Abstract
The main objective of a statistical mechanical calculation is drawing the phase diagram of a many-body system. In this respect, discrete systems offer the clear advantage over continuum systems of an easier enumeration of microstates, though at the cost of added abstraction. With this in mind, we examine a system of particles living on the vertices of the (biscribed) pentakis dodecahedron, using different couplings for first and second neighbor particles to induce a competition between icosahedral and dodecahedral orders. After working out the phases of the model at zero temperature, we carry out Metropolis Monte Carlo simulations at finite temperature, highlighting the existence of smooth transitions between distinct “phases”. The sharpest of these crossovers are characterized by hysteretic behavior near zero temperature, which reveals a bottleneck issue for Metropolis dynamics in state space. Next, we introduce the quantum (Bose-Hubbard) counterpart of the previous model and calculate its phase diagram at zero and finite temperatures using the decoupling approximation. We thus uncover, in addition to Mott insulating “solids”, also the existence of supersolid “phases” which progressively shrink as the system is heated up. We argue that a quantum system of the kind described here can be realized with programmable holographic optical tweezers.
Highlights
Investigating the behavior of a many-particle system has an undeniable charm: despite microscopic interactions being undirected, various forms of self-organization (“order”) can develop at the macroscale
We focus our attention on the pentakis dodecahedron [23] (PD, see Figure 1), a Catalan solid with 32 vertices obtained by augmenting the dodecahedron with 12 right pyramids on its pentagonal faces, in such a way that the resulting polyhedron is dual to the truncated icosahedron
We note that our phase diagram is symmetric around μ = 5/2
Summary
Investigating the behavior of a many-particle system has an undeniable charm: despite microscopic interactions being undirected, various forms of self-organization (“order”) can develop at the macroscale. Countless examples of emergent order have been described, each with its own practical realization, and many more can be devised by exploring through theory physical situations that are somehow atypical. These indications can stimulate new experimental work or be aimed to clarify and expand the scope of the theory itself. A way to produce novel, unconventional phase behaviors is to consider many-body systems under geometric constraints, since local interactions are frustrated and unusual ground states appear. If a toy model of classical particles on a finite mesh may look somewhat artificial and hardly corresponds to a real-world system, its quantum counterpart might be different.
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