Crystals of repulsively interacting cold ions in planar traps form hexagonal lattices, which undergo

a buckling instability towards a multi-layer structure as the transverse trap frequency is

reduced. The buckled structure is composed of three planes, whose separation increases continuously

from zero. In the work I will present [1], we study the effects of thermal and quantum fluctuations

by mapping this structural instability to the six-state clock model. A prominent implication of

this mapping is that at finite temperature T, fluctuations split the buckling instability into two

thermal transitions, accompanied by the appearance of an intermediate critical phase. This

phase is characterized by quasi-long-range order in the spatial tripartite pattern. It is manifested

by broadened Bragg peaks at new wave vectors, whose line-shape provides a direct

correlations in the critical phase. A quantum phase transition is found at the largest value of

the critical transverse frequency, where the critical intermediate phase shrinks to zero. Moreover,

within the ordered phase, we predict a crossover from classical to quantum behavior,

signifying the emergence of an additional characteristic scale for clock order. We discuss the experimental

realization, and propose that within accessible technology, such experiments can provide a direct probe of the rich phase diagram of the quantum clock model, not easily observable in condensed matter analogues. This highlights

the potential for ionic crystals to serve as simulators of complex models in

statistical mechanics and quantum field theory.

[1] Daniel Podolsky, Efrat Shimshoni, Giovanna Morigi and Shmuel Fishman, Phys. Rev. X

6, 031025 (2016).