The partition function depends on both $\tau$ and its complex conjugate $\overline{\tau }$. Following the mathematics literature, we write $Z(\tau )$ instead of $Z(\tau ,\overline{\tau })$. This convention applies to all modular functions throughout the Letter.

This is also consistent with Ref. [21]. We believe there is an error in the final result of Eq. (4.8). After correcting this error, the large-energy tail with integrand ${\mathcal{E}}^{-3/2}{e}^{({\beta }_H-\beta )\mathcal{E}}$ (while the final result in [21] corresponds to the integral of ${\mathcal{E}}^{+3/2}{e}^{({\beta }_H-\beta )\mathcal{E}}$) is finite but non-analytic at $\beta ={\beta }_H$ and divergent for $\beta <{\beta }_H$, consistent with our result in this work.

Note that even though there are infinitely many branch points in the range $[{\lambda }_m,\infty )$, they are not dense. The location of branch points ${\lambda }_{\gamma }$ can be written as ${\lambda }_{\gamma }=6/(\pi \overline{c})((c{\tau }_1+d{)}^2+c^2{\tau }_2^2)$ where $c$, $d$ are coprime integers, and it is then easy to show that there are finitely many ${\lambda }_{\gamma }$ small than $T$ for any $T>0$.