  We investigate tacitly assumed relationships between the concepts of
super-fluidity (-conductivity), long range order and entanglement. We prove
that the three are by no means equivalent, but that notwithstanding, some
rigorous implication can be established between them. This leads to three
different, albeit frequently related, notions of "criticality", all of which
are exemplified within the Hubbard model in the low density regime. We use
Peierls' method of twisted Hamiltonians to link the existence of entanglement
to superfluidity and (quasi)-long range order. As an application of our
formalism, we show that recent experiments with cold atoms already prove the
existence of the field theoretic, spatial entanglement in two dimensions. More
interestingly, the appearance of entanglement in these experiments seems to be
intimately related to the phase transition of the Kosterlitz Thouless type.

Quantum Physics