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                \begin{document}$$\phi \rightarrow \pi -\phi $$\end{document}$.

An illustration of how the interaction deforms the critical lines is shown in Fig.[2](#Fig2){ref-type="fig"}.Fig. 2Illustration of the deformation of the critical lines induced by the electron--electron interaction. The solid red curve corresponds to the non-interacting Haldane model with $\documentclass[12pt]{minimal}
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                \begin{document}$$t_1=1$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$t_2=0.1$$\end{document}$. The dotted and dashed-dotted lines correspond to the interacting case, with ultra-local (on-site) interaction and $\documentclass[12pt]{minimal}
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                \begin{document}$$U=\pm \,0.5$$\end{document}$; the lines are computed by truncating perturbation theory to first order, see \[[@CR19], Sect. III.E\] for details. The connected regions of the complement of the critical lines are labelled