1. Explores now also the key result, the length dynamics

1 - Validity of the way the Lindblad equation is used remains unclear 2- Applicability / generality of Lindblad approach misrepresented

I acknowledge that the author made several necessary corrections and improvements and also added useful and valid material in Section 6. One concern of mine was apparently misunderstood and remains only partially unanswered, another one remains unclear (see below). More importantly, the new version has new issues (see below). Overall, I can only partially agree with referee 1: The paper does concern an interesting discussion, i.e., deriving mathematically the LL equations from Lindblad is of value. But I concur with referee 2 that, in particular for physics, it should not leave key points unclear or implicit: If the mathematical manipulations cannot be identified with clear physical assumptions or ideas, then this of insufficient value for SciPost Physics. In this regard, the revised paper also sends an ambivalent message, e.g.,

“...link Lindbladian dynamics and LL(G) dynamics under <certain mild> conditions.” “... we find that the Landau-Lifshitz equation can be deduced from the Lindbladian under < mild and plausible> assumption. versus “Of course, the instantaneity <appears as a strong assumption> and future research should investigate which effects a relaxation of this assumption has.”

In my view, this remains a weak point of this work and I cannot recommend publication.

========================================== In response to my question :

"6. It is mentioned that the LL equation breaks down for λ>1. However, it is not clear or investigated whether the extended LL equation (7) is immune to any breakdown, even for λ<1, with the new term present for some initial magnitude of the |m|."

My concern was whether it was investigated whether the new equation shows a breakdown perhaps even in regimes where LL is <well> behaved. The modification of the paper regarding this is incorrect or confusing:

"… the instability of the LL equation for λ ≤ 1 also applies "

The breakdown of LL occurs for λ > 1 as the paper states?

My real concern was the following:

"Related: Is it possible that C=2S/|m|-1 changes sign during the dynamics, i.e., is it guaranteed that |m| < 2S with this term present?"

The author replied: "But from the inequality |m| ≤ S which is ensured for the quantum mechanical expectation value〈S 〉, it is evident that C ≥ 1."

My concern is that Eq. (9) [new version] is based on approximations which may have <broken this connection >between |m| and an expectation value of a spin vector. What the author says is what <should> be the case if all went well with the approximations. Breaking such ties is not uncommon: Apparently innocent approximations to quantum dynamics often result in non-positive dynamics of the density operator. Then probabilities become negative, even though these <should> be non-negative (expectations of positive projection operators) if no approximations were made. They approximated quantities have lost the tie to valid quantities of interest.

The author's statement that no such sign change occurred in the examples considered is of course valid and reassuring. However, the backward argument given above diverts from the relevant question about the validity of the “novel term” in the equation.

The new version has a number of new issues :

"In the weak-coupling derivation of the Lindblad formalism, one assumes that the rates γl are small relative to the time scales of H. But it is shown that the Lindblad structure (3) does not depend on the γl being small [31]."

"If one assumes that the Lindbladian is derived for weak-coupling limit [31], i.e., the coupling between the system and the bath is weak it is natural to assume that λ = γ/J is small. But, as indicated initially on the Lindblad formalism, this is not mandatory as far as the Lindblad formalism is concerned."

The paper here misrepresents “Lindblad formalism” as being more general than it is. If the decay rates are allowed to be larger, then they may no longer be time-independent or even positive! One is outside the Lindblad domain. The reference is to Breuer’s book where time-convolutionless (TCL) quantum master equations are discussed which are an entirely different thing which cannot be presented as “Lindbladian”:

• The superoperator structure of Eq. (3) is entirely due to the conservation of the hermicity of the density operator.This is confused with "Lindblad structure" in the paper. This holds for any dynamics, even with time dependent TCL generators, and is trivial to maintain. What is nontrivial are the contraints on the magnitudes and signs of the coefficients appearing there :

• However, specific to Lindblad dynamics are the positive rates $\gamma_l$. This is <not> general and derives from the Born-Markov-RWA assumptions that the paper mentions. This can break down even in simple models such as shown in Breuer’s book [31]. Models are even known for which one or several decay rates are negative for all times (“eternal non-Markovianity”).

Another statement that was added, also incorrectly suggests such generality: “... one can equally well resort o Eq. (5) which is based on a local mean-field approach, but does not require a particular energy hierarchy between internal and external magnetic energies, except that the bath is assumed to be internally fast.”

Here "except" does not state that the coupling is must be weak.

It is still not clear to me how the mean-field replacement affects the coupling to the reservoirs, i.e., whether it does not require some change of the Linbladian. The weak coupling transition rates (such as in the in Fermi’s Golden rule) depend on the transition energies of the many-body system (containing J!). A mean-field replacement <must> do something nontrivial with these, but this is not really addressed except for the statement:

“Eventually, one obtains Lindblad equations for each site..”

The paper seems to overstep this issue by always referring to one and the same Lindbladian, Eq. (3).

This following confusing: LL is correct but LLG is better suited? This is argued by whom?

“In accordance with the analytical derivation of the weak-field limit we find that the LL equation represents the correct limit of weak external fields best. It its argued that the LLG equation (1c) is better suited since it does not bear the risk of unphysical behavior.” ========================================== Appendix C: Caption Fig. 3 mentions lambda = 0.2, text mentions lambda=0.5.

I'm confused why Appendix C does not simply plot the LL and LLG predictions on in the same Fig. 3? That’s the comparison of interest. Instead LL and Lindblad are compared with a round about discussion why this is not LLG.

1- Correct statements about "beyond Lindblad" 2-Careful study of reduction of many-body Lindblad equation to self-consistent Lindblad equation

Reject

Some optional points that could be considered in addition:

Generically, the quantum spin couples via an exchange interaction to the local spin moment in the electron system. I do not see why spin friction mediated by phonons is discussed exclusively instead of conduction-electron spin and charge degrees of freedom.

Eq.8 is the self-consistency equation of a local mean-field theory. This immediately raises the question of how the resulting LL equation would look like, if nonlocal correlations were also included, e.g., as a first step, short-range correlations as captured by a cluster-mean-field-type approach. Is there a plausible expectation of how the spin dynamics would change with increasing cluster size?

It could be emphasized that the presented approach is fundamentally different from the usual ways to derive the damping term, as also pointed out by Referee 1. Those approaches lead directly to Eq. (1c) with the additional term interpreted as friction and are valid even for a single magnetic moment &lt;s&gt;, as long as it remains stable (no Kondo effect etc.). However, the resulting Ehrenfest-type dynamics is also a mean-field dynamics, so there are similarities that could be explored further.

Publish (meets expectations and criteria for this Journal)

Strengths: 1. mathematically technically clear 2. Mathematically valid 3. Reproducible

1. Key approximation not physically motivated sufficiently and convincingly
2. Overall motivation not sufficiently clear, especially not from physics point of view

In my previous report, I requested the following two changes: 1.Improve the discussion of why/how the Lindblad operator adapts to the field, indicate the regime of validity of this approximation, and discuss for which baths this approximation holds

1. Improve the motivation for this work. Argue convincingly why it is needed that the Landau-Lifshitz equation should follow from the Lindblad Master equations, and/or in which situations it should follow. Regarding point 1: the author argues that the Lindblad operator adapts to the field because then the entropy of the system increases. However, to me this sounds like using a macroscopic law (2nd law of thermodynamics) to argue for a microscopic description (the Lindblad operator) that then is tweaked to the macroscopic requirements but is not necessarily rooted in reality. I would think that the macroscopic dynamics should follow from the microscopic description without the need for such finetuning. In physical magnets, there are well-known baths (phonons for insulators, electrons for metals), with well-known microscopic descriptions which do not a priori “know” of the second law of thermodynamics, and - for this case also relevant - do not involve the magnetic field. The resulting dynamics, when coupling the magnetic order parameter to the baths, is such that energy flows to the bath in certain limits and approximations, and that entropy increases. Typically this is because the bath has a large amount of degrees of freedom when compared to the system of interest, the precessing magnet. In conclusion, regarding point 1 I do not find the argument of why the Lindblad operator adapts to the field convincing. Regarding point 2: The author maintains in the manuscript that “If the damping of a magnetic systems can be described by the LL(G) equations, which is governed by a single relaxation rate, the latter should be derivable from Lindbladian dynamics. So far, however, Lindblad dynamics and LL(G) dynamics are not linked by a mathematically rigorous derivation, except that for special systems where their outcomes are the same [15] for small deviations from equilibrium. This situation is unsatisfactory: spins are quantum objects so that the Lindblad approach is applicable. “ Moreover, the author adds in the response that “As for the motivation, it is in essence scientific curiosity which motivated me to search for a link between two seemingly very different formalisms. I do not think that this curiosity is a flaw.” I do not have a problem with curiosity per se, but I do think that the results can be misleading. I reiterate my statement that the LLG is intended to describe the collective classical transverse dynamics of the magnetic order parameter. Modelling it as arising from single quantum spins by 1) ad hoc – see point 1 - choosing a Lindblad operator and 2) coupling the spins in a mean-field approximation is very likely to be far from the physical reality for reasons I mention above and in my previous report. However, not all readers may be aware of this and may take these results to built upon. One can see also in the overinterpretation of results from atomistic spin simulations. In such simulations, each individual spin is modelled classically by an LLG equation. Results from such simulations have their merit for prediction low-frequency dynamics, but should be treated with caution when the predictions involve time and length scales that involve the exchange interactions. Similarly, the authors’ equation (5) is basically an LLG which involves the exchange interactions after insertion of Eq. (8), but is not intended to work at those energy/time scales. In conclusion, from my understanding of the author’s motivation in the previous response letter, one could view the results in the article as a mathematical way to derive the LLG with Landau-Lifshitz damping from a Lindblad formalism. However, I do not think that physically the modelling is well-founded (point 1 above), or that physically a close link between the LLG and the Lindblad formalism as applied to a single spin should necessarily exist (point 2). I think that without elaborate and convincing discussions of both points I mention at the beginning of this report, the paper should not be published.

1. Give elaborate discussion of range of validity of key assumption
2. Give elaborate discussion of motivation, especially from physics point of view

Reject