  A quasi one--dimensional system of trapped, repulsively interacting atoms
(e.g., an ion chain) exhibits a structural phase transition from a linear chain
to a zigzag structure, tuned by reducing the transverse trap potential or
increasing the particle density. Since it is a one dimensional transition, it
takes place at zero temperature and therefore quantum fluctuations dominate. In
[Fishman, et al., Phys. Rev. B 77, 064111 (2008)] it was shown that the system
close to the linear-zigzag instability is described by a $\phi^4$ model. We
propose a mapping of the $\phi^4$ field theory to the well known Ising chain in
a transverse field, which exhibits a quantum critical point. Based on this
mapping, we estimate the quantum critical point in terms of the system
parameters. This estimate gives the critical value of the transverse trap
frequency for which the quantum phase transition occurs, and which has a
finite, measurable deviation from the critical point evaluated within the
classical theory. A measurement is suggested for atomic systems which can probe
the critical trap frequency at sufficiently low temperatures T. We focus in
particular on a trapped ion system, and estimate the implied limitations on T
and on the interparticle distance. We conclude that the experimental
observation of the quantum critical behavior is in principle accessible.
