Critical behavior of lattice gauge theory Rydberg simulators from effective Hamiltonians

We consider multileg ladders of Rydberg atoms which have been proposed as quantum simulators for the compact Abelian Higgs model (CAHM) in 1+1 dimensions Y. Meurice, Phys. Rev. D 104, 094513 (2021) and modified versions of theses simulators such as triangular prisms. Starting with the physical Hamiltonian for the analog simulator, we construct translation-invariant effective Hamiltonians by integrating over the simulator high-energy states produced by the blockade mechanism when some of the atoms are sufficiently close to each other. Remarkably, for all the simulators considered, the effective Hamiltonians have the three types of terms present for the CAHM (electric field, matter charge, and current energy), but in addition, they have terms quartic in the electric field. For the two-leg ladder, these additional terms cannot be removed by fine-tuning the adjustable parameters of currently available devices. For positive detuning, the new terms create highly degenerate vacua, resulting in a very interesting phase diagram. Using numerical methods, we demonstrate the close correspondence between the physical simulator and the effective description for the ground-state energy and real-time evolution. We discuss the phase diagram at fixed geometry with variable Rabi frequency and detuning, and we show that a rich variety of phases can be reached with potential interest in the context of QCD at finite density. We illustrate how the effective description can be used to design simulators with desirable properties from the point of view of constructing hybrid event generators.