QUROPE Quantum Information Processing and Communication in Europe

A string of trapped interacting ions at zero temperature ($T=0$) exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or increasing the particle density. The transition is driven by transverse, short wavelength vibrational modes. We propose a quantum field--theoretical description of this transition by the one dimensional Ising model in a transverse field. Based on the mapping to this model, 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.