Landau-Lifshitz damping from Lindbladian dissipation in quantum magnets

Where necessary parts of the report are repeated and given in quotation marks.

"Strengths
1- Points out a basic feature of self-consistent use of Lindblad equation for single spin"
The formulation is slightly misleading because the self-consistency promotes the single-spin treatment to a local mean-field theory of an extended system

"One motivation is to provide a general approach for computing damping parameters from explicit descriptions of the reservoirs."
This sentence is not correct since the paper does not aim at an explicit description of the reservoirs, but establishes a fairly general link between a Lindbladian and the Landau-Lifshitz formalism of relaxation with a minimum of parameters.

"Although a basic calculation, this seems to be an interesting point."
Thank you for the interest in this result.

"However, the issue lies with motivating the equations and their use in self-consistent fashion. Here I concur with the first referee. As a result it remains unclear what precisely one can conclude from the (correct) analysis performed which aims to "fix a conceptual flaw". This leaves too many open ends to recommend publication in SciPost Physics. "
The modified version elucidates the arguments justifying the equations and removes some slightly inaccurate statements.

As for 1.
Equation (3) holds very generally; it is (almost) the text book version. In order to avoid confusion, I added explicit formulae for the Hamiltonians involved. It is true, that the statement on J was premature. It has been modified.

As for 2.
Even though it is very common to justify the Lindblad formalism in the weak-coupling limit its structure also holds for strong coupling. So, there is no need to impose a particularly small lambda. This is changed in the modified version. Moreover, the value J is indeed introduced later. Accordingly, a reference to J is postponed.

As for 3.
I agree that one has to assume this for the LL equation to be robust. But it is not the Lindblad formalism which requires it as clarified in the revised version.

As for 4.
"This does not justify it: it is just consistent with that approximation."
This is precisely what is meant: The approximation is justified a posteriori because the physics is consistent with its assumptions. The formulation has been adapted.

As for 5.
"- It is remarkable that in the case of saturation |m|⃗ = S the time scale 1/γ disappears completely and only the time scales 1/h0 and J/(γh0 ) survive."
I agree that this is truly remarkable making the presented finding an interesting and relevant result. As such it is not uncommon that certain fast energy scales drop out if one considers complex equations in certain limits.

"Unfortunately, this is not explored either but this should be easy with the numerics all set up. E.g. what special thing happens in the full self-consistent equations (5)+(6) for such initial lengths?"
I agree that the length change deserves investigation. Thus, the additional section 5 has been included discussing exemplary length changes on the basis of the extended formula.

As for 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|. "
The extended equation is not immune either because it becomes exactly the LL equation for saturated m. This is mentioned now in the revised version.

"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?"
If |m| is the spin expectation value a sign change cannot happen because |m|<=S rigorously. In the (approximate) numerics an overshooting of |m| cannot be excluded fully. But no such phenomenon has occurred in my simulations. This is now mentioned in the revised version.

As for 7.
The formulation was indeed slightly inaccurate: The fact that the magnetic order can have arbitrary length which changes in time is clearly a sign of quantum mechanics because a classical vector would have a fixed and constant length. In this point the local mean-field theory goes beyond the classical description. But I agree that it is conceivable that for other models the local mean-field treatment is still equivalent to a classical treatment. It does not per se imply quantum behavior.

As for 8.
Thank you for pointing these issues out. All of them have been accounted for in the revised version.

Requested changes are carried out.

Parts of the report are quoted to render it clear to which the reply refers.

"Weaknesses 1. Key approximation not physically motivated sufficiently" I provide further justification of the approximation here and in the revised version.

"2. Overall motivation not sufficiently clear" See below for my view on the overall motivation.

"The work by Uhrig ... latter contribution survives." By and large, I agree to the above summary, except for the last sentence since no high-frequency dynamics occur in the equation, neither in the Lindblad nor in the Landau-Lifshitz equation and thus, it is not damped out. Implicitly, however, there is some high-frequency dynamics going on in the bath which determines the Lindblad decay rate lambda.

"Regarding the approximations: the author’s key assumption is that the Lindblad operator adapts instantaneously to the external field. The question is, how does the bath do this? Particularly, a phonon bath does not couple to an external magnetic field and there seems to be no physical mechanism for the bath to “know” which angular momentum to absorb."

Thank you for this excellent question! Of course, the generic bath does not know about the orientation of a magnetic field. But this issue arises already in the first static case of a single spin. The answer is fundamental to thermodynamics: Boltzmann’s H theorem teaches us that the temporal evolution, at least after some coarse-graining of states, increases the total entropy. Hence, if there is a system in contact with a large bath at zero temperature the excess energy of in the system will flow into the bath and relax the system to its ground state in the long run. This is reflected in the Lindblad formalism in the condition that the Lindblad operator B_l increases the system’s energy. In this way, the bath “knows” the orientation of the magnetic field because only by aligning the spin to it the system’s energy is decreased, i.e., energy flows from the system into the bath incrementing its entropy.

The same holds for a slowly varying Hamiltonian if we assume that the bath is acting fast. The coupling to the bath will make the instantaneous energy in the system decrease.

The above reasoning has been included in the revised version for clarification.

"Regarding motivation: the author insists that it should be possible to derive the Landau-Lifshitz equation from Lindbladian dynamics because “… the spins are quantum objects and cannot have their own relaxation independent from the general quantum theory.” To be honest, I do not fully understand this sentence. I will assume that it is intended to mean that the since Landau-Lifshitz equation describes spins, and that since spins are quantum objects of which the relaxation should be described by the Lindblad equation, it should be possible to derive the Landau-Lifshitz equation from the Lindblad equation. " This is precisely the gist of the motivation, which I perhaps did not formulated optimally. I modified the formulation for improved clarity. I emphasize that the motivation in the paper is just that: a motivation for asking for a link between two formalisms out of scientific curiosity. The motivation is not meant to be a rigorous argument and I do not think that it would have to be since the following calculation realizes such a link.

"However, this motivation is unclear to me in the following sense: according to my understanding, the Landau-Lifshitz equation is intended to describe the low-frequency long wavelength dynamics of the magnetic order parameter far below the Curie temperature – so that amplitude fluctuations are irrelevant - and does not apply to an individual spin in a material. As such, the dynamics that the Landau-Lifshitz equation describes is inherently classical. " I agree as far as one restricts oneself to small deviations from equilibrium. But we address relaxation upon application of an external field and the system can have been manipulated before by coherent pulses to prepare non-equilibrium states. If it were in its unaltered ground state no relaxation would occur. But away from equilibrium it is a valid question to address fluctuations of the order parameter in the orientation and in the amplitude.

"Equations (5) and (6) suggest that an individual spin the material undergoes damped precession, even at the energy scales set by J, with a single damping parameter lambda. " Let me emphasize that the combination of equations (5) and (8) (formerly (6)) correspond to the self-consistent mean-field treatment of the order parameter, i.e., it describes the collective motion of the order parameter. Note that all local mean-field theories reduce an extended many body problem to an effective single site problem plus a self-consistency condition, see, e.g., the famous fermionic dynamic mean-field theory. Thus, the dynamics of the magnetization is not the dynamics of a single spin.

"However, in any realistic material, I do not think this is the case. If one would experimentally probe a single spin at high frequencies, one would find that it decays in a very different way than the Landau-Lifshitz equation predicts. It is only the collective motion of the spins that at small frequencies and long wavelengths that should be described by the Landau-Lifshitz-equation. " As pointed out above, it is precisely the dynamics of the order parameter which is described by the advocated equations in line with your argument.

"As a second, related, point, I do not think that the Landau-Lifshitz form of the damping is the appropriate one to focus on. The Gilbert damping form of the damping follows is in my opinion, more fundamental, as it does not depend on the field itself, but only on the frequency of the dynamics, similar to viscous friction forces in other systems. " The Referee knows more about LL and LLG treatments than I do. It is well possible that a link to the LLG equation would be even more desirable. Within the present line of argument, only the LL equation appeared although I looked for signs of LLG dynamics. Anyway, the found link between the Lindblad and Landau-Lifshitz is probably not yet the end of the story and further extensions are certainly possible.

"Finally, in the motivation, the author cites a paper, Ref. [15], where there is agreement between the Lindblad formalism and the Landau-Lifshitz equation. However, in this case the Lindblad formalism is applied to quantized excitations of the order parameter, magnons, rather than to microscopic spins." I quoted and summarized this nice paper of yours in the Introduction and I acknowledged that an agreement of both approaches was found, here and in the manuscript. If I am missing a point or mis-represent your results, please inform me so that I can improve on this point.

In my view, Ref. 15 goes beyond what my calculation is aiming at, but at the price of generality. The relaxation in Ref. 15 is dealt with not in a local mean-field theory, but on a spin wave level with magnons as damped harmonic oscillators. Thus, it is more microscopic, but less general: note that the agreement in Ref. 15 only occurs in leading order of the deviations from equilibrium and assuming special properties of the baths. In contrast, I only assume that the bath is fast and that the order parameter can point in any direction.

"In view of these unclarities in arguing for the key approximation and the overall motivation, I do not think the current form of the article meets SciPost’s criteria for publication." I hope to have provided sufficient explanations for the approximation. As for the motivation, I re-iterate that it is in essence scientific curiosity which indeed has led to an interesting link.

"Requested 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" This has been discussed here and in the revised manuscript.

"2. 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." 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.