  We investigate a quasi-two-dimensional system consisting of two species of
alkali atoms confined in a specific optical lattice potential [Phys. Rev. A 95,
053608 (2017)]. In the low-energy regime, this system is governed by a unique
$\mathbb{Z}_N$ gauge theory, where field theory arguments have suggested that
it may exhibit two exotic gapless deconfined phases, namely a dipolar liquid
phase and a Bose liquid phase, along with two gapped (confined and deconfined)
phases. We address these predictions numerically by using large-scale density
matrix renormalization group simulations. Our findings provide conclusive
evidence for the existence of a gapless Bose liquid phase for $N \geq 7$. We
demonstrate that this gapless phase shares the same critical properties as
one-dimensional critical phases, resembling weakly coupled chains of Luttinger
liquids. In the range of geometries and $N$ considered, the gapless dipolar
phase predicted theoretically is still elusive and its characterization will
probably require a full two-dimensional treatment.
