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Phase diagrams for continuous-spin models: an extension of the Pirogov- Sinai theory. (English) Zbl 0611.58025

Mathematical problems of statistical mechanics and dynamics, Collect. Surv., Math. Appl., Sov. Ser. 6, 1-123 (1986).
[For the entire collection see Zbl 0596.00023.]
S. A. Pirogov and Ya. G. Sinai [Theor. Math. Phys. 25, 1185- 1192 (1975); ibid. 26, 39-49 (1976)] have developed a powerful method for providing a complete picture of a phase diagram for a general class of discrete spin models at low temperatures. In the present work the authors extend the ideas in the Pirogov-Sinai theory to deal with a wider class of continuous spin models including Hamiltonians of the type: \[ H(x)=\sum_{s,t}| x_ t-x_ s|^ 2+\sum_{t} U(x_ t) \] where \(x_ t,x_ s\in {\mathbb{R}}^ k\), \(t,s\in {\mathbb{Z}}^{\nu}\) (k\(\geq 1)\), \(\nu\geq 2)\) and the first sum is taken over all pairs of nearest neighbour spins. The potential U contains a massive barrier between two wells, the second is less deep but wider than the first.
For extending the Pirogov-Sinai theory, it is necessary to develop new mathematical tools: the authors give an admirable self-contained exposition of these in Section 2 of their work. They also state some limitations as well as some advantages of their method.
The authors draw attention to several alternative methods developed by other authors [see, for example, J. Z. Imbrie, Commun. Math. Phys. 82, 261-304, 305-343 (1982); E. I. Dinaburg and Ya. G. Sinai, Proc. Conf. Math. Phys. Dubna (1984)] which can be used to study the same problem. A comparative study of these different methods is not yet available.
Reviewer: C.S.Sharma

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
81Q15 Perturbation theories for operators and differential equations in quantum theory
82B26 Phase transitions (general) in equilibrium statistical mechanics

Citations:

Zbl 0596.00023