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In quantum many-body system, dimensionality plays a critical role on type ofthe quantum phase transition. In order to study the quantum system duringdimensional crossover, we studied the Bose-Hubbard model on cubic lattice withanisotropic hopping by using the high order symbolic strong coupling expansionmethod. The analytic series expanded boundaries between the Mott-insulator andsuperfluid phase up to eighth order are calculated. The critical exponents areextracted by Pad\'{e} re-summation method, which clearly shows the dimensionalcrossover behavior. Meanwhile, the critical points at commensurate filling canalso be obtained, and they match well with the prediction of renormalizationgroup theory. The scaling of the gap energy and whole phase diagram are givenat last, and they can be taken as the benchmark for experiment and numericalsimulations in the future study.

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