  Quantum link models (QLMs) have attracted a lot of attention in recent times
as a generalization of Wilson's lattice gauge theories (LGT), and are
particularly suitable for realization on quantum simulators and computers.
These models are known to host new phases of matter and act as a bridge between
particle and condensed matter physics. In this article, we study the Abelian
$U(1)$ lattice gauge theory in $(3+1)$-d tubes using large-scale exact
diagonalization (ED). We are then able to motivate the phase diagram of the
model with finite size scaling techniques (FSS), and in particular propose the
existence of a Coulomb phase. Furthermore, we introduce the first models
involving fermionic quantum links, which generalize the gauge degrees of
freedom to be of fermionic nature. We prove that while the spectra remain
identical between the bosonic and the fermionic versions of the
$U(1)$-symmetric quantum link models in $(2+1)$-d, they are different in
$(3+1)$-d. We discuss the prospects of realizing the magnetic field
interactions as correlated hopping in quantum simulator experiments.
