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In [1], starting with the bosonic Hamiltonian (Eqn. 1) for the dice lattice model with half flux density (with Ahronov-Bohm phases incorporated),
\begin{equation}
H=-t\sum_{\langle j,\mu\rangle}(a^\dagger_\mu a_je^{iA_{\mu j}}+\text{h.c.})+\frac{U_\Delta}{2}\sum_{\mu \in \Delta,\nabla}a^\dagger_\...
