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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics