Experiments on chains of Rydberg atoms appear as a playground to study quantum phase transitions in 1D. As a natural extension, we report a quantitative ground-state phase diagram of Rydberg atoms arranged in a two-leg ladder interacting via van der Waals potential. We address this problem numerically, using the density matrix renormalization group algorithm. Our results suggest that, quite remarkably, Zk crystalline phases, with the exception of the checkerboard phase, appear in pairs characterized by the same pattern of occupied rungs but distinguishable by a spontaneously broken Z2 symmetry between the two legs of the ladder. Within each pair, the two phases are separated by a continuous transition in the Ising universality class, which eventually fuses with the Zk transition, whose nature depends on k. According to our results, the transition into the Z2 - Z2 phase changes its nature multiple times, including an Ashkin-Teller transition that is surprisingly stable over an extended interval, followed by the Z4-chiral transition, and finally in a two step-process mediated melting via the floating phase. The transition into the Z3 phase with resonant states on the rungs belongs to the three-state Potts universality class at the commensurate point, to the Z3-chiral Huse-Fisher universality class away from it, and eventually it is through an intermediate floating phase. The Ising transition between Z3 and Z3 - Z2 phases, entering the floating phase, opens the possibility to realize lattice supersymmetry in Rydberg quantum simulators.@en

The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.@en