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Supersolid phase of the extended Bose-Hubbard model with an artificial gauge field

We examine the zero and finite temperature phase diagrams of soft-core bosons of the extended Bose-Hubbard model on a square optical lattice. To study various quantum phases and their transitions we employ single-site and cluster Gutzwiller mean-field theory. We have observed that the Mott insulator phase vanishes above a critical value of nearest-neighbour interaction and the supersolid phase occupies a larger region in the phase diagram. We show that the presence of artificial gauge field enlarges the domain of supersolid phase. The finite temperature destroys the crystalline structure of the supersolid phase and thereby favours normal fluid to superfluid phase transition. The presence of an envelope harmonic potential demonstrates coexistence of different phases and at $z~k_{B}T\geqslant V$, thermal energy comparable and higher to the long-range interaction energy, the supersolidity of the system is destroyed.