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

We examine the Bose-Hubbard model in the Penrose lattice based on inhomogeneous mean-field theory. Since the averaged coordination number in the Penrose lattice is four, the 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 the 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 the cooperative effect between quasiperiodicity in the Penrose lattice and criticality at the phase transition.