Buckling Transitions and Clock Order of Two-Dimensional Coulomb Crystals

Crystals of repulsively interacting ions in planar traps form hexagonal lattices, which undergo a buckling instability towards a multilayer structure as the transverse trap frequency is reduced. Numerical and experimental results indicate that the new structure is composed of three planes, whose separation increases continuously from zero. 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, 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 measurement of the temperature-dependent exponent $\eta (T)$ characteristic of the power-law correlations in the critical phase. A quantum phase transition is found at the largest value of the critical transverse frequency: Here, 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 experimental realizations with trapped ions and polarized dipolar gases, 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. Therefore, this work highlights the potential for ionic and dipolar systems to serve as simulators for complex models in statistical mechanics and condensed matter physics.

Popular Summary

Advances in techniques to cool and manipulate interacting atomic systems have yielded new access to a rich variety of phases of matter. Scientists now have the exciting opportunity to conduct controlled studies of classical and quantum phase transitions whose nature, remarkably, is predicted to depend only on fundamental properties such as the symmetries and dimensionality of the system. Two-dimensional systems, in particular, support critical phases that are intermediate between disordered and fully ordered. This behavior arises because of the key role played by fluctuations in the form of topological excitations. However, these phases are typically difficult to observe in solid-state systems. Here, we demonstrate that two-dimensional ionic crystals can serve as a playground to realize and directly probe such critical phases.

We theoretically show that, as the potential that confines the ions to two dimensions weakens, the ions exhibit an instability toward the formation of a buckled pattern in which the new structure has points on three separate planes. By considering the effects of fluctuations in the ionic positions arising from both thermal and quantum effects, we are able to map the problem into the six-state clock model. This finding implies that the buckling instability is split into two separate thermal transitions accompanied by the appearance of an intermediate critical phase. The temperatures necessary to recover these effects are within reach of current cooling systems. In the limit of zero temperature, the critical phase shrinks, and the two thermal transitions merge into a single quantum critical point.

We expect that our results will stimulate novel experimental exploration of two-dimensional critical phenomena using, for example, trapped ions or polarized dipolar gases.

Figure 1

Ions in a planar trap form a hexagonal lattice, which undergoes a mechanical instability as a function of the temperature and/or the transverse frequency. Left panel: Height pattern in the ordered phase. The hexagonal lattice is split into three sublattices, one of which stays in the $z=0$ plane (gray circles), one is raised above it (red), and one is lowered below it (blue). There are six inequivalent configurations, corresponding to the $3!$ possible height assignments to the sublattices. Right panel: The first Brillouin zone of the undistorted hexagonal lattice is shown, together with the wave vectors $\mathbf{K}$ and ${\mathbf{K}}^{*}=-\mathbf{K}$ corresponding to the height pattern.

Figure 2

Phase diagram of the structural instability of a planar ion crystal in the parameter space determined by the temperature $T$ and the coefficient $r$, which is a function of the ratio $\mathcal{K}/C$. The external line at $T_{\mathrm{KT}}$ indicates the transition from a planar structure to a critical phase, where the ions are distributed along three layers and exhibit quasi-long-range order in the transverse direction. The second line at $T_6$ indicates the transition to a long-range ordered buckled phase, where the ions form three hexagonal lattices in the three layers, which are phase locked as shown in Fig. 1. The value $r_c<0$ indicates the critical point at $T=0$. The dashed line at $T_6^{*}$ separates the quantum from the classical critical behaviors. See text for further details.

Figure 3

Interlayer separation $h(T)$ as a function of temperature, based on Eq. (4.9). The zero-temperature value, $h_0$, is of order 1 micron, as shown in Eq. (4.4). Note that the functional form of $h(T)$ is sensitive to the value of $m_0$, even at temperatures well above $m_0$ itself. Throughout, the onset temperature for clock order is taken to be $T_6=10\mathrm{mK}$.