  In this work we present a set of microscopic U(1)xU(1) models which realize
insulating phases with a quantized Hall conductivity \sigma_{xy}. The models
are defined in terms of physical degrees of freedom, and can be realized by
local Hamiltonians. For one set of these models, we find that \sigma_{xy} is
quantized to be an even integer. The origin of this effect is a condensation of
objects made up of bosons of one species bound to a single vortex of the other
species. For other models, the Hall conductivity can be quantized as a rational
number times two. For these systems, the condensed objects contain bosons of
one species bound to multiple vortices of the other species. These systems have
excitations carrying fractional charges and non-trivial mutual statistics. We
present sign-free reformulations of these models which can be studied in Monte
Carlo, and we use such reformulations to numerically detect a gapless boundary
between the quantum Hall and trivial insulator states. We also present the
broader phase diagrams of the models.
