Doped resonating-valence-bond states: Robustness of the spin-ice phases in three-dimensional Rydberg arrays

Rydberg blockade effect provides a convenient platform for simulating locally constrained many-body systems, such as quantum dimer models and quantum loop models, especially their novel phases like topological orders and gapless quantum spin ice (QSI) phases. To discuss the possible phase diagram containing different QSIs in three-dimensional (3D) Rydberg arrays, we have constructed an extended Rokhsar-Kivelson (RK) Hamiltonian with equal weight superposition ground state in different fillings at the RK point. Therefore, the perfect QSIs with fixed local dimer filling and their monomer-doped states can be simulated directly by Monte Carlo sampling. Using single-mode approximation, the excitations of dimers and monomers have also been explored in different fillings. We find that, in the thermodynamical limit, even doping a small amount of monomers can disrupt the topological structure and lead to the existence of off-diagonal long-range order. However, in a finite size (as in cold-atom experiment), the property of QSI will be kept in a certain region like a crossover after doping. The phase diagram containing different QSIs and off-diagonal order phases is proposed.