  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
others. Remarkably, for all the simulators considered, the effective
Hamiltonians have the three types of terms present for the CAHM (Electric
field, matter charge and currents energies) but, in addition, 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 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.
