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**The statistical mechanics of quantum lattice systems. A path integral approach.**
*(English)*
Zbl 1178.82001

EMS Tracts in Mathematics 8. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-070-8/hbk). xiii, 379 p. (2009).

This book deals with a mathematically rigorous study of equilibrium statistical mechanics for lattice systems of interacting quantum anharmonic oscillators by means of path integral. This subject are interesting not only as a model of quantum solid state physics, but also with its connection to quantum field theory. For these models on infinite lattices, the method of the KMS condition on the operator algebra does not work to describe the equilibrium, because of the unboundedness of the Hamiltonian of finite subsystems. The strategy adopted in this book is the Euclidean path integral approach.

That is to say: The Høegh-Krohn’s process (periodic Ornstein-Uhlenbeck process of period \(\beta\): the inverse temperature) is introduced for a unbounded spin on each site in the lattice. For a finite subset \(\Lambda\) of the lattice, the local Euclidean Gibbs measure for \(\Lambda\) is constructed as the path integral of exponentiated interaction terms in \(\Lambda\) by the finite product of measures of the corresponding processes. Then the thermodynamic limit yields the global Euclidean Gibbs measure on the whole lattice. The Matsubara functions (the Euclidean Green functions) are represented as the integrals in terms of the global Gibbs measure.

The book is divided into two parts. Part I, which consists of three chapters, is devoted to the mathematical foundations of the method. In Part II, some physical properties are described on the basis developed in the Part I. Chapter I is the longest chapter which amounts to about 140 pages. Here, presentation of the local Euclidean Gibbs measures is the goal. Firstly, conditions of the self-adjointness of the local Hamiltonians of interacting anharmonic oscillators are discussed. The gap estimate for the eigenvalues of the single anharmonic oscillator is also discussed. Secondly, on the framework of operator algebra, the analytic properties of the Green functions and the Matsubara functions for the local Gibbs states are considered. Then the local Euclidean Gibbs measures are introduced. The integral representations of the Matsubara functions are given with their analytic properties. In Chapter 2, based on the correlation inequalities for classical spin models, their generalizations to the local Euclidean Gibbs measures are proved. Chapter 3 is devoted to the discussion of the thermodynamic limit of the local Euclidean Gibbs measures. The DLR technique is adapted here. It reflects this book’s standpoint that the subject can be understood as the classical lattice system of infinite-dimensional spins.

In Chapter 4, the first chapter of Part II, the classical limit and the high temperature uniqueness of the Euclidean Gibbs measure are given. Chapter 5 deals with the thermodynamic pressure on the lattice \(\mathbb{Z}^d\). In Chapter 6, phase transitions of the reflection positive models and the hierarchical models are studied. The final Chapter 7 is devoted to the mechanism of suppression of phase transition due to quantum effects. Every chapter is closed with “Comments and Bibliographic Notes”, where historical and experimental backgrounds as well as recent progress on the theme of the chapter are mentioned with related references.

The arguments of the book are based mainly on the mathematical facts which contains the spectral theory on unbounded self-adjoint operators, stochastic analysis up to the Feynman-Kac formula, basics of measure theory on the Polish space, rudiments of operator algebra and complex analysis of several variables, and the correlation inequalities for the ferromagnetic spin systems. The text include concise introductions to basic notions in these subjects. Some of the important theorems in them are even proved. And references are indicated on every mathematical facts from which the arguments of the book are developed. However, background knowledges of the above subjects will help the reader’s understanding.

The text is highly recommended to all mathematicians who are interested in this subject, especially expert researchers in related fields and graduate students who are aiming for this area.

That is to say: The Høegh-Krohn’s process (periodic Ornstein-Uhlenbeck process of period \(\beta\): the inverse temperature) is introduced for a unbounded spin on each site in the lattice. For a finite subset \(\Lambda\) of the lattice, the local Euclidean Gibbs measure for \(\Lambda\) is constructed as the path integral of exponentiated interaction terms in \(\Lambda\) by the finite product of measures of the corresponding processes. Then the thermodynamic limit yields the global Euclidean Gibbs measure on the whole lattice. The Matsubara functions (the Euclidean Green functions) are represented as the integrals in terms of the global Gibbs measure.

The book is divided into two parts. Part I, which consists of three chapters, is devoted to the mathematical foundations of the method. In Part II, some physical properties are described on the basis developed in the Part I. Chapter I is the longest chapter which amounts to about 140 pages. Here, presentation of the local Euclidean Gibbs measures is the goal. Firstly, conditions of the self-adjointness of the local Hamiltonians of interacting anharmonic oscillators are discussed. The gap estimate for the eigenvalues of the single anharmonic oscillator is also discussed. Secondly, on the framework of operator algebra, the analytic properties of the Green functions and the Matsubara functions for the local Gibbs states are considered. Then the local Euclidean Gibbs measures are introduced. The integral representations of the Matsubara functions are given with their analytic properties. In Chapter 2, based on the correlation inequalities for classical spin models, their generalizations to the local Euclidean Gibbs measures are proved. Chapter 3 is devoted to the discussion of the thermodynamic limit of the local Euclidean Gibbs measures. The DLR technique is adapted here. It reflects this book’s standpoint that the subject can be understood as the classical lattice system of infinite-dimensional spins.

In Chapter 4, the first chapter of Part II, the classical limit and the high temperature uniqueness of the Euclidean Gibbs measure are given. Chapter 5 deals with the thermodynamic pressure on the lattice \(\mathbb{Z}^d\). In Chapter 6, phase transitions of the reflection positive models and the hierarchical models are studied. The final Chapter 7 is devoted to the mechanism of suppression of phase transition due to quantum effects. Every chapter is closed with “Comments and Bibliographic Notes”, where historical and experimental backgrounds as well as recent progress on the theme of the chapter are mentioned with related references.

The arguments of the book are based mainly on the mathematical facts which contains the spectral theory on unbounded self-adjoint operators, stochastic analysis up to the Feynman-Kac formula, basics of measure theory on the Polish space, rudiments of operator algebra and complex analysis of several variables, and the correlation inequalities for the ferromagnetic spin systems. The text include concise introductions to basic notions in these subjects. Some of the important theorems in them are even proved. And references are indicated on every mathematical facts from which the arguments of the book are developed. However, background knowledges of the above subjects will help the reader’s understanding.

The text is highly recommended to all mathematicians who are interested in this subject, especially expert researchers in related fields and graduate students who are aiming for this area.

Reviewer: Hiroshi Tamura (Kanazawa)

### MSC:

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

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |