Dilute random-field Ising models and uniform-field antiferromagnets

Abstract

The order-parameter susceptibility χ of dilute Ising models with random fields and dilute antiferromagnets in a uniform field are studied for low temperatures and fields with use of low-concentration expansions, scaling theories, and exact solutions on the Cayley tree to elucidate the behavior near the percolation threshold at concentration $p_c$. On the Cayley tree, as well as for d>6, both models have a zero-temperature susceptibility which diverges as ‖ln($p_c$-p)‖. For spatial dimensions 1d$p_c$-p${)}^{\mathrm{-}\left({\gamma }_p\mathrm{-}{\beta }_p\right)/2\mathrm{}}$, where ${\gamma }_p$ and ${\beta }_p$ are percolation exponents associated with the susceptibility and order parameter. At d=6, the susceptibilities diverge as ‖ln($p_c$-p)${\Vert }^{9/7}$. For d=1, exact results show that the two models have different critical exponents at the percolation threshold. The (finite-length) series at d=2 seems to exhibit different critical exponents for the two models. At p=$p_c$, the susceptibilities diverge in the limit of zero field h as χ∼h-(${\gamma }_p$-${\beta }_p$)/(${\gamma }_p$+${\beta }_p$ ) for d${\Vert }^{9/7}$ for d=6, and as χ∼‖lnh‖ for d>6.