B. Drossel et al., Spin glasses without time-reversal symmetry and the absence of a genuine structural glass transition, PHYS REV E, 62(6), 2000, pp. 7690-7699
We study the three-spin model and the Ising spin glass in a held using the
Migdal-Kadanoff approximation. The flows of the couplings and fields indica
te no phase transition, but they show even for the three-spin model a slow
crossover to the asymptotic high-temperature behavior for large values of t
he coupling. We have also evaluated a quantity that is a measure of the deg
ree of non-self-averaging, and we found that it can become large for certai
n ranges of the parameters and the system sizes. For a spin glass in a fiel
d the maximum of non-self-averaging follows a line for given system size th
at resembles the de Almeida-Thouless Line. We conclude that non-self-averag
ing found in Monte Carlo simulations cannot be taken as evidence for the ex
istence of a low-temperature phase with replica symmetry breaking. Models s
imilar to the three-spin model have been extensively discussed in order to
provide a description of structural glasses. Their theory at mean-field lev
el resembles the mode-coupling theory of real glasses. At that level the ap
proach via one-step replica symmetry breaking predicts two transitions, the
first transition being dynamic and the second thermodynamic. Our results s
uggest that in real finite-dimensional glasses there will be no genuine tra
nsitions at all, but that some features of mean-field theory could still pr
ovide some useful insights.