Ultracold Bosons on a Regular Spherical Mesh

Abstract

Here, the zero-temperature phase behavior of bosonic particles living on the nodes of a regular spherical mesh (“Platonic mesh”) and interacting through an extended Bose-Hubbard Hamiltonian has been studied. Only the hard-core version of the model for two instances of Platonic mesh is considered here. Using the mean-field decoupling approximation, it is shown that the system may exist in various ground states, which can be regarded as analogs of gas, solid, supersolid, and superfluid. For one mesh, by comparing the theoretical results with the outcome of numerical diagonalization, I manage to uncover the signatures of diagonal and off-diagonal spatial orders in a finite quantum system.

Bose-Hubbard model on polyhedral graphs

“…In fact, we have already commented in Ref. [31] that the right quantity to look at is the superfluid density ρ SF (red curve in Fig. 5b), which indeed compares well with φ 2 i .…”

“…A preliminary study of the extended BH model on the graph of a regular polyhedron has been given in Ref. [31]. There, we have employed the decoupling approximation [8,32] (DA, a kind of mean-field theory) to sketch the phase behavior at T = 0, finding that DA is already reliable for a graph as simple as that of a cube.…”

“…Here, we carry out a similar analysis for more complex graphs, choosing the skeleton of two Catalan solids for demonstration. As in [31] we make the further simplification that multiple node occupancy is forbidden, which corresponds to a system of hard-core bosons. With this assumption, the dimensionality of the Hilbert space is reduced to such a degree that in one case the DA can be validated against exact diagonalization.…”

“…In Ref. [31] we have studied model (2.1) at T = 0 on a polyhedral graph with M nodes, focusing on those Platonic polyhedra (i.e., the cube and the dodecahedron) where a subset of vertices forms itself a regular polyhedron. Besides a number of insulating "phases", crystalline or not, the ground-state diagram contains a wide superfluid basin and, only in the dodecahedral case, a small supersolid region.…”

“…In Ref. [31], where in the cases investigated the vertex subsets are two, the strategy put forward was to diagonalize a two-site Hamiltonian, hence a 4 × 4 matrix. Here, we find easier to divide the same task in as many one-site problems as are the vertex types, which are three for both TH and PD graphs.…”