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Identifying Chern numbers of superconductors from local measurements
by Paul Baireuther, Marcin Płodzień, Teemu Ojanen, Jakub Tworzydło, Timo Hyart
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Paul Baireuther · Timo Hyart |
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| Preprint Link: | https://arxiv.org/abs/2112.06777v2 (pdf) |
| Date accepted: | 2023-11-09 |
| Date submitted: | 2023-06-05 20:30 |
| Submitted by: | Baireuther, Paul |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Fascination in topological materials originates from their remarkable response properties and exotic quasiparticles which can be utilized in quantum technologies. In particular, large-scale efforts are currently focused on realizing topological superconductors and their Majorana excitations. However, determining the topological nature of superconductors with current experimental probes is an outstanding challenge. This shortcoming has become increasingly pressing due to rapidly developing designer platforms which are theorized to display very rich topology and are better accessed by local probes rather than transport experiments. We introduce a robust machine-learning protocol for classifying the topological states of two-dimensional (2D) chiral superconductors and insulators from local density of states (LDOS) data. Since the LDOS can be measured with standard experimental techniques, our protocol contributes to overcoming the almost three decades standing problem of identifying the topological phase of 2D superconductors with broken time-reversal symmetry.
Author comments upon resubmission
Resubmission Letter
We thank the editor for considering our manuscript and for providing additional references to put our work in a larger context.
We thank the referee for his valuable feedback based on which we have revised the manuscript.
Response to referee report
Dear Prof. Del Maestro,
Thank you very much for your valuable feedback, based on which we have revised our manuscript. We have addressed most of your comments and suggestions directly in the manuscript and have summarized those changes in the "List of changes". In addition, in the manuscript we have provided references to the source code we used to train and evaluate the neural network ensembles as well as to the Shiba lattice model dataset. In the following, we would like to address the remaining points.
Regarding our choice of model parameters, and their connection to real engineered quantum systems:
1.1 In [1], it was shown that the Shiba lattice model phase diagrams are qualitatively similar for a range of values of \(\lambda\). We only require that \(\lambda\) is not too small, because it controls the size of the bulk energy gap. Otherwise, the exact value of \(\lambda\) is not important.
1.2 While in typical experiments the lattice constant is shorter than the coherence length, we choose this slightly longer lattice constant to obtain a simple phase diagram containing large patches of topological phases. Moreover, in experiments the magnetic atoms are typically placed on the surface of a 3D superconductor, reducing the coupling between the Shiba states in comparison to our 2D model calculations. Therefore, the magnitudes of the effective couplings between the Shiba states in our model are probably quite realistic although we set the distance between the impurity atoms larger than in experiments. We have also included this paragraph in the manuscript.
1.3 For our numerical experiments it is important, that the system size is larger than the localization length of the edge modes. In addition, we believe that \(24 \times \xi\) is a system size which does not pose a significant experimental challenge.
1.4 The energy window we are considering for the local density of states is quite realistic. We are also considering quite large disorder strengths so that if experiments would achieve lower levels of disorder the performance could even be better.
As you suggested, we have included clarifying sentences about the disorder potential and spin-orbit coupling at an earlier part of the manuscript. Overall, we would like to keep the separation of technical details in order to appeal to a broader audience who are not necessarily interested in the details of the models.
[1] J. Röntynen and T. Ojanen, Chern mosaic: Topology of chiral superconductivity on ferromagnetic adatom lattices, Phys. Rev. B 93, 094521 (2016).
List of changes
1. We have re-run our numerical experiments systematically and updated the figures and numbers accordingly. The results have slightly improved which we attribute to a :math:`\sim20\%` increase in training dataset size.
2. In the references, we provided a link to the source code that we used to generate the training data as well as to train and evaluate the neural network ensembles.
3. In the references, we provided a link to the Shiba lattice model dataset on which we evaluated the performance of the neural network ensembles.
4. In the introduction, we added a paragraph citing the works suggested by the editor along with some complementary references from the machine learning assisted quantum phase classification literature. We conclude this paragraph by explaining why our work is novel in this context.
5. We added an explanation to the caption of Fig. 1, as to why the types of images we use as input to the neural networks can be expected in an experiment: "The LDOS is averaged over a symmetric energy window around zero energy :math:`[-\overline{\Delta}_{\text{shiba}} / 6, \overline{\Delta}_{\text{shiba}} / 6]`, and therefore, the tunneling density of states is proportional to the quasiparticle density of states. Hence, STM measurements in the weak coupling limit will produce the type of pictures illustrated here."
6. We now introduce the disorder potential scale :math:`V_0` at an earlier point in the manuscript.
7. We reformulated our claims in a more modest wording in abstract, introduction and discussion. In addition, we further highlighted the scope of our numerical experiments by adding "In the future it would be important to extend testing of the protocol beyond the single model and restricted parameter space used in this work." to the discussion section.
8. We have expanded the discussion of our choice of model parameters and now explain that "The qualitative features of the phase diagram do not depend on the exact value of :math:`\lambda` [1]." and argue that our system size is realistic, because we expect that "manufacturing systems of size :math:`24\xi` does not pose a significant experimental challenge". With respect to the question why we set the superconducting coherence length equal to the impurity lattice spacing, we now clarify that "While in typical experiments the lattice constant is shorter than the coherence length, we choose this slightly longer lattice constant to obtain a simple phase diagram containing large patches of topological phases. Moreover, in experiments the magnetic atoms are typically placed on the surface of a 3D superconductor, reducing the coupling between the Shiba states in comparison to our 2D model calculations. Therefore, the magnitudes of the effective couplings between the Shiba states in our model are probably quite realistic although we set the distance between the impurity atoms larger than in experiments.".
9. In our previous numerical experiments, we had used a slightly different threshold for the minimum bulk gap in training and test data which made the manuscript difficult to read. Therefore, we have unified this in our updated numerical experiments to :math:`3.5 / N_x \approx 0.15`.
10. We have revised the methods section, added further technical details, and provided a reference to the source code that we used to train and evaluate the neural network ensembles.
11. We fixed the spelling and formatting errors pointed out by the referee.
[1] J. Röntynen and T. Ojanen, Chern mosaic: Topology of chiral superconductivity on ferromagnetic adatom lattices, Phys. Rev. B 93, 094521 (2016).
Published as SciPost Phys. Core 6, 087 (2023)
Reports on this Submission
Report
The manuscript studies whether a supervised ML algorithm can be useful in identifying the Chern number of a Shiba lattice using LDOS. The manuscript correctly states that LDOS in principle does not contain sufficient information to identify the Chern number, and therefore argues that using a combination of knowing the conceptual model describing the Shiba lattice together with a trained ML model is up to the task. This is supported by examining the generalizability of the model outside of the training dataset. This validation, however, is insufficient for the algorithm to be applicable to a real world scenario: a real life system may have more dramatic biases, such as different spin order, terms omitted from an effective Hamiltonian, spatial inhomogeneity, and a significantly larger disorder. Furthermore, the manuscript does not address the question how one would determine whether in a specific experiment one could defend that the proposed approach has a sufficient generalizability. For that reason I believe that the manuscript details a useful advance in ML identification of topological phases. It, however, does not address the problem it is set to resolve—namely the identification of the topological phases based on the real world data. Therefore I consider the manuscript suitable for SciPost Physics Core rather than SciPost Physics.