  The transformation of the free-energy landscape from smooth to hierarchical
is one of the richest features of mean-field disordered systems. A well-studied
example is the de Almeida-Thouless transition for spin glasses in a magnetic
field, and a similar phenomenon--the Gardner transition--has recently been
predicted for structural glasses. The existence of these
replica-symmetry-breaking phase transitions has, however, long been questioned
below their upper critical dimension, d_u=6. Here, we obtain evidence for the
existence of these transitions in d<d_u using a two-loop calculation. Because
the critical fixed point is found in the strong-coupling regime, we corroborate
the result by resumming the perturbative series with inputs from a three-loop
calculation and an analysis of its large-order behavior. Our study offers a
resolution of the long-lasting controversy surrounding phase transitions in
finite-dimensional disordered systems.

Statistical Mechanics;Disordered Systems and Neural Networks;High Energy Physics - Theory