Kibble–Zurek mechanism in the Ising Field Theory

We are grateful to the Referee for the careful reading of the manuscript and his valuable feedback. We are pleased by his opinion that our paper presents a systematic analysis in a very clear way.

Let us address his questions and comments in order.

1. and 3.: Indeed, our ramp definition was not fully precise. Now in Sec. 2.2 we describe the various quench protocols more precisely, eliminating previous inconsistencies.

2. We included the masses of the single-particle states of the $E_8$ model and commented on its exact S-matrix, citing the relevant references.

3. This mismatch is absent from the integrated quantities like the overall defect density, the excess heat density, and their respective cumulants. A remote analogy can be drawn with the form factor series expansion calculation of the central charge in integrable perturbed conformal field theories, where the result of the sum over multiparticle states is fixed by the $c$-theorem, while the separate terms vary greatly due to the details of the interaction. To improve our presentation, we merged the two concluding paragraphs of Sec. 2.3.1 with the concluding paragraph of Sec. 3.2, where we discuss the applicability of the APT in light of the above discrepancy.

4. The reasoning leading to Eq. (4.3) assumes that due to the critical slowing down the only relevant time scale is set by the "freeze-out" time. The domain of validity for this assumption is in the "impulse" regime, i.e. for -$\tau_\text{KZ} < t < \tau_\text{KZ}$ (see in detail in Ref. [31]). Hence, in general we do not expect Eq. (4.3) to apply for the late-time dynamics, with a notable exception given by the energy density, as discussed around Eq. (4.4).

We believe that the above modifications, based on the Referee's advice, have substantially improved the manuscript.

We thank the Referee for their question which allows us to further clarify this issue.

First, we would like to stress that we did not attempt to extract numerically accurate values using adiabatic perturbation theory (APT), but we used the formalism to support the scaling laws. However, the Referee's question is of course relevant, as the validity of the APT would be questioned by a huge numerical deviation from the exact results.

So let us address the 60% discrepancy mentioned by the Referee. We believe it probably comes from a typo in Ref. [24] (ref. number in our resubmitted manuscript), namely in Eq. (4.89) (or Eq. (89) in the arXiv version). However, comparing Eqs. (4.26), (4.88) and (4.28), we find
\[ |\alpha_k|^2 = \pi^2/9 e^{-2\pi k^2/\delta}. \]
Then integrating with respect to $k$ as in Eq. (4.89) one obtains
\[ \pi^2/9 \cdot 1/(2\pi) \sqrt{\delta/2} = 0.1234... \sqrt{\delta}. \]
This essentially agrees with what is written in the review [26] by Dziarmaga below Eq. (121). We suspect that the first two digits in Eq. (4.89) of Ref. [24] got swapped accidentally. We note that Ref. [24] presents the most detailed derivation of the APT kink density after the ramp. In particular, it is more accurate than the estimate in Ref. [7]. While the numerical values of the prefactor differ in Refs. [7, 24, 26], a careful examination of the details in Ref. [24] show that the discrepancy with respect to the exact prefactor $1/2\sqrt{2}\approx0.1125$ is given by the factor $\pi^2/9$, which amounts to a discrepancy around 10% instead of 60%.

As discussed in Ref. [24], the reason why APT overestimates the density of defects is that the first order approximation does not take into account the fermionic nature of the particles which prevents them from occupying the same momentum state. (For bosons, the APT result underestimates the true value.) This is manifest in that $|\alpha_k|^2 > 1$ is permitted by the APT despite the fact that it represents a fermionic occupation number of the mode with momentum $k$. However, even if the numerical prefactor has some error, the scaling dependence on the ramp velocity is correctly captured and this is what we exploit.

Finally, let us emphasise that while we use APT to facilitate some of the scaling arguments, we always put these scaling laws to numerical test as the core of our work is of numerical nature.

We thank the Referee for reading our manuscript and for their useful remarks. We are glad that the Referee finds our work well argumented and scientifically sound. We found their observations very useful and in response to them, we implemented the following changes:

1. In order to highlight our key results more clearly, we expanded the abstract and modified the Conclusions.

2. To avoid unnecessary or unclear references to CFT, we expanded and partially rewrote Sec. 2.2 that outlines the model. Instead of referring to the CFT, we use the expressions "quantum critical point" and "free massless Majorana theory" which are much more accessible to a broader audience. Some CFT concepts lie at the heart of our numerical method and thus are indispensable. The remaining appearances of CFT and integrable field theory concepts support the internal coherence of the text, articulating the field theoretical nature of our work.

3. We added a paragraph to the Conclusions discussing possible experiments with cold atomic gases and Rydberg atoms in which our predictions could be investigated.

We believe that the weaknesses pointed out by the Referee are eliminated by the refined presentation.