### Abstract

I use Israel's methods to prove new theorems of "ubiquitous pathology" for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ρ{variant} be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ_{k}} of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ρ{variant}_{k}, such that {ρ{variant}_{k}} converges to ρ{variant}. In certain situations the perturbations Φ_{k}-Φ can be chosen to lie in a cone of "antiferromagnetic pair interactions." I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r^{2} interaction (Thouless effect).

Original language | English (US) |
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Pages (from-to) | 327-336 |

Number of pages | 10 |

Journal | Communications in Mathematical Physics |

Volume | 86 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1982 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*86*(3), 327-336. https://doi.org/10.1007/BF01212172

**More surprises in the general theory of lattice systems.** / Sokal, Alan D.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 86, no. 3, pp. 327-336. https://doi.org/10.1007/BF01212172

}

TY - JOUR

T1 - More surprises in the general theory of lattice systems

AU - Sokal, Alan D.

PY - 1982/9

Y1 - 1982/9

N2 - I use Israel's methods to prove new theorems of "ubiquitous pathology" for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ρ{variant} be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φk} of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ρ{variant}k, such that {ρ{variant}k} converges to ρ{variant}. In certain situations the perturbations Φk-Φ can be chosen to lie in a cone of "antiferromagnetic pair interactions." I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r2 interaction (Thouless effect).

AB - I use Israel's methods to prove new theorems of "ubiquitous pathology" for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ρ{variant} be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φk} of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ρ{variant}k, such that {ρ{variant}k} converges to ρ{variant}. In certain situations the perturbations Φk-Φ can be chosen to lie in a cone of "antiferromagnetic pair interactions." I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r2 interaction (Thouless effect).

UR - http://www.scopus.com/inward/record.url?scp=0042104688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042104688&partnerID=8YFLogxK

U2 - 10.1007/BF01212172

DO - 10.1007/BF01212172

M3 - Article

AN - SCOPUS:0042104688

VL - 86

SP - 327

EP - 336

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -