# Classical and Quantum Gases on a Semiregular Mesh
## 1 Introduction  
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. In the last century, 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 simply 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 then appear. A classic example is a (finite) system of hard particles confined in the surface of a sphere [1]. The sphere topology forces an excess of five-fold coordinated particles over seven-fold ones, leading to high-density packings with defects [2, 3, 4, 5, 6, 7, 8]. We note that bosonic atoms confined in thin spherical shells [9] have already been realized [10, 11] and are currently studied in microgravity [12, 13]. In other cases, frustration is directly embodied in the interaction law--like in spin glasses or in antiferromagnets on a triangular lattice [14, 15].  
In this paper, we consider a discrete system of particles (\"lattice gas\") on a spherical mesh of points, which is chosen such that a rich interplay arises between distinct \"phases\" having the symmetries of a Platonic solid. Clearly, on a finite mesh well-definite phases only exist at zero temperature (\(T=0\)), since for \(T>0\) any phase transition will be smearedout, i.e., replaced by a smooth crossover region. Using a finite mesh, we greatly reduce the computational effort without however making the phase behavior trivial.  
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. The last decades have witnessed a considerable progress in the manipulation of quantum atoms at low temperature, opening the way to a systematic study of correlation effects in many-body systems [16; 17; 18; 19]. While optical lattices [20] are routinely employed in numerous laboratories worldwide as a tool for confinement of quantum atoms, in the last few years a laser technology has been invented, based on the use of optical tweezers [21; 22], which allows virtually any type of structure (not necessarily a lattice) to be realized with cold atoms. We are thus encouraged to consider the quantum (Bose-Hubbard) counterpart of the lattice gas on a spherical mesh, with the explicit purpose to compare their thermal behaviors. In particular, we devise a quantum variational theory that predicts supersolid phases and returns the results of a classical mean-field theory when quantum tunneling is precluded.  
The rest of the paper is organized as follows. In Section 2 we introduce our model and the methods used to investigate its phase behavior. Next, we present our results, first at zero temperature (Section 2.1) and then at finite temperatures (Section 2.2). In Section 3 we deal with the quantum extension of the model in Section 2. Using the decoupling approximation, we not only work out the ground-state diagram (Section 3.1) but also a few finite-temperature properties (Section 3.2). Lastly, we give our concluding remarks in Section 4.