Evidence that the AT 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/r^2σ. In the presence of a random vector-field of variance h_r^2 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 h_AT^2 ∼ (2/3 -σ) implying that the AT line does not exist below 6 dimensions and that the Parisi scheme is not appropriate for spin glasses in three dimensions.