  We study the crossover from classical to quantum phase transitions at zero
temperature within the framework of $\phi^4$ theory. The classical transition
at zero temperature can be described by the Landau theory, turning into a
quantum Ising transition with the addition of quantum fluctuations. We perform
a calculation of the transition line in the regime where the quantum
fluctuations are weak. The calculation is based on a renormalization group
analysis of the crossover between classical and quantum transitions, and is
well controlled even for space-time dimensionality $D$ below 4. In particular,
for $D=2$ we obtain an analytic expression for the transition line which is
valid for a wide range of parameters, as confirmed by numerical calculations
based on the Density Matrix Renormalization Group. This behavior could be
tested by measuring the phase diagram of the linear-zigzag instability in
systems of trapped ions or repulsively-interacting dipoles.
