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Quantum phase transition of the Bose-Hubbard model with anisotropic hopping on a cubic lattice

In a quantum many-body system, dimensionality plays a critical role in determining the type of quantum phase transition. To study the quantum system during dimensional crossover, we studied the Bose-Hubbard model on a cubic lattice with anisotropic hopping by using the high-order symbolic strong-coupling expansion method. The analytic series expanded boundaries between the Mott insulator and the superfluid phase up to eighth order are calculated. The critical exponents are extracted by the Pad\'e resummation method, which clearly shows dimensional crossover behavior. Meanwhile, the critical points at commensurate filling can also be obtained, and they match well with the prediction of renormalization group theory. Finally, the scaling of the gap energy and the whole phase diagram are given, and they can be taken as the benchmark for experiment and numerical simulations in future study.