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Journal of Mathematical Physics

Introduction to Special Issue: Integrable Quantum Systems and Solvable Statistical Mechanics Models

Alexander Rudol'fovich Its, Michio Jimbo, Jean Michel Maillet

Published Aug 27, 2009• 1 citation
Publisher
DOI
10.1063/1.3216475

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A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.
Development in the theory of solvable (integrable) models is reviewed. It covers from basic knowledge on completely integrable systems to recent work by the authors. First, soliton theory is briefly summarized. Through the inverse scattering method and its quantum extension, a central concept, commuting transfer matrices, and a key relation, the Yang-Baxter relation, are introduced. Second, it is shown that there exists at least ∞ × ∞ number of solvable models in two-dimensional statistical mechanics. Third, quantum spin chains corresponding to solvable statistical mechanical models are discussed. In particular, finite temperature extension of Baxter's formula is given. Fourth, a new approach is presented to the classification problem of knots and links. It is shown that link polynomial, topological invariant for knots and links, can be associated with any solvable model in statistical mechanics. In the presentation, universality of the soliton picture in field theory, spin systems and statistical mechanics is observed.