  Perovskites like LaAlO3 (or SrTiO3) undergo displacive structural phase
transitions from a cubic crystal to a trigonal (or tetragonal) structure. For
many years, the critical exponents in both these types of transitions have been
fitted to those of the isotropic three-components Heisenberg model. However,
field theoretical calculations showed that the isotropic fixed point of the
renormalization group is unstable, and renormalization group iterations flow
either to a cubic fixed point or to a fluctuation-driven first-order
transition. Here we show that these two scenarios correspond to the cubic to
trigonal and to the cubic to tetragonal transitions, respectively. In both
cases, the critical behavior is described by slowly varying effective critical
exponents, which exhibit universal features. For the trigonal case, we predict
a crossover of the effective exponents from their Ising values to their cubic
values (which are close to the isotropic ones). For the tetragonal case, the
effective exponents can have the isotropic values over a wide temperature
range, before exhibiting large changes en route to the first-order transition.
New renormalization group calculations near the isotropic fixed point in three
dimensions are presented and used to estimate the effective exponents, and
dedicated experiments to test these predictions are proposed. Similar
predictions apply to cubic magnetic and ferroelectric systems.
