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Zero temperature superconductor - edge metal - insulator transition in $2D$ bosonic systems

Motivated by the recent experimental observation of an intermediate bosonic metallic state in the two-dimensional superconductor-insulator transition at $T=0$, we study an extended Bose Hubbard model in the limit of large number of particles per site. Using a representation of this in terms of two coupled $XY$ models, we find, in addition to an insulating phase and a $(2+1)D$ superfluid phase, two other phases. One phase is a $2D$ superfluid phase where a crossover from $(2+1)D$ to $2D$ has taken place as a result of incipient charge ordering, signaled by $θ$ disordering, and which is closely related to a supersolid phase. The other new phase is an edge metal state characterized by zero superfluid stiffness, zero charge ordering, and zero bulk compressibility. However, the edge compressibility of the system is nonzero. While we do not find any intermediate state with $2D$ metallic conductivity, we are able to connect these results to STM experiments on $\mathrm{MoS_2}$ showing brims of finite density of states around the entire edge of $2D$ $\mathrm{MoS_2}$ samples.