# Classical and Quantum Gases on a Semiregular Mesh
## 3 Hard-Core Bosons on a Spherical Mesh: Extended Bose-Hubbard Model
### Numerical Results  
Until now, the value of \(V_{2}\) was arbitrary. For the sake of example, we henceforth take \(\gamma=V_{2}/V_{1}=2/3\). We note that the case \(\gamma=1\) was considered in [45]. We have first solved Equation (23) numerically for \(T=0\) and various \(\mu\) values, being careful that the minimum \(\Omega\) solution is picked out in each case. The resulting phase diagram is shown in Figure 9. In addition to the \"empty\" phase (\(\rho_{\rm A}=\rho_{\rm B}=\rho_{\rm C}=0\)) and the \"full\" phase (\(\rho_{\rm A}=\rho_{\rm B}=\rho_{\rm C}=1\)), we observe an icosahedral phase (\(\rho_{\rm A}=1,\rho_{\rm B}=\rho_{\rm C}=0\)) and a dodecahedral phase (\(\rho_{\rm A}=0,\rho_{\rm B}=\rho_{\rm C}=1\)). All these phases are insulating (\(\phi_{\rm A}=\phi_{\rm B}=\phi_{\rm C}=0\)) and incompressible (i.e., the density is constant throughout the phase). Upon increasing \(t\) at fixed \(\mu\) a supersolid phase invariably appears, characterized by \(\rho_{\rm A}\
eq\rho_{\rm B}=\rho_{\rm C}\). We also note that our phase diagram is symmetric around \(\mu=5/2\). Indeed, we see from Equation (23) that \(\rho_{\rm A,B}(\mu)=1-\rho_{\rm A,B}(5-\mu)\) and \(\phi_{\rm A,B}(\mu)=\phi_{\rm A,B}(5-\mu)\), and an analogous symmetry property holds for (22). It is worth stressing the similarities and differences between Figure 9 and the phase diagram of hard-core bosons on a triangular lattice [33; 47; 48]: The overall structure is the same, but the nature of the condensed phase at large \(t\) is different, being herein supersolid rather than superfluid--owing to the frustration effect associated with the existence of inequivalent nodes. We actually distinguish four different supersolid phases (see Figure 10, where the OPs are plotted for two representative values of \(t\)). While \(\phi_{\rm B}\) is always slightly larger than \(\phi_{\rm A}\), in the region between the ICO and DOD lobes (where, in particular, \(t<0.266\)) we find \(\rho_{\rm A}>\rho_{\rm B}\) for \(\mu<5/2\) (supersolid 1b) and \(\rho_{\rm A}<\rho_{\rm B}\) for \(\mu>5/2\) (supersolid 2b). Outside the lobes, we instead find \(\rho_{\rm A}<\rho_{\rm B}\) for \(\mu<5/2\) (supersolid 1) and \(\rho_{\rm A}>\rho_{\rm B}\) for \(\mu>5/2\) (supersolid 2), namely the densities are in reverse order with respect to the reference \"solid\" phase. For \(t\lesssim 0.278\), the values of \(\rho_{\rm A}\) and \(\rho_{\rm B}\) jump discontinuously at \(\mu=5/2\), signaling that the phase transitions along this line are first-order. A further first-order line runs vertically near \(t=0.266\), separating the supersolid phases 1b and 2b from the supersolid phases 1 and 2, respectively. Finally, there are four second-order transition lines: the two lines separating \"empty\" and \"full\" from the adjacent supersolid, the descending part of the boundary between ICO and supersolid 1b, and the ascending part of the boundary between supersolid 2b and DOD. Figure 9: DA phase diagram of the extended BH model at \(T=0\), for \(\gamma_{T}=2/3\). The open dots mark transition points. The dashed red curves are the continuous-transition loci derived in the text (cf. Equation (27)). The black dashed line marks the passing from supersolid 1 to 2 for \(t\gtrsim 0.278\). Here the system is superfluid (see right panel of Figure 10). The remaining black lines represent first-order transitions. Imposing B-C symmetry, we may simplify Equation (23) and then determine the equations for the continuous-transition loci, following the same procedure as illustrated in Ref. [45]. We eventually find:  
\[\mu = -\frac{3+\sqrt{69}}{2}t\] (\"empty\"-supersolid boundary) ; \[\mu = 5V_{1}+\frac{3+\sqrt{69}}{2}t\] (\"full\"-supersolid boundary) ; \[\mu = \frac{3V_{1}-3t\pm\sqrt{9V_{1}^{2}-18V_{1}t-51t^{2}}}{2}\] (ICO-supersolid boundary) ; \[\mu = \frac{7V_{1}+3t\pm\sqrt{9V_{1}^{2}-18V_{1}t-51t^{2}}}{2}\] (DOD-supersolid boundary).