  Fundamental forces of Nature are described by field theories, also known as
gauge theories, based on a local gauge invariance. The simplest of them is
quantum electrodynamics (QED), which is an example of an Abelian gauge theory.
Such theories describe the dynamics of massless photons and their coupling to
matter. However, in two spatial dimension (2D) they are known to exhibit gapped
phases at low temperature. In the realm of quantum spin systems, it remains a
subject of considerable debate if their low energy physics can be described by
emergent gauge degrees of freedom. Here we present a class of simple
two-dimensional models that admit a low energy description in terms of an
Abelian gauge theory. We find rich phase diagrams for these models comprising
exotic deconfined phases and gapless phases - a rare example for 2D Abelian
gauge theories. The counter-intuitive presence of gapless phases in 2D results
from the emergence of additional symmetry in the models. Moreover, we propose
schemes to realize our model with current experiments using ultracold bosonic
atoms in optical lattices.
