Contours of constant χ_sG in the H-T plane: Mean-field versus droplet theories of Ising spin glasses

One of the central features of the mean-field theory of Ising spin glasses is the de Almeida-Thouless phase transition line in the H-T plane, where the spin-glass susceptibility χ_SG diverges. Contours of constant χ_SG in the paramagnetic phase go to high fields as T→0 in mean-field theory. In contrast, in the droplet theory for short-ranged spin glasses, χ_SG remains finite in a field and the contours go to H=0 as T→0. We have investigated the constant χ_SG contours both in the SK model and for d-dimensional short-ranged spin glasses. For the latter we use transfer matrix methods in d=1 and high-temperature expansions and Monte Carlo simulations in higher d. The results are in good accord with droplet theory for d=1 and 2. For d=3 the evidence remains ambiguous, although extrapolations of high-temperature series are qualitatively different from d=2.