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### Description

Recent developments in experimental condensed matter physics allow for realization of Abelian gauge Hamiltonians on routinely controllable systems (Josephson junction arrays, optical lattices, etc). This opened the prospect of simulation of the phenomena of compact quantum electrodynamics and quantum chromodynamics (e.g, quark confinement) in the laboratory setup.

We consider $Z_N$ symmetric generalized Bose-Hubbard Hamiltonian on a two-dimensional lattice. However, field theory predicts that it exhibits the confined phase, the gapped phase, one-dimensional Bose liquid phase, or gapless dipolar liquid phase. The latter two are particularly interesting in view of AdS/CFT correspondence (and the latter one is believed to exist only at large $N$ and small coupling $g$). Our goal is search for these two phases (gapless dipolar liquid is most intriguing) by numerical methods.

The Hamiltonian is re-written in the second order of the perturbation theory and transformed so that it acts on a dual cylinder lattice. Its ground state is approximated by matrix product state (MPS) and evaluated using infinite-size density-matrix renormalization group (iDMRG). We calculate the von Neumann entanglement entropy as function of $g$ for each $N$. The discontinuities of its first derivative signal the phase transitions, and the type of these transitions (or entire phases) is identified by the central charge: $S = (c/6) \log ξ + S_0$.

Our calculations for $N = 2$ indicate only one Ising-type phase transition with central charge $c = 1/2$, and the system is in a gapped phase at large $g$. On the contrary, at $N > 2$ the ground state of the system is highly degenerate at large $g$, and we have found no evidence of two phase transitions for $N < 6$. The calculations for $N = 6$ or larger are underway.