Ground state of weakly repulsive soft-core bosons on a sphere

We study a system of penetrable bosons embedded in a spherical surface. Under the assumption of weak interaction between the particles, the ground state of the system is, to a good approximation, a pure condensate. We employ thermodynamic arguments to investigate, within a variational ansatz for the single-particle state, the crossover between distinct finite-size “phases” in the parameter space spanned by the sphere radius and the chemical potential. In particular, for radii up to a few interaction ranges we examine the stability of the fluid phase with respect to a number of crystal-like arrangements having the symmetry of a regular or semiregular polyhedron. We find that, while quantum fluctuations keep the system fluid at low density, upon compression it eventually becomes inhomogeneous, i.e., particles gather together in clusters. As the radius increases, the nature of the high-density aggregate varies and we observe a sequence of transitions between different cluster phases (“solids”), whose underlying rationale is to maximize the coordination number of clusters, while ensuring at the same time the proper distance between each neighboring pair. We argue that, at least within our mean-field description, every cluster phase is supersolid.

Figure 1

Scatter plot of the surface $\mathcal{S}:r=T_0^0+\beta T_6^0(\theta ,\varphi )$, for $\beta =0.1$ (left) and 0.3 (right), which clearly reveals the icosahedral symmetry. As $\beta$ increases, clusters become more localized on the sphere. These plots were obtained by picking ${10}^4$ vectors $(x,y,z)$ at random over the unit sphere and multiplying each by the respective value of $T_0^0+\beta T_6^0$. The surface $\mathcal{S}$ is the envelope of the free ends of the vectors. The red (blue) points are those $(x,y,z)$ where $T_6^0>0$ ($T_6^0<0$, respectively).

Figure 2

Excess energy (units of $e_0$) of the PSM “solid” relative to the fluid, plotted as a function of $\beta$ for $R=1.45\sigma$ and three densities $\rho$ (from now on reported in reduced units ${\sigma }^{-2}{e}_0/\epsilon$): from top to bottom, $\rho =14.5$ (black), 15 (blue), and 15.5 (red).

Figure 3

Ten circumscribable polyhedra considered in this work. Each of them provides the underlying skeleton of a possible $T=0$ phase in a system of spherical bosons (i.e., clusters are centered at the vertices of the polyhedron). First row: the five Platonic solids (from left to right: tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Second row: three Archimedean solids (from left to right: cuboctahedron, rombicuboctahedron, and snub cube) and two Catalan solids (tetrakis hexahedron and pentakis dodecahedron).

Figure 4

PSM bosons on a sphere at $T=0$: an example of the determination of the phase-transition point (this case refers to an inhomogeneous phase having the symmetry of a snub cube, for $R=1.9\sigma$). Left: excess energy (units of $e_0$) vs $\alpha$ for a number of reduced densities in the range from 12 to 17 [each plotted point is the result of an average over 20 independent estimates of the energy given by Eq. (4.1)]. In the inset, we show a magnification of the $\alpha$ interval from 3 to 7, made in order to highlight the magnitude of the error bars. Full lines are spline interpolants. Right: the Maxwell-like construction (fluid, red crosses; solid, blue circles) allowing one to determine the exact transition threshold.

Figure 5

PSM bosons on a sphere at $T=0$: excess energy $\mathrm{\Delta }\mathcal{E}$ (units of $e_0$) of the icosahedral cluster phase, plotted as a function of $\alpha$ for $R=1.4\sigma$ and $\rho =14$ (units of ${\sigma }^{-2}{e}_0/\epsilon$). The full lines were obtained from Eq. (4.3) by including in the two series only terms up to $l=l_{\max }$. The data points are results from MC integration.

Figure 6

PSM bosons on a sphere at $T=0$: fluid-solid transition lines according to variational theory [notice that the same wave function (3.2) was also used in the two “Catalan” cases, despite the vertices of the latter polyhedra being of two different kinds]. Continuous freezing is marked by a dashed line, whereas full lines indicate first-order freezing. The horizontal purple line marks two-dimensional freezing [19]. The pink region is where the fluid is unstable (see text).

Figure 7

PSM bosons on a sphere at $T=0$: excess energy $\mathrm{\Delta }\mathcal{E}$ (units of $e_0$) of the cubic cluster phase, plotted as a function of $\alpha$ for $R=1.05\sigma$ and a number of densities in the range enclosing the two-step transition. The lines are “exact” results obtained from Eq. (4.3) by including in both series all terms up to $l=16$. The data points are results from MC integration, relative to a reduced density of $\rho =23.5$ (notice the difference in energy scale between this picture and Fig. 4).

Figure 8

PSM bosons on a sphere at $T=0$: we highlight here the region of low $R$ values, where the stable cluster phase has tetrahedral symmetry (orange). Its melting line consists of a continuous portion (dashed line) and a first-order portion (full line). There is a narrow range of radii where freezing proceeds in two steps. The emergence of tetrahedral ordering at high density is not exclusive of soft particles, since it is also found in hard particles [58].

Figure 9

PSM bosons on a sphere at $T=0$: freezing and melting lines according to variational theory. The colorful shadow regions represent fluid-solid coexistence regions, while dashed lines indicate continuous freezing. The horizontal purple stripe marks the region of coexistence between the planar fluid and the triangular cluster crystal [19].

Figure 10

PSM bosons on a sphere at $T=0$. Left: $\alpha$ value at melting for the various solid phases, plotted as a function of the radius (the full lines represent fourth-order polynomial interpolants through the data points). We stress that the imprecise determination of the location of the energy minimum only barely affects the estimate of the minimum itself. Right: average number $N_{\mathrm{cl}}$ of particles per cluster at melting. The purple straight line at 25.827 represents the value of $N_{\mathrm{cl}}$ for the triangular cluster crystal [19].

Figure 11

PSM bosons on a sphere at $T=0$: phase diagram according to variational theory. As $R$ increases, the stable cluster phase changes accordingly, successively taking the symmetry of a tetrahedron (T), octahedron (O), cube (C), icosahedron (I), tetrakis hexahedron (TH), snub cube (SC), and pentakis dodecahedron (PD). Solid-solid lines were drawn on the basis of the few points (two or three) which we have been able to locate on iso-$R$ lines through a comparison between grand potentials. Since in the pink regions the fluid is unstable, there are ranges of $R$ where the stable high-density phase of the system is actually unknown.

Figure 12

Profile of ${\mu }_{\mathrm{inst}}\left(R\right)$, representing the upper stability threshold of the fluid phase, for three distinct interactions (PSM, GEM-8, and GEM-4). In preparing this figure, only $l$ values from 2 to 12 were considered; for each $l$, a specific minimum arises in ${\mu }_{\mathrm{inst}}\left(R\right)$, as marked below the graph for the PSM case.