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                \begin{document}$$t_1$$\end{document}$ describes nearest neighbor hopping on the hexagonal lattice. The term proportional to $\documentclass[12pt]{minimal}
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                \begin{document}$$t_2$$\end{document}$ describes next-to-nearest neighbor hopping, with the complex phases $\documentclass[12pt]{minimal}
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                \begin{document}$$e^{\pm \, i\phi }$$\end{document}$ modeling the effect of an external, transverse, magnetic field. The term proportional to *W* describes a staggered potential, favoring the occupancy of the *A* or *B* sublattice, depending on whether *W* is negative or positive. Finally, the term proportional to $\documentclass[12pt]{minimal}
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                \begin{document}$$\mu $$\end{document}$ is the chemical potential, which controls the average particle density in the Gibbs state. See Fig. [1](#Fig1){ref-type="fig"}. Concerning the many-body interaction, we assume it to be