Phase diagram of Rydberg atoms in a two-leg rectangular ladder

Using the density matrix renormalization group algorithm, we map the ground-state phase diagram of a two-leg Rydberg ladder array with lattice spacings ${a}_{x}=2{a}_{y}$. We identify various density wave phases that spontaneously break the translational symmetry or the top-bottom reflection symmetry within the ladder. By increasing the laser detuning from zero, where the system is in a disordered phase that preserves all symmetries, we observe density wave orders with spontaneous breaking of the translational ${\mathbb{Z}}_{p}$ symmetries at intermediate detuning values, while the reflection symmetry is preserved. These orders exhibit nonzero bond orders with positive expectation values on every $p\mathrm{th}$ rung, thus labeled as ${\mathbb{Z}}_{p}^{+}$ phases. At larger detuning values, another spontaneous breaking of the reflection symmetry, which disrupted the bond orders on the rungs, occurs via an Ising phase transition. In these phases, either the top or the bottom site is occupied in a staggered way on every $p\mathrm{th}$ rung, breaking the translational ${\mathbb{Z}}_{2p}$ symmetry, thus labeled by ${\mathbb{Z}}_{2p}$ phases. We locate and characterize the 3-state Potts point and Ashkin-Teller point along the commensurate lines, as well as the direct chiral phase transitions between the disordered phase and the ${\mathbb{Z}}_{p}^{+}$ $(p=3,4)$ phases. Critical exponents $\ensuremath{\nu}$ and $z$ are calculated for both conformal and chiral phase transition points. We finally identify two types of floating phases in the phase diagram: one characterized by a quasi-long-range incommensurate bond-order wave, and the other by a quasi-long-range incommensurate wave of density differences in the rungs. Our work motivates further applications of Rydberg atom arrays in quantum simulation.