The venerable phenomena of Anderson localization, along with the more recent many-body localization (MBL), both depend crucially on the presence of disorder. Here we introduce a family of simple translationally invariant models of fermions locally coupled to spins, which have a disorder-free mechanism for localization. This mechanism is due to a local Z$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ gauge symmetry and we uncover the connection to lattice gauge theories. We diagnose the localization through long-time memory of initial conditions after a global quantum quench.

One of the defining features of the models that we study is the binary nature of the emergent disorder, related to the Z$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ degrees of freedom. This results in a qualitatively different behaviour in the strong effective disorder limit compared to typically studied models of localization. For example it gives rise to the possibility of a delocalization transition via quantum percolation in higher than one dimension.

In connection to the recently proposed quantum disentangled liquid (QDL) we also study the entanglement properties of our models. The QDL provides an alternative to both complete localization and to the eigenstate thermalization hypothesis. Our models highlight the subtlety of defining a QDL and we offer new insights into their entanglement properties.

While the simplest models we consider can be mapped onto free fermions, we also include interactions which leads to MBL-like behaviour characterised by logarithmic entanglement growth. We further consider interactions that generate dynamics for the conserved charges, which give rise to only transient localization behaviour, or quasi-MBL.

Finally, we present a proposal for the experimental measurement of gauge field correlators for our model in two-dimensions. This proposal is based on interferometric techniques which are feasible using current experimental capabilities. Furthermore, the interacting generalizations of our models can be similarly implemented in experiments, providing access to the dynamics of strongly interacting lattice gauge theories, beyond what can be simulated on a classical computer.