Mapping a dissipative quantum spin chain onto a generalized Coulomb gas

I thank the author for clarifying on page 7 that the $\nu\to0$ limit cannot be taken trivially from the derived equations. The author also discusses the implications of the $\nu\to0$ limit in Section 6. Because this discussion is rather technical, the meaning is not fully clear to me. As I understand this part, the Coulomb gas picture applies here as well, with the only difference that the interaction in Eq. (23) becomes local in space (and remains nonlocal in time). Is that correct?

In the introduction, the author reviews how the Coulomb-gas picture had been applied to generalized XY models, but I do not find any discussion of applications to dissipative systems. In the past, dissipation effects had also been studied in the context of Josephson junctions. For example, the Coulomb gas picture is mentioned for a single dissipative junction [Schmid, Phys. Rev. Lett. 51, 1506 (1983)] or for arrays of dissipative junctions [Bobbert, Fazio, Schön, Zimanyi, Phys. Rev. B 41, 4009 (1990)], but probably also in other works. It would be fair to review also relevant work including dissipation and to check if there are any similarities to the present work.

Ask for minor revision

same as in previous report

In my previous report, I had already recommended publication of the manuscript in SciPost. A few misprints numbered 2-4 were pointed out by the other referee and have been corrected in the revised version.
The second referee also raised the issue of the limit $\nu \to 0$ (point 1) and the authors have responded that the Green's function of the bath operator $(-\nu^2 \partial_x^2 -\partial_\tau^2+\Omega^2)^{-1}$ does not deform confinuously into the one on the local bath operator $(-\partial_\tau^{2} +\Omega^2)^{-1}$. They have also inserted a footnote in the manuscript making that argument.

I am not sure the argument of the authors is the correct one.
If we insert Eqs. (5) into Eq. (12) we find
\begin{eqnarray}
\label{eq:douze}
\mathcal{K}(x,\tau)=\frac{1}{\pi^2 \nu} \int_0^{\tau_c^{-1}} d\Omega \alpha \tau_c^{s-1} \Omega^{s+1} K_0\left(\Omega \sqrt{\tau^2 + (x/\nu)^2} \right)
\end{eqnarray}
If we let $\nu \to 0$ for $x\ne 0$, because of the exponential decay of the modified Bessel function for large argument, we obtain
\begin{equation}
\lim_{\nu \to 0} \mathcal{K}(x\ne 0,\tau) =0,
\end{equation}
while it is obvious that for $x=0$,
\begin{equation}
\lim_{\nu \to 0} \mathcal{K}(0,\tau) =+\infty.
\end{equation}
To verify that $\mathcal{K}(x,\tau)$ behaves as a Dirac delta distribution, we only need to calculate the weight
\begin{eqnarray}
\label{eq:weight}
\int dx \mathcal{K}(x,\tau) &=& \frac{1}{\pi^2} \int_0^{\tau_c^{-1}} d\Omega \alpha \tau_c^{s-1} \Omega^{s+1} \int_{-\infty}^{+\infty} K_0\left(\Omega \sqrt{\tau^2 + (x/\nu)^2} \right) \frac{dx}{\nu}, \\
&=& \frac{\alpha \tau_c^{s-1}}{\pi^2 \tau^{s+1}} \int_0^{\tau/\tau_c} dw w^{s+1} \int_{-\infty}^{+\infty} du K_0 (w \sqrt{1+u^2}),
\end{eqnarray}
and note that it is independent of $\nu$. Moreover, when $\tau \gg \tau_c$, we can extend the $w$ integration to $+\infty$ to find
\begin{eqnarray}
\label{eq:asymp-w}
\int dx \mathcal{K}(x,\tau\gg \tau_c) &\sim& \frac{\alpha \tau_c^{s-1}}{\pi^2 \tau^{s+1}} \int_{-\infty}^{+\infty} \frac{du}{(1+u^2)^{1+s/2}} \int_0^{+\infty} dv v^{s+1} K_0(v) \\
&\sim& \frac{\alpha \tau_c^{s-1}}{\pi^2 \tau^{s+1}} \\
&\sim& \frac{\alpha \tau_c^{s-1}\Gamma(s+1)}{\pi \tau^{s+1}},
\end{eqnarray}
and recover for $s=1$ the decay $1/\tau^2$ expected in an ohmic bath.
So even though the bath operators do not deform continuously into each other,
it is possible to recover the correct limit for $\nu \to 0$ directly from Eq. (12).
In other words, the comment 1 of the other referee was incorrect, and the authors
have been too cautious in their response and footnote.

I suggest the authors replace their footnote with a derivation of the delta distribution limit of the kernel from Eq. (12) and that final version of the manuscript gets published in SciPost.

Publish (easily meets expectations and criteria for this Journal; among top 50%)