Discontinuity of the magnetization in one-dimensional 1/absolute value x-y/sup 2/ Ising and Potts models

Journal Article · Fri Jan 01 04:00:00 EST 1988 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01022985· OSTI ID:7180343

Aizenman, M; Chayes, J T; Chayes, L; Newman, C M

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/sub x,y/ approx. = const/absolute value x-y/sup 2/. For translation-invariant systems with well-defined lim/sub x ..-->.. infinity/x/sup 2/J/sub x/ = J/sup +/ (possibly O or infinity) we establish: (1) there is no long-range order at inverse temperatures ..beta.. with ..beta..J/sup +/ less than or equal to1. (2) If ..beta..J/sup +/ > q, then by sufficiently increasing J/sub 1/ the spontaneous magnetization M is made positive. (3) In models with 0 < J/sup +/ < infinity the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys M(..beta../sub c/) less than or equal to 1/(..beta../sub c/J/sup +/)/sup 1/2. (4) For Ising (q =2) models with J/sup +/ < infinity, it is noted that the correlation function decays as (..beta..) approx. = c(..beta..)/absolute value x-y/sup 2/ whenever ..beta.. < ..beta../sub 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.

• Research Organization: New York Univ., NY (USA)
• OSTI ID: 7180343
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 50:1-2; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

656002* -- Condensed Matter Physics-- General Techniques in Condensed Matter-- (1987-)
657002 -- Theoretical &amp; Mathematical Physics-- Classical &amp; Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ANGULAR MOMENTUM
CORRELATION FUNCTIONS
CRYSTAL MODELS
DISTANCE
FERROMAGNETISM
FUNCTIONS
HAMILTONIANS
INTERACTION RANGE
INVARIANCE PRINCIPLES
ISING MODEL
MAGNETISM
MAGNETIZATION
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MECHANICS
ONE-DIMENSIONAL CALCULATIONS
ORDER PARAMETERS
ORDER-DISORDER TRANSFORMATIONS
PARTICLE PROPERTIES
PHASE TRANSFORMATIONS
QUANTUM MECHANICS
QUANTUM OPERATORS
SPIN
STATISTICAL MECHANICS