  In the previous works, we proposed atomic quantum simulations of the U(1)
gauge-Higgs model by ultra-cold Bose gases. By studying extended Bose-Hubbard
models (EBHMs) including long-range repulsions, we clarified the locations of
the confinement, Coulomb and Higgs phases. In this paper, we study the EBHM
with nearest-neighbor repulsions in one and two dimensions at large fillings by
the Gutzwiller variational method. We obtain phase diagrams and investigate
dynamical behavior of electric flux from the gauge-theoretical point of view.
We also study if the system exhibits glassy quantum dynamics in the absence and
presence of quenched disorder. We explain that the obtained results have a
natural interpretation in the gauge theory framework. Our results suggest
important perspective on many-body localization in strongly-correlated systems.
They are also closely related to anomalously slow dynamics observed by recent
experiments performed on Rydberg atom chain, and our study indicates existence
of similar phenomenon in two-dimensional space.
