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Mean-field study of the Bose-Hubbard model in Penrose lattice

We examine the Bose-Hubbard model in the Penrose lattice based on inhomogeneous mean-field theory. Since averaged coordination number in the Penrose lattice is four, mean-field phase diagram consisting of the Mott insulator (MI) and superfluid (SF) phase is similar to that of the square lattice. However, the spatial distribution of Bose condensate in the SF phase is significantly different from uniform distribution in the square lattice. We find a fractal structure in its distribution near the MI-SF phase boundary. The emergence of the fractal structure is a consequence of cooperative effect between quasiperiodicity in the Penrose lattice and criticality at the phase transition.