Analyticity of the Gibbs state for a quantum anharmonic crystal: no order parameter
type: Item
instance of: scholarly article
title: Analyticity of the Gibbs state for a quantum anharmonic crystal: no order parameter
zbMATH Open document ID: 1016.82005
published in: Annales Henri Poincaré
publication date: 2003-08-07T00:00:00Z
review text: This work may be considered as a continuation of the research described in the article by \textit{R. A. Minlos, A. Verbeure} and \textit{V. A. Zagrebnov} [Rev. Math. Phys. 12, 981-1032 (2000; Zbl 0976.82010)]. The authors consider the following model of the physical system called by them quantum anharmonic crystal. To every point of the \(\nu\)-dimensional simple cubic lattice there is attached a quantum particle of mass \(m\) performing one-dimensional anharmonic oscillations. The particles interact with each other, the formal Hamiltonian describing this system is taken in the form \[ H = - {\hbar \over 2m} \sum_{x} {\partial^2 \over \partial q_x^2} + \varepsilon \sum_{\langle x, y\rangle}(q_x - q_y)^2 + \sum_x V(q_x) - h\sum_x q_x , \] where \(\varepsilon >0\) is the interaction intensity, \(q_x\) describes displacements of the particle attached to \(x\) and the second sum is taken over nearest neighbors. The one-particle potential is a polynomial \[ V(q) = a_0 q^{2s} + a_1 q^{2s-2} + \cdots + a_{2s-2} q^2 , \quad s>1 , \;a_0 >0. \] By means of a suitable modification of the cluster expansion method developed for this model in the paper mentioned above, the authors obtain the following result. There exists a positive constant \(m_0\), depending possibly on \(\varepsilon\), \(\nu\) and the parameters of \(V\) such that, for \(m\in (0, m_0)\), the thermodynamic limits obtained under the zero boundary conditions for the free energy density and expectation values of local observables are holomorphic functions of the external field \(h\) in the vicinity of the origin, the radius of which depends on \(m\). This effect holds for all temperatures including zero.
reviewed by: Yuri Kozitsky
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 82D45
Mathematics Subject Classification ID: 81V70
zbMATH DE Number: 1854193
zbMATH Keywords: structural phase transition
zbMATH Keywords: Gibbs state
zbMATH Keywords: uniqueness
zbMATH Keywords: analyticity
zbMATH Keywords: external field
zbMATH Keywords: double-well anharmonic oscillator
zbMATH Keywords: ground state
author: R. A. Minlos
author: Eugene Pechersky
author: V. A. Zagrebnov
MaRDI profile type: Publication
full work available at URL: s00023-002-8642-5
OpenAlex ID: W1979623310
DOI: 10.1007/S00023-002-8642-5
description: scientific article; zbMATH DE number 1854193