Evidence that the de Almeida–Thouless transition disappears below six dimensions

One of the key predictions of Parisi's broken replica symmetry theory of spin glasses is the existence of a phase transition in an applied field to a state with broken replica symmetry. This transition takes place at the de Almeida–Thouless (AT) line in the h-T plane. We have studied this line in the power-law diluted Heisenberg spin glass in which the probability that two spins separated by a distance r interact with each other falls as 1/r2σ. In the presence of a random vector field of variance hr2 the phase transition is in the universality class of the Ising spin glass in a field. Tuning σ is equivalent to changing the dimension d of the short-range system, with the relation being d=2/(2σ‑1) for σ<2/3. We have found by numerical simulations that hAT2∼(2/3‑σ) implying that the AT line does not exist below six dimensions and that the Parisi scheme is not appropriate for spin glasses in three dimensions.