For the Bose-Hubbard model, we know that there are the Mott insulator phase with $\langle a_i \rangle = 0$ and the superfluid phase with $\langle a_i \rangle \neq 0$. However, when we are trying to solve the Bose-Hubbard model numerically, we often assume a fixed site number and a fixed total particle number. But if $\langle a_i \rangle \neq 0$, the total particle number is not conserved. Does that mean we cannot really get the true superfluid phase numerically, thus we don't really have a phase transition if we solve the Bose-Hubbard model numerically?
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