### Abstract

Results from percolation theory are used to study phase transitions in one-dimensional Ising and q-state Potts models with couplings of the asymptotic form J_{x,y}≈ const/|x-y|^{2}. For translation-invariant systems with well-defined lim_{x→∞}x^{2}J_{x}=J^{+} (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperatures β with βJ^{+}≤1. (2) If βJ^{+}>q, then by sufficiently increasing J_{1} the spontaneous magnetization M is made positive. (3) In models with 0<J^{+}<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys M(β_{c})≥1/(β_{c}J^{+})^{1/2}. (4) For Ising (q=2) models with J^{+}<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x-y|^{2} whenever β<β_{c}. Points 1-3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values of q.

Original language | English (US) |
---|---|

Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Journal of Statistical Physics |

Volume | 50 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1988 |

### Fingerprint

### Keywords

- 1/r interactions one dimension
- discontinuous transition
- Fortuin-Kasteleyn representation
- Ising model
- percolation
- Potts models
- Thouless effect

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*50*(1-2), 1-40. https://doi.org/10.1007/BF01022985

**Discontinuity of the magnetization in one-dimensional 1/|x-y|2 Ising and Potts models.** / Aizenman, M.; Chayes, J. T.; Chayes, L.; Newman, C. M.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 50, no. 1-2, pp. 1-40. https://doi.org/10.1007/BF01022985

}

TY - JOUR

T1 - Discontinuity of the magnetization in one-dimensional 1/|x-y|2 Ising and Potts models

AU - Aizenman, M.

AU - Chayes, J. T.

AU - Chayes, L.

AU - Newman, C. M.

PY - 1988/1

Y1 - 1988/1

N2 - Results from percolation theory are used to study phase transitions in one-dimensional Ising and q-state Potts models with couplings of the asymptotic form Jx,y≈ const/|x-y|2. For translation-invariant systems with well-defined limx→∞x2Jx=J+ (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperatures β with βJ+≤1. (2) If βJ+>q, then by sufficiently increasing J1 the spontaneous magnetization M is made positive. (3) In models with 0<J+<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys M(βc)≥1/(βcJ+)1/2. (4) For Ising (q=2) models with J+<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x-y|2 whenever β<βc. Points 1-3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values of q.

AB - Results from percolation theory are used to study phase transitions in one-dimensional Ising and q-state Potts models with couplings of the asymptotic form Jx,y≈ const/|x-y|2. For translation-invariant systems with well-defined limx→∞x2Jx=J+ (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperatures β with βJ+≤1. (2) If βJ+>q, then by sufficiently increasing J1 the spontaneous magnetization M is made positive. (3) In models with 0<J+<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys M(βc)≥1/(βcJ+)1/2. (4) For Ising (q=2) models with J+<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x-y|2 whenever β<βc. Points 1-3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values of q.

KW - 1/r interactions one dimension

KW - discontinuous transition

KW - Fortuin-Kasteleyn representation

KW - Ising model

KW - percolation

KW - Potts models

KW - Thouless effect

UR - http://www.scopus.com/inward/record.url?scp=0002205186&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002205186&partnerID=8YFLogxK

U2 - 10.1007/BF01022985

DO - 10.1007/BF01022985

M3 - Article

AN - SCOPUS:0002205186

VL - 50

SP - 1

EP - 40

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -