  A string of trapped ions at zero temperature exhibits a structural phase
transition to a zigzag structure, tuned by reducing the transverse trap
potential or the interparticle distance. The transition is driven by
transverse, short wavelength vibrational modes. We argue that this is a quantum
phase transition, which can be experimentally realized and probed. Indeed, by
means of a mapping to the Ising model in a transverse field, we estimate the
quantum critical point in terms of the system parameters, and find a finite,
measurable deviation from the critical point predicted by the classical theory.
A measurement procedure is suggested which can probe the effects of quantum
fluctuations at criticality. These results can be extended to describe the
transverse instability of ultracold polar molecules in a one dimensional
optical lattice.
