Exact ground states of one-dimensional long-range random-field Ising magnets

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability p ∼r^-1-σ, where r is the distance between these spins and σ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to L =2¹⁹ via graph theoretical algorithms for four different values of σ ∈{0.25,0.4,0.5,1.0} while varying the strength h of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values h_c(σ) as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for σ =1/2 the data is compatible with a critical random-field strength h_c>0.