Reply to "Anonymous Report 1 on 2019-6-17 Invited Report"

i) At the beginning of Sec. 4, it is said " We give below an alternative -- and somewhat direct -- derivation of this solution."

So far I can see, the presented derivation is exactly the same as that presented previously in ref. [27]. Well, this is just the simplest Riemann-type hydrodynamic approximation where one neglects the dispersive term and solves the 'Riemann flow' subjected to suitable boundary conditions. This derivation is nothing deep really, merely a short exercise. I do not think it is legitimate to proclaim this derivation as "alternative" or "more direct"...

The referee is right, the derivation of this result is indeed quite simple. Apart from the fact that we use polar coordinates and that we do not make explicit use of the Riemann invariant $r_\pm$, the derivation we present is essentially the same as that given by Gamayun et al. The formulation has been changed in the revised version.

ii) At the beginning of Sec. 6, we can read " As for the LL problem, the z magnetization problem was observed to freeze at long times [27, 28], and to be described by a soliton solution of the easy-axis LL equation." One should notice here that ref. [27] obtained the exact spectral data for the domain-wall initial profile. In the easy-axis regime, the spectrum involves a static kink configuration (in general there are additional radiative modes and/or breathers). The absence of spin transport is thus attributed exclusively to the stability of a localized kink, which an exact (and rigorous) statement. Saying "was observed to freeze" may read as an understatement.

We agree with the referee (and changed the text accordingly).

1) I think it would be beneficial to clarify a bit better how much can one strictly infer from the energy conservation argument. Just by knowing that asymptotic regions will remain ferromagnetic does not automatically ensure that the interface region centered at the origin will be free of genuine quantum effects (and therefore effectively governed by the classical ferromagnetic field theory), does it? In my reading, the argument here relies on the empirical evidence (based on e.g. [26] or other simulations) that the magnetization gradient vanishes as t→∞ on the characteristic length R(t). Naively, one would then expect the semi-classical description should work outside of the dynamical interface [-R(t),R(t)]. But then what guarantees you that long-wavelength semi-classical theory w.r.t. ferromagnetic vacuum remains applicable in the vicinity of the origin (even at Δ=1 to begin with)?

Yes, knowing that that asymptotic regions will remain ferromagnetic does not automatically ensure that the interface region centered at the origin will be free of genuine quantum effects. But the energy conservation argument we put forward is different. We start from the assumption that $R(t)\to\infty$ (which is the case for $|\Delta|\leq 1$). Then this implies that the total energy -- which is $\mathcal{O}(1)$ gets "diluted" across the system. As a consequence, for any fixed position $r$ (including the center at $r=0$), the energy density $\epsilon(r,t)$ goes to zero when $t\to\infty$. Then, if $\Delta=1$ the vanishing energy density implies a ferromagnetic state (at least locally).

2) In order to observe the quantum-classical correspondence at the quantitative level one has to probe the dynamics in the weakly anisotropic regime. Then I am curious whether one could also detect classical breather modes (which "live" in the strongly anisotropic regime of the Landau-Lifshitz magnet) in the non-equilibrium dynamics of the quantum spin chain?

If one goes deeply in the easy-axis regime of the quantum spin chain, the magnetization profile hardly evolves before it rapidly freezes, and the width of the asymptotic profile is close two the initial one, that is one lattice spacing. So, the short answer is no. As far as we can judge from our numerics, one does not detect the quantum analog of the classical breather modes.

3) If the authors could give some quantitative estimates on the "quantum corrections" (when moving away from the Δ→1±), that would be very nice.

The referee is certainly right, it would be nice to have some better understanding of these quantum corrections, and we are planning to investigate this further in the future. At the moment we must however admit that we do not yet have any result, except for the observation that some quantities do not match perfectly between the Landau-Lifshitz problem and the quantum spin chain (see for instance Fig. 10, 12 and 13).

The presentation of the left plot from Fig. 9 can be probably be improved. In particular, it is hard to distinguish the blue and yellow curves by naked eye. Moreover, I have a question here: how does one determine the fitting window? Is there a way to estimate the relevant transient time-scale after which the asymptotic law become effectively valid I can imagine that by some ad-hoc manipulations (e.g. sliding the window to larger times), the green fit for the diffusive curve will presumably look much better. Perhaps some clarifications along these line can be helpful.

We have improved the readability of the left panel of Fig. 9 (by a better choice of colors, and by plotting the Landau-Lifshitz data using crosses instead of a full line).

The fitting window was chosen here simply to illustrate the fact that some fitting functions [(b), (c) and (d)] continue to agree with the data beyond (and before) the window, while the simple "diffusive" function ((a), without logarithmic correction) only agrees inside the window. The window is otherwise somewhat arbitrary. Sliding the fitting window would indeed improve the agreement of the fit (a) up to time $\sim 800$, but it would deteriorate at shorter times, and the fitting function would presumably still strongly depart from the data at longer times. It is not obvious if one can determine a finite time-scale beyond which the asymptotic law become effectively valid. Indeed, if we consider the fitting function (d) [Fig. 9], it appears that the 1/t term becomes smaller than the constant term if $t\gtrsim 1.8$. But for the constant term to become smaller than the (dominant) logarithm, one needs to reach $t\sim 9000$.

Reply to "Anonymous Report 2 on 2019-6-30 Invited Report"

1) A tanh-sinh profile similar to the initial data for the LL can also be implemented in the spin chain. Did the authors look whether the results presented here remain valid for such initial states?

No, we have only considered the "sharp" domain wall state, where $S^z=\pm1/2$. One attracting feature of this state is the absence of any "parameter" and its invariance under rotations about the $z$ axis, but it is clear that several other states seem worth investigating, including those proposed by the referee.

2) Regarding the corrections to the linear behavior discussed at the end of page 10. It is not clear to me, because of this, how the agreement between LL and XXZ can be seen as quantitative in a precise sense. Also, it appears that the agreement is not perfect at the isotropic point either. While I understand that the match between the various models is qualitatively remarkable, it is not clear how this may be turned into exact statements, such as the one for the density profile stemming from generalized hydrodynamics.

From our numerics (Fig. 3), the quantitative agreement between LL and XXZ appear to be very good for $\Delta=0.8,0.9$ and 0.95, if we omit the region corresponding to the tip of the front, where the profile has not yet converged to its asymptotic form (for $\Delta=0.99$ the result is expected to be good too, but for the accessible times the profiles are still far from "converged"). In fact for $\Delta<1$ we know from GHD and from the LL equation that both (quantum and classical) profiles will become linear at long times, with two slopes which become identical when $\Delta\to 1^-$. So, in the weakly easy-plane regime there is no doubt that the $z$ magnetization profiles become identical. The agreement for the in-plane component is also excellent (right panel of Fig. 2). However, as noted by the referee, it is at the isotropic point that the agreement is only semi-quantitative (Fig. 10 for instance).

3) Regarding the discussion of the entanglement entropy in the isotropic case. Does the log(R(t)) argument still underestimates the entanglement entropy upon inclusion of multiplicative logarithm corrections to R(t)?

Yes, even with the the multiplicative logarithm, log(R(t)) underestimates the entanglement entropy.

4) Page 13, last paragraph. 'exponential decay in r/t... This exponential decay is however cut for r/t=1'. I do not understand this sentence. Wouldn't it be more correct to say that there is no exponential decay in the quantum case, since e−1=cst ?

Yes, the referee is right, our formulation was incorrect.

6) Section 7.3. It is not clear to me how the LL and XXZ chain calculations should match, since the assumption of ferromagnetic correlations breaks down at very large distances. Usually when dealing with hydrodynamics descriptions of quantum systems, it is necessary to 're-quantize' the classical solution in order to access long-range correlations. This is what is done in Luttinger liquid theory, for instance. Could the authors comment on that?

The referee is right, at a given time we should not expect the quantum correlations to match the LL result beyond a certain distance. Even if the spins have locally some ferromagnetic correlations, at any finite time these correlations are not perfectly ferromagnetic. Nevertheless, we have some indication that, at a fixed distance $r$, the agreement between the XXZ correlations and the LL magnetization improves with time. This can for instance be seen in the right panel of Fig. 2, where the XXZ data at time $t=100$ appear to be closer to the LL result than the XXZ data at $t=70$.

Anonymous Report 3 on 2019-7-3 Invited Report

The main issue: I have serious doubts about the correctness of the heuristic argument presented by the authors to interpret the entanglement entropy growth. Indeed, they essentially invoke only the U(1) symmetry, which is present for arbitrary values of Δ. However, the authors themselves make a footnote that the argument does not apply away from Δ=1. But why? In particular, the same argumentation would give R(t)∼t and thus S(t)∼ln(t) for Δ<1 which is not the correct prefactor.

The first version of the manuscript was probably not clear enough on that point, and we thank the referee for raising this issue.

In fact our argument does not apply away from Δ=1, because it does not only rely on the U(1) symmetry. It is also necessary that the spins acquire perfect ferromagnetic correlations, and this is only the case at fixed position when $t\to \infty$ and Δ=1 (cf. energy conservation argument). If we instead consider Δ slightly smaller than 1, the nearest neighbor correlations will not converge everywhere to that of a triplet. There will be small deviations from $\langle \vec S_r\cdot \vec S_{r+} \rangle=1/4$. Although these small deviations may only weakly affect the local magnetization of the energy density, it can change completely the entanglement entropy. This can be illustrated, for instance, by considering a simpler case. Consider the ground-state of the xx chain, with a magnetization per site $m$ close (but not equal) to $-1/2$. The spin-spin correlations can be made arbitrarily close to that of a ferromagnet by taking $m$ to $-1/2$. Nevertheless the quantum state is a (low but finite density) Fermi sea. In such a case the entropy of a segment of length $l$ will scale as $\sim \log(l)/3$. This is clearly very different from the $m=-1/2$ case (which is a product state).

I believe, that this argument is just too naive to be able to interpret the entanglement. In fact, having a look at Fig. 11, one even observes that the ansatz log(t)/2 gives a rather poor description of the numerical data.

The referee is right, in the sense that log(t)/2 is only the "classical" contribution to the entanglement entropy, corresponding to a symmetrized classical state. And we also agree on the fact that the numerics indicate that the actual entropy is larger than this classical contribution (this has been made more clear in the new version, including, in particular, in the caption of Fig. 11).

(3) In Fig. 8 is I(t) the local (at r=0) or the global current plotted? How comes that it is nonzero at t=0?

The current is indeed zero at t=0, as it should. But the initial short-time increase of $I$ is quite abrupt at the scale of these plots, and it was no therefore visible. It is corrected now.

# From "Report 1"

> I do not think it is legitimate to proclaim
this derivation as "alternative" or "more direct"...

* The formulation has been changed in the revised version.

* We have improved the readability of the left panel of Fig. 9 (by a
better choice of colors, and by plotting the Landau-Lifshitz data
using crosses instead of a full line).

> Minor remarks: Bad spelling: "insure" should be "ensure" (end of
page 23). Just a suggestion which may slightly improve readability:
one could try use explicit symbols to the terms of the sort 1/4−S⋅S
and 1/4−Sz⋅Sz, e.g. using Hiso and V

* Done.

> (...) Saying "was observed to freeze" may read as an understatement.

* We agree with the referee (and changed the text accordingly).

# From "Report 2"

> (...) Wouldn't it be more correct to say that there is no
exponential decay in the quantum case, since e−1=cst ?

* Corrected.

> 5) Page 21, caption to figure 11. 't=185' contradicts the claim
't=210' at the end of page 19. Also, inconsistent notations τ vs dt.

* Corrected (we updated this plot, now the entropy data are available up
to t=400)

> 7) Section 7.6, last paragraph. Perhaps it would be better to state
more clearly that the divergences are precisely the logarithmic
divergences that make up for the logarithmically enhanced diffusion
discussed before.

* Done.

> Typos:
> Page 2, 'form' -> 'from'.
> Page 4, line 12. 'and in characterize'.
> Page 4, the last sentence of the introduction starts with 'And'.
> Page 10, last line: 'r/r' ->'r/t'.
> Page 18, second paragraph in 7.1. 'As can be seen in in' -> 'As can be seen in'
> Page 26, line before equation (39). 'the following the filament' ->'the following filament'.
> Also, the filament function is 'ψ(x,t)' not u(x,t), so the sentence reads awkwardly.
> Page 26, two equations are not numbered.

* Corrected, thanks.

# From Report 3

* We have improved the discussion on the entropy, at the end of Sec. 7.2.
We have explained why the semi-classical entropy argument does not
apply away from Δ=1.

> The smaller corrections:
>
> (1) In Fig. 4 the labels (1) and (2) should
> be shifted in order not to overlap with the x axis. There is also an
> extra parenthesis in the y axis label of (2).
> (2) In Fig. 7, there is a typo in the legend, Ωx should read Ωz.

* Done

* Improved Fig. 8, to make visible the fact that the current vanishes at t=0.

> (4) Before Eq. (39): “filament function” ψ(r,t) I guess this should
read u(r,t).
> (5) Typo later —> latter (appears many times in text)

* Corrected