Quantum Zigzag Transition in Ion Chains

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.

Figure 1

Phase diagram for a linear-zigzag transition, according to the mapping to the quantum Ising transition in 1D, where $T$ is the sample temperature and the dimensionless parameter $\epsilon$ is tuned by the confining potential or the interparticle distance. The classical instability of the linear chain [13] occurs at $\epsilon =0$. The quantum critical point, at ${\epsilon }_c>0$, separates the linear from the zigzag phase at $T=0$. For $0<\epsilon <{\epsilon }_c$ quantum fluctuations dominate, and the crystal is in the linear (disordered) phase. The dashed lines indicate the boundaries of the quantum critical region.