We study the three-spin model and the Ising spin glass in a field using
Migdal-Kadanoff approximation. The flows of the couplings and fields indicate
no phase transition, but they show even for the three-spin model a slow
crossover to the asymptotic high-temperature behaviour for strong values of the
couplings. We also evaluated a quantity that is a measure of the degree of
non-self-averaging, and we found that it can become large for certain ranges of
the parameters and the system sizes. For the spin glass in a field the maximum
of non-self-averaging follows for given system size a line that resembles the
de Almeida-Thouless line. We conclude that non-self-averaging found in
Monte-Carlo simulations cannot be taken as evidence for the existence of a
low-temperature phase with replica-symmetry breaking. Models similar to the
three-spin model have been extensively discussed in order to provide a
description of structural glasses. Their theory at mean-field level resembles
the mode-coupling theory of real glasses. At that level the one-step replica
symmetry approach breaking predicts two transitions, the first transition being
dynamical and the second thermodynamical. Our results suggest that in real
finite dimensional glasses there will be no genuine transitions at all, but
that some features of mean-field theory could still provide some useful
insights.Comment: 11 pages, 11 figure
