  The exploration of phase diagrams of strongly interacting gauge theories
coupled to matter in lower dimensions promises the identification of exotic
phases and possible new universality classes, and it facilitates a better
understanding of salient phenomena in Nature, such as confinement or
high-temperature superconductivity. The emerging new techniques of quantum
synthetic matter experiments as well as efficient classical computational
methods with matrix product states have been extremely successful in one
spatial dimension, and are now motivating such studies in two spatial
dimensions. In this work, we consider a $\mathrm{U}(1)$ quantum link lattice
gauge theory where the gauge fields, represented by spin-$\frac{1}{2}$
operators are coupled to a single flavor of staggered fermions. Using matrix
product states on infinite cylinders with increasing diameter, we conjecture
its phase diagram in $(2+1)$-d. This model allows us to smoothly tune between
the $\mathrm{U}(1)$ quantum link and the quantum dimer models by adjusting the
strength of the fermion mass term, enabling us to connect to the well-studied
phases of those models. Our study reveals a rich phase diagram with exotic
phases and interesting phase transitions to a potential liquid-like phase. It
thus furthers the collection of gauge theory models that may guide future
quantum-simulation experiments.
