Universal upper bounds on the Bose-Einstein condensate and the Hubbard star

TL;DR

This paper establishes universal upper bounds on Bose-Einstein condensate for hard-core bosons on arbitrary lattices, linking maximal condensation to delocalized states and minimal mode entanglement, with implications for the bosonic Hubbard star.

Contribution

It proves a universal upper bound on condensate for hard-core bosons and relates maximal condensation to ground states of a bosonic Hubbard star.

Findings

Universal upper bound on condensate: N_max=(N/d)(d-N+1).

Maximal condensation occurs in delocalized states with equal site amplitudes.

Maximal local condensation requires minimal mode entanglement.

Abstract

For $N$ hard-core bosons on an arbitrary lattice with $d$ sites and independent of additional interaction terms we prove that the hard-core constraint itself already enforces a universal upper bound on the Bose-Einstein condensate given by $N_{max}=(N/d)(d-N+1)$. This bound can only be attained for one-particle states $|\varphi\rangle$ with equal amplitudes with respect to the hard-core basis (sites) and when the corresponding $N$-particle state $|\Psi\rangle$ is maximally delocalized. This result is generalized to the maximum condensate possible within a given sublattice. We observe that such maximal local condensation is only possible if the mode entanglement between the sublattice and its complement is minimal. We also show that the maximizing state $|\Psi\rangle$ is related to the ground state of a bosonic `Hubbard star' showing Bose-Einstein condensation.