Dynamical correlation functions in the Ising field theory

We thank the Referee for the thorough reading of our manuscript. We are pleased by their recommendation for the publication of our paper in SciPost Physics. The Referee made some interesting comments which we answer below.

1- For the sake of clarity, we decided to take the scaling limit while keeping the volume of the system, $L=Na$, fixed. As a result, the momenta are still quantised even in the field theory. However, in the rest of the paper, we indeed work in the thermodynamic limit, where such sums become integrals. The Referee's comment made us realise that this was not discussed clearly, so we have added a sentence to the beginning of the paragraph below Eq. (2.7).

2- This is an interesting question. The non-analytic behaviour in the form of cusps appears in the correlation length as a function of the temperature or of $\zeta=t/x$. Taking time derivatives does not modify the non-analytic behaviour in temperature but one may wonder what happens with the $\zeta$ dependence. Importantly, the correlation length describes the asymptotic behaviour in the $x\to\infty$ limit taken at a fixed value of $\zeta$. The time derivative of the function giving this asymptotic behaviour does not give in general the asymptotic behaviour of the time derivative of the correlation function. Thus the correlation function itself can be, and we believe is, continuous in $x$ and $t$.

3- We are not aware of transfer matrix approaches that would yield the asymptotic behaviour for fixed $\zeta$, let alone in the field theory. But it is of course possible that such an approach exists or will be constructed in the future. In this case, it is quite possible that the cusps we found correspond to crossing eigenvalues of a transfer matrix.

4- The notation $\zeta=x/t$ is probably more common, however, we used $\zeta=t/x$ as it was more natural and convenient because we discussed the equal time case ($\zeta=0$) but not the autocorrelation function ($\zeta=\infty$).

We thank the Referee for his careful reading of our manuscript and we are glad that he recommends the publication of our paper in SciPost Physics. Below we provide our answers to the insightful questions raised by the Referee:

1. This is a very interesting question. To our knowledge, the Fredholm determinant representation can be used to derive the long distance (or time) asymptotic behavior. We do not know if it is possible to obtain the short distance behavior but it would certainly be interesting. However, we refrained from such analytic derivations as they are beyond the scope of our paper. The analytic derivation of both the large and short distance asymptotics directly from the Fredholm determinant representations is an exciting line of future research.

2. Thank you for this thoughtful question. The non-analyticity indeed stems from the switch between the leading exponential terms. However, it only arises when taking the limit x→ $\infty$ asymptotic limit, so finding a clear crossover behavior between the two leading exponentials in a finite regime is challenging. The correlators themselves are smooth functions of the ray parameter. Regardeless, one could investigate whether including more sub-leading terms in the expansion improves the approximation and smoothens the transition. To explore this, we performed additional simulations at $m\beta = 5$ and $\zeta = 0.75$, where the leading term corresponds to $k_0 = 1$ (see Eq. (5.13)). The attached plot shows the simulated data, the leading decay corresponding to this $k = 1$ value, and a refinement including the $k = 0$ and $k = 2$ terms as well. We can clearly see that keeping more terms makes the approximation better for smaller mx values, which might be phrased as a crossover in the spatial separation.

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