. To perform averaging over disorder in this case one has to average the quenched free energy F rather than the partition sum Z itself. The standard method for performing such an average is the replica method. After averaging the free energy becomes a function of the order parameters depending on replica indices: α
1 x = N
αβ
α
F = F ( x , q ),
q
αβ
1 = N
α
αβ
N
∑U
N
∑U
α i ,
i=1
α β i Ui .
i=1
The free energy F(x , q ) has an extremum at replica symmetric (RS) solution when all qαβ are equals. However thi
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