One-dimensional spin-glass model with long-range random interactions

We consider an Ising chain with Hamiltonian H=Σi>j^(εᵢⱼSᵢSⱼ)(a|i-j|)^σ, where the εᵢⱼ are independent random variables. We find a phase transition for 1/2<σ<1. For 1/2<σ<2/3 the critical exponents exhibit mean-field classical behavior. Near σ=1 we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an ε expansion around σ=1 and 2/3.