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                \begin{document}$$(W,\phi )=(0,0),(0,\pi )$$\end{document}$; see \[[@CR19]\] for the proof.

We remark that the proof of Theorem [2.1](#FPar1){ref-type="sec"} is constructive: therefore, a patient reader can extract from it an explicit bound on $\documentclass[12pt]{minimal}
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                \begin{document}$$U_0$$\end{document}$. Such a bound would certainly be far from optimal; optimizing it would be a non-trivial, interesting, exercise, requiring a computer-assisted proof (at least if one is interested in getting a physically significant bound). In any case, conceptually, the only important requirement should be that *U* is sufficiently small, compared to the bandwidth of $\documentclass[12pt]{minimal}
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                \begin{document}$${\mathcal {H}}_0$$\end{document}$, see the definition after ([2.7](#Equ7){ref-type=""}).

Finally, concerning the model: we expect that the specific choice of the interacting Haldane model is not crucial for the validity of the result. The proof extends straightforwardly to strictly related models, such as the spin-conserving Kane--Mele model. An appropriate adaptation should apply, more generally, to any interacting Hamiltonian of the form $\documentclass[12pt]{minimal}
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