Absence of phase transitions in certain one-dimensional long-range random systems
type: Item
instance of: scholarly article
title: Absence of phase transitions in certain one-dimensional long-range random systems
zbMATH Open document ID: 0642.60108
DOI: 10.1007/BF01007972
published in: Journal of Statistical Physics
publication date: 1985-01-01T00:00:00Z
review text: An Ising chain is considered with a potential of the form \(J(i,j)/| i-j|^{\alpha}\), where the J(i,j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if \(\alpha >1\). This is to be contrasted with a ferromagnetic interaction which requires \(\alpha >2\). The basic idea is to prove that the ``surface'' free energy between two half-lines is finite, although the ``surface'' energy may be unbounded. For d-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if \(\alpha >(1/2)d\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 4046872
zbMATH Keywords: long-range interactions
zbMATH Keywords: relative entropy
zbMATH Keywords: free energy
zbMATH Keywords: phase transition
zbMATH Keywords: ferromagnetic interaction
author: A. C. D. van Enter
author: J. Leo van Hemmen
MaRDI profile type: Publication
cites work: Existence of a phase-transition in a one-dimensional Ising ferromagnet
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cites work: Q5563135
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cites work: Ergodic theory. Introductory lectures
full work available at URL: https://doi.org/10.1007/bf01007972
OpenAlex ID: W2099263471
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description: scientific article; zbMATH DE number 4046872