Interacting Langevin Equation And A Microscopic Mechanism For Kinks In Trapped Ions

The transition between the linear phase and zigzag phase in an ion trap is widely used to study the mechanism of the second-order continuous phase transition. For a linear ion chain, during the quenching across the critical point, kinks are formed. The relation between the density of kinks and quenching rate can be described by the Kibble-Zurek mechanism. In this work, we consider a one-dimensional trapped-ion chain with the motion on the phase-transition plane. Using the interacting Langevin equation, we show that when one ion is kicked out from the chain, the off-axis action caused by this ion will propagate to others one by one with a defined velocity. This velocity will be the same as the sound velocity in some extreme cases but has obviously different meaning. In the presence of finite temperature, which is modeled using a many-body stochastic Langevin equation to unveil a microscopic mechanism for kink formations in this model, we find that the kink can be formed when the diffusion radius is larger than the mean displacement of the ions along the radial direction. This criterion provides an alternative mode for the formation of kinks in realistic experiments. The predictions based on this mechanism can be qualitatively consistent with those from the Kibble-Zurek mechanism in homogeneous structures; however, the microscopic details, such as kink formation and kink disappearance can be seen much clearer in our simplified model.