Critical lines and ordered phases in a Rydberg-blockade ladder

Arrays of Rydberg atoms in the blockade regime realize a wealth of strongly correlated quantum physics, but theoretical analysis beyond the chain is rather difficult. Here we study a tractable model of Rydberg-blockade atoms on the square ladder with a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry and at most one excited atom per square. We find $D_4$, $\mathbb{Z}_2$ and $\mathbb{Z}_3$ density-wave phases separated by critical and first-order quantum phase transitions. A non-invertible remnant of $U(1)$ symmetry applies to our full three-parameter space of couplings, and its presence results in a larger critical region as well as two distinct $\mathbb{Z}_3$-broken phases. Along an integrable line of couplings, the model exhibits a self-duality that is spontaneously broken along a first-order transition. Aided by numerical results, perturbation theory and conformal field theory, we also find critical Ising$^2$ and three-state Potts transitions, and provide good evidence that the latter can be chiral.