Discontinuity of the magnetization in one-dimensional 1/absolute value x-y/sup 2/ Ising and Potts models
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), Vol. 50:1-2
- Country of Publication:
- United States
- Language:
- English
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SUPERCONDUCTIVITY AND SUPERFLUIDITY
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CRYSTAL MODELS
MAGNETIZATION
ORDER-DISORDER TRANSFORMATIONS
CORRELATION FUNCTIONS
FERROMAGNETISM
HAMILTONIANS
INTERACTION RANGE
INVARIANCE PRINCIPLES
ISING MODEL
ONE-DIMENSIONAL CALCULATIONS
ORDER PARAMETERS
QUANTUM MECHANICS
SPIN
STATISTICAL MECHANICS
ANGULAR MOMENTUM
DISTANCE
FUNCTIONS
MAGNETISM
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MECHANICS
PARTICLE PROPERTIES
PHASE TRANSFORMATIONS
QUANTUM OPERATORS
656002* - Condensed Matter Physics- General Techniques in Condensed Matter- (1987-)
657002 - Theoretical & Mathematical Physics- Classical & Quantum Mechanics