  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.
