$\mathbb{Z}_N$ lattice gauge theories with matter fields

Motivated by the exotic phenomenology of certain quantum materials and recent advances in programmable quantum emulators, we here study fermions and bosons in $\mathbb Z_N$ lattice gauge theories. We introduce a family of exactly soluble models, and characterize their orthogonal (semi-)metallic ground states, the excitation spectrum, and the correlation functions. We further study integrability breaking perturbations using an appropriately derived set of Feynman diagrammatic rules and borrowing physics associated to Anderson's orthogonality catastrophe. In the context of the ground states, we revisit Luttinger's theorem following Oshikawa's flux insertion argument and furthermore demonstrate the existence of a Luttinger surface of zeros in the fermionic Green's function. Upon inclusion of perturbations, we address the transition from the orthogonal metal to the normal state by condensation of certain excitations in the gauge sectors, so-called ``$e$-particles''. We furthermore discuss the effect of dynamics in the dual ``$m$-particle'' excitations, which ultimately leads to the formation of charge-neutral hadronic $N$-particle bound states. We present analytical arguments for the most important phases and estimates for phase boundaries of the model. Specifically, and in sharp distinction to quasi-1D $\mathbb Z_N$ lattice gauge theories, renormalization group arguments imply that the phase diagram does not include an emergent deconfining $U(1)$ phase. Therefore, in regards to lattice QED problems, $\mathbb Z_N$ quantum emulators with $N<\infty$ can at best be used for approximate solutions at intermediate length scales