Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions

We are grateful to the three referees for their careful reading of the manuscript and helpful comments. All three referees note the significant progress made in our work in calculating (dynamical) correlation functions of semi-local operators at finite temperature and after quantum quenches and recommend publication.

In the following we respond to the various points raised.

---Reply to the second referee---

We thank the referee for their report and constructive comments. The referee's main concern is that the presentation of our manuscript is rather technical. On the other hand, as they go on to point out, given the nature of the subject matter this is difficult to avoid. In light of the referee's comments we have made additional efforts to improve the readability of our manuscript. In particular we have added a summary of our main results at the end of the introduction, which hopefully will make it easier for readers to navigate their way through the technical parts that follow. We have also improved the text in the Appendix.

---Reply to the third referee---

We thank the referee for their report and constructive comments. The referee did not request specific changes, but raised several questions to which we now reply in turn.

-The referee notes that ''Interestingly but unfortunately, the formalism does not seem to be extendable for calculating local operators even for integrable cases in the presence of interactions.'' and considers this a weakness of our work. Here we beg to differ: this is not a weakness, but a simply consequence of fundamental differences in the properties of Lehmann representations of local and semi-local operators. Which states dominate the spectral sums of response functions is ultimately a physical property, and these are simply different between local and semi-local operators. As is already mentioned in our manuscript, local operators will be the subject of a separate publication.

-The referee expresses ''a small confusion regarding the interpretations of the finite-T result.'' We discuss two completely separate physical problems: (i) Finite-temperature dynamical (time-dependent) spin-spin correlation functions and (ii) Time evolution of the order parameter after a quantum quench of the transverse field (from $h_0$ to $h$), where the initial state is taken to be the ground state at field $h_0$. These two unrelated problems can both be formulated in terms of spectral representations, and the calculations required to extract the dynamics turn out to be technically similar. The spectral representation in the quench case, and the resulting averaging in the steady state are based on the quench-action approach co-developed by one of us. There is no problem with applying it to problems where the initial density matrix is thermal, but we do not do this here.

• The referee wonders if the formalism can be extended to calculate (in a perturbative sense) semi-local operators for non-integrable interacting systems. We agree that this is an interesting question that should be addressed in future work. The current state of the art is the methodology developed in our manuscript, and in our view the most immediate application/generalization is to interacting integrable theories such as the Lieb-Liniger model.

---Reply to the fourth referee---

We thank the referee for their report, pointing out a number of typos, and constructive comments. We have corrected the typos mentioned by the referee (as well as some others). We have followed the referee's suggestion and added an extended outline of the paper and summary of our key results at the end of the introduction. We hope this will help in making our work more accessible.

The referee comments that ''The description is so technical that it is difficult for non-experts on this subject to read.'' We are well aware of this issue. However, as the content of our paper is inherently technical we are afraid there is a limit to how accessible we can make our work to a wider audience.