SciPost Submission Page
Simulating the interplay of dipolar and quadrupolar interactions in NMR by spin dynamic mean-field theory
by Timo Gräßer, Götz S. Uhrig
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Timo Gräßer · Götz S. Uhrig |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2507.17720v1 (pdf) |
| Code repository: | https://doi.org/10.17877/TUDODATA-2025-MD4EYWOL |
| Date submitted: | Aug. 1, 2025, 2:57 p.m. |
| Submitted by: | Timo Gräßer |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
The simulation of nuclear magnetic resonance (NMR) experiments is a notoriously difficult task, if many spins participate in the dynamics. The recently established dynamic mean-field theory for high-temperature spin systems (spinDMFT) represents an efficient yet accurate method to deal with this scenario. SpinDMFT reduces a complex lattice system to a time-dependent single-site problem, which can be solved numerically with small computational effort. Since the approach retains local quantum degrees of freedom, a quadrupolar term can be exactly incorporated. This allows us to study the interplay of dipolar and quadrupolar interactions for any parameter range, i.e., without the need for a perturbative treatment. We highlight the relevance of local quantum effects by a comparison with the classical analogue system.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1) The manuscript is well written and structured, mostly self-contained and pedagogical. It should hence be accessible even for non-specialists. 2) The comparison of the results obtained from quantum and classical treatment of the spins allows to appreciate the importance of the dynamical treatment brought about by spinDMFT.
Weaknesses
1) Even though the limitations of the technique due to its inherent approximations are clearly stated, it is less clear how well the technique performs for calculating spin autocorrelations as relevant to NMR measurements. Since this is the motivation of the present manuscript, an exemplary model system with dipole-dipole and quadrupolar interaction that can be directly discussed in comparison to existing NMR measurements would be helpful in this context.
Report
However, even though the work is motivated by NMR, a direct (even qualitative) comparison to experimental data is unfortunately absent. This is why I would not go as far as qualifying it as a breakthrough on a long-standing research stumbling block as listed in the author indications on fulfilling the journal expectations.
All in all, I recommend the manuscript for publication in SciPost Physics after minor revisions, see requested changes below.
Requested changes
1) Statistical errors of the Monte Carlo (MC) simulation are mentioned explicitly in Fig. 2, but they should lead to error bars for most of the shown quantities. The authors might want to either include error bars in the respective figures or state clearly that those error bars would be smaller than the symbol size of the data points. 2) In Sec. III.B the authors fit exponentials to the longitudinal correlations, stating in Table I that numerical errors are in the order of $10^{-2}-10^{-3}$. They might want to clarify whether these errors refer to the quality of the fits or whether they also include the MC error. 3) Why is the spinDMFT broadening Gaussian and not e.g. Lorentzian? Is it rooted in the use of a Gaussian probability functional? What is the interpretation of this ‘Gaussian prediction’ of the line shape and has it been observed experimentally via NMR? The authors might want to comment on these questions in the manuscript. 4) The steps in between Eqns. (27a-c) should be explained in more detail.
Minor details: 5) Eqn. (22) lacks two right brackets ")". 6) In Fig. 8, the authors should mention the dashed orange line (analytic result of eqn. (27)) in the caption. 7) I would also like to ask the authors to check again the prefactor of Eqn. (27b). I may be wrong, but I think it lacks a factor $2/\vert \omega\vert$.
Recommendation
Ask for minor revision
Author: Timo Gräßer on 2025-10-30 [id 5969]
(in reply to Report 1 on 2025-09-19)General reply:
First of all, we thank the Referee for the very thorough reading of our manuscript and the constructive comments. We are very glad that the paper is perceived as "well written" and recommended for publication in SciPost Physics.
Furthermore, we fully understand the Referee's concern regarding the lack of a comparison to experiment. However, we would like to emphasize that the purpose of this work is a proof of principle. Our main goal was to demonstrate that quadrupolar interactions can be directly included in spinDMFT without changing the approach itself. Of course, a comparison to experiment is desirable, but unfortunately, we did not find a suitable and simple benchmark system with available experimental data in the literature. Measurements on quadrupolar spins in monocrystals are rarely performed and nowadays NMR experiments are rather complex (involving for example magic-angle spinning), which requires one to include many further details in the simulations complicating a direct comparison.
1. requested change:
Statistical errors of the Monte Carlo (MC) simulation are mentioned explicitly in Fig. 2, but they should lead to error bars for most of the shown quantities. The authors might want to either include error bars in the respective figures or state clearly that those error bars would be smaller than the symbol size of the data points.
Reply:
We fully agree that numerical errors should be discussed more thoroughly. The Monte-Carlo errors were already smaller than any linewidths except for the logarithmic plot. However, the error from finite time discretization, which we did not yet consider in detail, turned out to be relevant for large Omega. We performed additional simulations to obtain error estimates for this. Unexpectedly, we obtained changes in Fig. 2 and 3 and Tab. 1 for large Omega, when making the time discretization finer. The new plots suggest a saturation of the decay time for large Omega, which has not been discussed in the literature before. We assign this saturation to an effective dynamics that can be accessed by average Hamiltonian theory in the limit of large quadrupolar interactions.
Action taken:
We included error estimates or error bars in the respective figures and tables. We added a short paragraph about the effective dynamics in the limit of large quadrupolar interactions.
2. requested change:
In Sec. III.B the authors fit exponentials to the longitudinal correlations, stating in Table I that numerical errors are in the order of 10−2−10−3. They might want to clarify whether these errors refer to the quality of the fits or whether they also include the MC error.
Reply:
The errors referred to the quality of the fits. Indeed, we changed the fitting procedure in Fig. 2 from a two-parameter to a one-parameter fit which turned out to be significantly more robust at the price of yielding a slightly worse agreement for small values of Omega.
Action taken:
We changed the fitting procedure as described above. We decided to include estimates of the numerical error, which is dominated by the time discretization, instead of estimates of the fit quality.
3. requested change:
Why is the spinDMFT broadening Gaussian and not e.g. Lorentzian? Is it rooted in the use of a Gaussian probability functional? What is the interpretation of this ‘Gaussian prediction’ of the line shape and has it been observed experimentally via NMR? The authors might want to comment on these questions in the manuscript.
Reply:
We thank the referee for pointing this out. The Gaussian probability functional results from the central limit theorem for the mean-fields and it does not directly imply that the line shapes are Gaussian. Yet, the separation of time scales of transverse and longitudinal spin dynamics matters: it is induced here by the anisotropy of the dipolar interaction as well as by the purely longitudinal quadrupolar term. Thus, the longitudinal mean-fields are dominant and have a very long correlation time, i.e., they are rather static. This makes the dynamics comparable to that of an Ising model, where the line shapes are exactly Gaussian if there are many interaction partners so that the central limit theorem applies. We found a reference measuring NMR spectra in aluminium nitride, where indeed Gaussian dipolar line broadening is seen. However, we stress that the scenario is not equivalent to our model as it involves two spin types and autocorrelation and FID spectra are distinct from each other.
Action taken:
We added a paragraph about this in the manuscript.
4. requested change:
The steps in between Eqns. (27a-c) should be explained in more detail.
Reply:
We agree that the derivation was too brief.
Action taken:
We added further explanation sentences and an intermediate calculational step.
5. requested change:
Minor details: 5) Eqn. (22) lacks two right brackets ")". 6) In Fig. 8, the authors should mention the dashed orange line (analytic result of eqn. (27)) in the caption. 7) I would also like to ask the authors to check again the prefactor of Eqn. (27b). I may be wrong, but I think it lacks a factor 2/|ω|.
Action taken:
We thank the referee for the corrections and fixed all minor issues.
Attachment:
main_redline.pdf