Complexity of Bose-Hubbard Model : Quantum Phase Transition

The operator approach is applied to investigate the complexity of Bose-Hubbard model. We present a systematic method to expand the quantum complexity in series of coupling constant. We first study 2-sites system. For the ground state we can find the exact value of complexity which is the summation of all order. The complexity is divergent at critical value of coupling constant that indicates the quantum phase transition at this point. We then generalize the method to the N-sites closed chain and any dimensional system. The found properties are similar to those in 2-sites system. We also study the excited state and present the general formulas of Bose-Hubbard model complexity, which shows a similar form as that in $\lambda \phi^4$ theory studied in our previous paper.