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Confinement is an ubiquitous phenomenon when matter couples to gauge fields,which manifests itself in a linear string potential between two static charges.Although gauge fields can be integrated out in one dimension, they can mediatenon-local interactions which in turn influence the paradigmatic Luttingerliquid properties. However, when the charges become dynamical and theirdensities finite, understanding confinement becomes challenging. Here we showthat confinement in 1D $\mathbb{Z}_2$ lattice gauge theories, with dynamicalmatter fields and arbitrary densities, is related to translational symmetrybreaking in a non-local basis. The exact transformation to this string-lengthbasis leads us to an exact mapping of Luttinger parameters reminiscent of aLuther-Emery re-scaling. We include the effects of local, but beyond contact,interactions between the matter particles, and show that confined mesons canform a Mott-insulating state when the deconfined charges cannot. While thetransition to the Mott state cannot be detected in the Green's function of thecharges, we show that the metallic state is characterized by hiddenoff-diagonal quasi-long range order. Our predictions provide new insights tothe physics of confinement of dynamical charges, and can be experimentallyaddressed in Rydberg-dressed quantum gases in optical lattices.

Confinement and Mott transitions of dynamical charges in 1D lattice gauge theories