The statistical mechanics of quantum lattice systems. A path integral approach.
type: Item
instance of: scholarly article
title: The statistical mechanics of quantum lattice systems. A path integral approach.
zbMATH Open document ID: 1178.82001
DOI: 10.4171/070
published in: EMS Tracts in Mathematics
publication date: 2009-07-10T00:00:00Z
review text: 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. \noindent 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…
reviewed by: Hiroshi Tamura
Mathematics Subject Classification ID: 82-02
Mathematics Subject Classification ID: 81-02
zbMATH DE Number: 5577516
zbMATH Keywords: Euclidean Green function
zbMATH Keywords: Euclidean Gibbs measure
zbMATH Keywords: interacting quantum anharmonic oscillators
zbMATH Keywords: phase transition
zbMATH Keywords: path integral approach
zbMATH Keywords: quantum lattice system
zbMATH Keywords: quantum statistical mechanics
author: S. Albeverio
author: Yuri Kondratiev
author: Yuri Kozitsky
author: Michael Röckner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/070
OpenAlex ID: W1687361011
description: scientific article; zbMATH DE number 5577516