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Learning crystal field parameters using convolutional neural networks

by Noah F. Berthusen, Yuriy Sizyuk, Mathias S. Scheurer, Peter P. Orth

Submission summary

As Contributors: Peter P. Orth
Preprint link: scipost_202012_00008v1
Code repository:
Data repository:
Date submitted: 2020-12-11 11:43
Submitted by: Orth, Peter P.
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Computational


We present a deep machine learning algorithm to extract crystal field (CF) Stevens parameters from thermodynamic data of rare-earth magnetic materials. The algorithm employs a two-dimensional convolutional neural network (CNN) that is trained on magnetization, magnetic susceptibility and specific heat data that is calculated theoretically within the single-ion approximation and further processed using a standard wavelet transformation. We apply the method to crystal fields of cubic, hexagonal and tetragonal symmetry and for both integer and half-integer total angular momentum values $J$ of the ground state multiplet. We evaluate its performance on both theoretically generated synthetic and previously published experimental data on PrAgSb$_2$ and PrMg$_2$Cu$_9$, and find that it can reliably and accurately extract the CF parameters for all site symmetries and values of $J$ considered. This demonstrates that CNNs provide an unbiased approach to extracting CF parameters that avoids tedious multi-parameter fitting procedures.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-2-21 Invited Report


The manuscript entitled “Learning crystal field parameters using convolutional neural networks” by Berthusen et al presents a machine learning algorithm to extract crystal field parameters from experimental thermodynamic data obtained in rare-earth-based materials. The authors apply the method to cubic, tetragonal, and hexagonal site symmetries as well as integer and half-integer total angular momentum, J, ions. Finally, the authors evaluate their algorithm against experimental data on PrAgSb2 and PrMg2Cu9.
Using convolutional neural networks to solve this important inverse problem in condensed-matter physics is a worthwhile idea, which is presented in a very pedagogical way that will benefit early-career researchers in the field. Extracting crystal field parameters is known to be time consuming, and the fact that the algorithm is provided as an open source will be useful to the community. Overall, this manuscript meets SciPost’s criteria for publication. A few issues need to be addressed, however, before the manuscript is published.
1. One general limitation of performing fits of thermodynamic data to a crystal field Hamiltonian is the possibility of reaching a local minimum that reproduces the data reasonably well but that does not necessarily contain the correct Steven parameters. In particular, this is more likely to happen when a large number of parameters is taken into account. The authors address the issue of over-parametrization, but could the authors comment on whether their algorithm has encountered local minima (of e.g. MSE) during the development of this work? In particular, one would naively think that local minima may be why “the CNN can accurately predict the Stevens parameters for the majority of the data points that it was tested on”, but not all data points.
2. To address the issue above, it might be very enlightening to apply this algorithm to data sets of materials in which neutron scattering measurements have been performed. Two examples that come to mind are CeAgSb2 (from the same family of PrAgSb2) and CeRhIn5 (at high temperatures T>T_K, the Kondo temperature). One interesting aspect here is that CEF fits and neutron data agree well for CeAgSb2, whereas the agreement is worse for CeRhIn5. This could obviously be due to the Kondo effect, but nonetheless it would be a very informative comparison.
3. I further encourage the authors to consider Cerium instead of Praseodymium because the former has only one f-electron, whereas the latter has two, which could give rise to J mixing. This has been shown, for instance, in the case of Pr2Sn2O7 via neutron spectroscopy [PRB 88, 104421 (2013)]. Even if the authors are absolutely convinced there is no J mixing in the Pr compounds investigated in their work, it is worth mentioning this general possibility in the manuscript to inform the reader of this potential issue.
4. On the Kondo effect, the authors correctly point out in the Introduction that the crystal field scheme can also have important ramifications for the nature of the Kondo effect in the system. In a simple local picture, this can be understood by considering that the shape/anisotropy of the ground state wavefunction is related to the overlap between f electrons and conduction electrons, which in turn determines the Kondo hybridization. However, the citations provided [11-14] are not particularly general, and they mostly focus on U-based materials, for which the LS coupling may not be valid as mentioned in point #3. To address SciPost’s criterion to “Provide citations to relevant literature in a way that is as representative and complete as possible” – and the fact that the community is considering crystal field effects more strongly in the recent past – the authors are encouraged to provide more citations to the relevant literature.
5. It could be useful to mention that the leading term, B20, is proportional to the difference in Weiss temperatures along different axes. Even though the method introduced by the authors is supposed to be unbiased, there are tricks that could be useful for training data.
6. Regarding the outlook of this work, including magnetic interactions will be very valuable. The easiest way of doing this is by adding a molecular field term as employed in JPSJ 70, 877 (2001), for example. The next step would be to include a mean-field Hamiltonian with one exchange constant and so on.
7. This is a minor point, but it could stimulate readers to actually use the open software in Ref. [47]: one of the main motivations of this work is to circumvent a time-consuming fitting procedure, but if one is using a conventional minimization method, a significant portion of the overall time is also spent i) preparing/converting the data and ii) waiting for iterations. Could the authors comment on the overall time spent during their approach (machine time and researcher time)?
8. A couple of final points regarding input data: i) similar to the subtraction of the phonon contribution to obtain the magnetic part of the specific heat, one also has to subtract an enhanced Pauli susceptibility from magnetic susceptibility data (if applicable); ii) trying to fit experimental data at low temperatures (eg M(H) at 1.7K) may be very hard, even if there is no magnetic order because of the effects of magnetic exchange. I wonder whether this is causing the disagreement in Fig 8d. This could also be an indication that, although the CF splitting is predicted correctly, the ground state wavefunctions are not. Therefore, it might be more useful to only fit magnetic susceptibility and magnetization data at higher temperatures, e.g., T>25K or so.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2021-2-3 Invited Report


- Well-posed physical and realistic inverse problem adapted to machine learning
- Useful for experiments
- Open source code associated
- Well-written paper


- No weaknesses


This manuscript presents a neural network analysis to infer crystal field properties (here in terms of Steven parameters) for thermodynamic measurements in f-electrons materials . This is a typical well-posed inverse problem in experimental physics, where the ‘deduction’ of crystal field parameters can become quite cumbersome.
This problem is well suited to a machine learning approach as the number of outputs (Steven parameters) to find is not too large (of the order O(1) to O(10)) and it is easy to generate training data in the single-ion approximation.
The approach chosen is through a 2d convolutional neural network of the wavelet-transformed thermodynamic data. At first glance, this seems a bit complicated and it is not clear why this approach is taken; later in the paper we understand that other simpler approaches (simpler feed-forward neural networks, 1d CNN) do not give good results.

The efficiency of solving this inverse problem is measured through a metrics in the Steven parameters space (distance between measured and true parameters) and in the thermodynamic data. Convincing results are presented for standard symmetries for synthetic data, as well as by using input thermodynamical data from experiments on two compounds to obtain Steven parameters and then recompute thermodynamics (which are found to reproduce the main features of the experimentally obtained one).

While not revolutionary, this is a simple yet useful example of how the machine learning approach can reduce a somewhat tedious task often performed manually in experimental labs. The paper is very clearly written, and is accompanied by an open source repository where the corresponding code can be directly used to compute Steven parameters from thermodynamic measurements. This clearly will be useful for future experiments.

This paper can be accepted as it is, and I have only basic questions / suggestions.

- To improve quality of the results for the hexagonal and tetragonal point group, it is suggested to add other input data such as a second magnetisation curve (obtained from a field aligned in a different direction). I guess this is quite easy to add in the training data and approach, and I would suggest to try it to see if it indeed improves the results. Same question for a larger temperature range.

- When comparing to experimental data, it would be interesting to see how good/bad is this approach when one of the experimental input is missing (as could happen in a lab, if e.g. specific heat measurements are not available).

- Typo : the from (2nd column, page 4)

Requested changes

- No changes requested, but the authors could consider the two suggestions above

  • validity: top
  • significance: good
  • originality: ok
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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