1 Correction to: Commun. Math. Phys. 376, 649–679 (2020). https://doi.org/10.1007/s00220-019-03523-3

We correct an error in the Appendix of [1]. The quantity \(c_{v}\), which is defined in (A.19) in the proof of Lemma A.2, does not vanish in the limit \(v\searrow 0\), contrary to the claim in (A.12). The reason is that the assumptions of Lemma A.2 allow for energies \(E \in [-2,2]\), which is a too big interval.

The problem has a simple solution which consists of restricting Lemma A.2 to energies \(E \in [-v,v]\). Accordingly, the maximum over energies in the definition of \(c_{v}\) in (A.19) is to be taken only over the smaller interval \([-v,v]\), too, and the vanishing of \(c_{v}\) as \(v\searrow 0\) then holds.

We point out that this change in assumptions in Lemma A.2 does not cause a problem for its applications in Lemma A.5 and in the proof of Theorem 3.4: these applications require only energies in the critical window \(E \in {\mathcal {W}}_{L} := [-L^{-(\alpha +1/2)}, L^{-(\alpha +1/2)}]\) with \(\alpha >0\). Since both Lemma A.5 and Theorem 3.4 involve a minimal length scale above which their statements hold, all that needs to be done is to ensure that the minimal length scale is larger than \(v^{-2/(2\alpha +1)}\) so that \({\mathcal {W}}_{L} \subset [-v,v]\), and Lemma A.2 is applicable.

Finally, and not related to the above, we remark that the right-hand side of (A.41) should not read \(\mathrm{e}^{3c_{v}}[-1,1]\) but \([\mathrm{e}^{-3c_{v}}, \mathrm{e}^{3c_{v}}]\).