  We study a system of two hardcore bosonic Hubbard models weakly coupled with
each other by tunneling. Assuming that the single uncoupled model exhibits
off-diagonal long-range order, we prove that the coupled system exhibits
spontaneous symmetry breaking (SSB) in the infinite volume limit, in the sense
that the two subsystems maintain a definite relative U(1) phase when the
tunneling is turned off. Although SSB of the U(1) phase is never observable in
a single system, SSB of the relative U(1) phase is physically meaningful and
observable by interference experiments. The present theorem is made possible by
the rigorous theory of low-lying states and SSB in quantum antiferromagnets
developed over the years.
