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Quantum link models (QLMs) have attracted a lot of attention in recent timesas a generalization of Wilson's lattice gauge theories (LGT), and areparticularly suitable for realization on quantum simulators and computers.These models are known to host new phases of matter and act as a bridge betweenparticle and condensed matter physics. In this article, we study the Abelian$U(1)$ lattice gauge theory in $(3+1)$-d tubes using large-scale exactdiagonalization (ED). We are then able to motivate the phase diagram of themodel with finite size scaling techniques (FSS), and in particular propose theexistence of a Coulomb phase. Furthermore, we introduce the first modelsinvolving fermionic quantum links, which generalize the gauge degrees offreedom to be of fermionic nature. We prove that while the spectra remainidentical 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 fieldinteractions as correlated hopping in quantum simulator experiments.

Exploring Bosonic and Fermionic Link Models on $(3+1)-$d tubes