We investigate the temperature dependence of the low-frequency vibrational spectrum of an ion Coulomb crystal near the linear to zigzag transition. For zero temperature, the linear to zigzag transition is a second-order phase transition that exhibits, amongst other things, a soft mode, which is a vibrational mode with zero frequency at the transition. Experimentally and numerically, a non-zero frequency of the soft mode is measured for T>0. In the zigzag phase a fast switching between the two possible ground states is observed near the transition.
Based on this observation, we develop a simple analytical model for the frequency of the soft mode, that extents the harmonic approximation of the linear chain with averaged higher-order corrections from high-frequency vibrational modes. This model agrees well with numerical and experimental findings and might be extendable to other symmetry-breaking transitions at finite temperature. Our analysis finds that ground-state laser cooling of a soft mode near a phase transition requires cooling of high-frequency modes, as well.
The publication can be found here.