Discontinuity of the magnetization in one-dimensional 1/|x-y|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_x,y≈ const/|x-y|². For translation-invariant systems with well-defined lim_x→∞x²J_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 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(β_c)≥1/(β_cJ⁺)^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.