# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh
### Zero-Temperature Phases  
At zero temperature, the stable phase at fixed \(\mu\) is the one minimizing the grand potential \(\Omega\). We expect the absolute minimum \(\Omega\) to be reached in one of a few microstates/configurations, chosen among those exhibiting a homogeneous occupancy of equivalent sites. To be clear, there are five ways to select--out of 20 dodecahedral sites--8 sites forming a cube [24] (similarly, each cube is the union of 2 tetrahedra). Then, the PD vertices are naturally grouped in three sets of equivalent nodes: we call A the set of  
Figure 1: The pentakis dodecahedron has 12 icosahedral vertices (red dots) and 20 dodecahedral vertices (yellow dots). There are five distinct ways to choose eight yellow dots forming a cube—then, the other 12 dodecahedral vertices are said to form a “co-cube”. The short edges of the PD mesh are colored in grey and the long edges in blue. The couplings entering the model Hamiltonian (1) are indicated.  
icosahedral sites, B any set of cubic sites, and C the set comprising the remaining dodecahedral sites (\"co-cubic\" sites). For example, in the icosahedral phase (ICO) only A sites are occupied at \(T=0\); in the dodecahedral phase (DOD) only B and C sites are occupied. In our setting, ICO and DOD play a role analogous to two distinct crystalline phases. By a straightforward count of neighbors, the grand potential of the most relevant phases is readily calculated: in an obvious notation, \(\Omega_{\rm empty}=0\), \(\Omega_{\rm ICO}=-12\mu\), \(\Omega_{\rm COC}=6V_{2}-12\mu\), \(\Omega_{\rm DOD}=30V_{2}-20\mu\), and \(\Omega_{\rm full}=60V_{1}+30V_{2}-32\mu\). The 10-fold-degenerate tetrahedral phase (TET, \(\Omega=-4\mu\)) and the 5-fold-degenerate cubic phase (CUB, \(\Omega=-8\mu\)) are never stable, since they are less stable than \"empty\" or ICO. Further possibilities are configurations where, in addition to icosahedral sites, also a selection of dodecahedral sites are occupied: \(\Omega_{\rm ICO+TET}=12V_{1}-16\mu\), \(\Omega_{\rm ICO+CUB}=24V_{1}-20\mu\), and \(\Omega_{\rm ICO+COC}=36V_{1}+6V_{2}-24\mu\).  
By a lengthy (though elementary) calculation, we arrive at the \(T=0\) phase diagram depicted in Figure 2. Here, PQ, PR, and RS are straight lines with equations \(\mu^{*}=3+3\gamma\), \(\mu^{*}=9-6\gamma\), and \(\mu^{*}=3+(3/2)\gamma\), respectively; the two half-lines departing from the origin, namely OT and OU, have equations \(\mu^{*}=(15/4)\gamma\) and \(\mu^{*}=(3/2)\gamma\), respectively. All the phase boundaries are first-order (since the number of particles changes discontinuously from one phase to the other), except for RT (since both the particle number \(N\) and the energy \(E\) change continuously through it). Looking at Figure 2, we note that: (i) the ICO phase is only stable for \(\gamma>0\), in a region delimited by the \(\mu^{*}=0\) and \(\mu^{*}=3\) lines; (ii) the DOD phase is stable in a wide region of the \(\gamma\)-\(\mu\) plane, bounded from the right by \(\gamma=4/5\) and from the above by \(\mu^{*}=5\); (iii) ICO+CUB and ICO+COC are each stable in an unbounded set of positive \(\gamma\) values, and coexist along the RS line; (iv) for each \(\gamma\), \"empty\" and \"full\" are respectively stable for all values of \(\mu\) that are sufficiently small or sufficiently large.  
Figure 2: Zero-temperature phase diagram of the lattice gas on a PD mesh. The phase boundaries are reported in the main text.