Potts models and related problems in statistical mechanics.

*(English)*Zbl 0734.17012
Series on Advances in Statistical Mechanics, 5. Singapore etc.: World Scientific. xiii, 344 p. $ 19.00 (1991).

In recent years we are witnessing a rapid development of fascinating subjects of research generated by the confluence of conformal field theory, theory of integrable systems (classical and quantum), topological field theory and quantum groups. Subtle algebraic, analytic and topological methods intertwine with advanced physical applications forming an extremally attractive tapestry. As usual, anyone intending to master this new field is faced with the problem of taming and selecting the abundance of information. The need for good monographs covering at least some part of the subject is therefore urgent.

The book under consideration is a well written introduction to the algebraic methods of statistical mechanics. The author had a fortunate idea to select one class of models, namely the Potts models, and discusses the main ideas and methods using these models as an example. For the justification of the choice of Potts models let us quote the author: “The Potts models are a special and easily defined class of statistical mechanical models... Nonetheless, they are richly structured enough to illustrate almost every conceivable nuance of the subject.... We are fortunate that all problems in statistical mechanics seem to be related to Potts models... this means that a general discussion of the subject can be couched in Potts model terms. These models are invaluable in that they allow a ready understanding of their own basic physical significance and, compared to many of their more purely mathematically motivated counterparts, exhibit a robust insensivity to boundary conditions away from the critical region”.

The book consists of thirteen chapters. The first one is of introductory character. In the second chapter the transfer matrices are introduced and their basic properties are discussed. The commuting transfer matrices and solvable models are discussed in chapters 3 and 4. Chapter 5 provides a very concise introduction to some algebraic methods or, rather, terminology. In the next three chapters a compendium of useful results for Temperley-Lieb algebras is collected. Several representations are described and used to determine the structure of the algebras. In chapter 9 the Hecke algebras are discussed. The generalized Clifford algebras are defined in chapter 10 and used to construct the representations of graph Temperley-Lieb algebras appropriate for the three dimensional Potts lattice gauge theories. The zeros of the partition functions and the physical picture of phase transitions are discussed in chapter 11. The next one is devoted to the study of vertex models. Finally, in the last chapter the structure of algebras obtained by cabling quotients of the braid group algebra is determined.

The book under consideration is a well written introduction to the algebraic methods of statistical mechanics. The author had a fortunate idea to select one class of models, namely the Potts models, and discusses the main ideas and methods using these models as an example. For the justification of the choice of Potts models let us quote the author: “The Potts models are a special and easily defined class of statistical mechanical models... Nonetheless, they are richly structured enough to illustrate almost every conceivable nuance of the subject.... We are fortunate that all problems in statistical mechanics seem to be related to Potts models... this means that a general discussion of the subject can be couched in Potts model terms. These models are invaluable in that they allow a ready understanding of their own basic physical significance and, compared to many of their more purely mathematically motivated counterparts, exhibit a robust insensivity to boundary conditions away from the critical region”.

The book consists of thirteen chapters. The first one is of introductory character. In the second chapter the transfer matrices are introduced and their basic properties are discussed. The commuting transfer matrices and solvable models are discussed in chapters 3 and 4. Chapter 5 provides a very concise introduction to some algebraic methods or, rather, terminology. In the next three chapters a compendium of useful results for Temperley-Lieb algebras is collected. Several representations are described and used to determine the structure of the algebras. In chapter 9 the Hecke algebras are discussed. The generalized Clifford algebras are defined in chapter 10 and used to construct the representations of graph Temperley-Lieb algebras appropriate for the three dimensional Potts lattice gauge theories. The zeros of the partition functions and the physical picture of phase transitions are discussed in chapter 11. The next one is devoted to the study of vertex models. Finally, in the last chapter the structure of algebras obtained by cabling quotients of the braid group algebra is determined.

Reviewer: P.Kosiński (Łódź)

##### MSC:

17B81 | Applications of Lie (super)algebras to physics, etc. |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

00A79 | Physics (Use more specific entries from Sections 70-XX through 86-XX when possible) |

82B23 | Exactly solvable models; Bethe ansatz |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

15A90 | Applications of matrix theory to physics (MSC2000) |