  A system of neutral atoms trapped in an optical lattice and dispersively
coupled to the field of an optical cavity can realize a variation of the
Bose-Hubbard model with infinite-range interactions. This model exhibits a
first order quantum phase transition between a Mott insulator and a charge
density wave, with spontaneous symmetry breaking between even and odd sites, as
was recently observed experimentally [Landig \emph{et. al.}, Nature {\bf 532}
(2016)]. In the present paper we approach the analysis of this transition using
a variational model which allows us to establish the notion of an energy
barrier separating the two phases. Using a discrete WKB method we then show
that the local tunneling of atoms between adjacent sites lowers this energy
barrier and hence facilitates the transition. Within our simplified
description, we are thus able to increment the phase diagram of the model with
information concerning the height of the barrier separating the metastable
minima from the global minimum in each phase, which is an essential aspect for
the understanding of the reconfiguration dynamics induced by a quench across a
quantum critical point.
