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

Abstract
We consider an Ising chain with Hamiltonian H=Σi>j(εijSiSj)(a|ij|)σ, where the εij are independent random variables. We find a phase transition for 12<σ<1. For 12<σ<23 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 23.