On a \(C^*\)-algebra approach to phase transition in the two-dimensional Ising model
type: Item
instance of: scholarly article
title: On a \(C^*\)-algebra approach to phase transition in the two-dimensional Ising model
zbMATH Open document ID: 0535.46046
DOI: 10.1007/BF01206017
author: David E. Evans
author: Huzihiro Araki
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: The phase transition of the two-dimensional Ising model is investigated in terms of the state on the \(C^*\)-algebra of Pauli spins on a one- dimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two- dimensional Ising model (with nearest neighbour interaction) through the transfer matrix method. It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary is derived. It is also shown that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operators \(E_ 1\) and \(E_ 2\) (on the direct sum of two copies of the basic Hilbert space) satisfy the following two conditions: (1) \(E_ 1-E_ 2\) is in the Hilbert-Schmidt class, (2) \(E_ 1\wedge(1-E_ 2)\) has an even dimension, where the even- oddness of \(\dim E_ 1\wedge(1-E_ 2)\) is called \({\mathbb{Z}}_ 2\)-index of \(E_ 1\) and \(E_ 2\) and is continuous in \(E_ 1\) and \(E_ 2\) relative to the norm topology (as long as the condition (1) is satisfied).
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 82B10
zbMATH DE Number: 3849889
zbMATH Keywords: CAR algebra
zbMATH Keywords: Fermion algebra
zbMATH Keywords: \(Z_ 2\)-index
zbMATH Keywords: quasifree states
zbMATH Keywords: phase transition of the two-dimensional Ising model
zbMATH Keywords: state on the \(C^*\)- algebra of Pauli spins on a one-dimensional lattice
zbMATH Keywords: thermodynamic limit of the Gibbs ensemble
zbMATH Keywords: Pauli spin algebra
zbMATH Keywords: Fock state of the Fermion algebra
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01206017
OpenAlex ID: W2166594857
description: scientific article; zbMATH DE number 3849889