SciPost logo

SciPost Submission Page

Identifying Chern numbers of superconductors from local measurements

by Paul Baireuther, Marcin Płodzień, Teemu Ojanen, Jakub Tworzydło, Timo Hyart

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Paul Baireuther · Timo Hyart
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-04-21 17:00
Submitted by: Baireuther, Paul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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 overcomes the almost three decades standing problem of identifying the topology of 2D superconductors with broken time-reversal symmetry.

Current status:
Has been resubmitted

Reports on this Submission

Report 1 by Adrian Del Maestro on 2022-6-21 (Invited Report)

  • Cite as: Adrian Del Maestro, Report on arXiv:2112.06777v1, delivered 2022-06-21, doi: 10.21468/SciPost.Report.5263


1. Introduction is well written, with a concise yet useful discussion of the main focus of the paper -- the utility of the Chern number in characterizing quantized thermal conductance in topological superconductors.
2. 95% success rate of supervised learning approach in identifying topological phases of the Shiba lattice.
3. Ensemble ML approach utilizing multiple networks is used, with the outputs averaged giving a collective predictive vector identifying the probability of a Chern number that is robust against strong fluctuations during training due to rare signals in the training data.
4. The machine learning protocol, especially the use of an ensemble approach is well described, with careful analysis of the statistical accuracy of ensemble predictions.
5. The manuscript includes some interesting ideas for addressing issues (non-uniformity, sparse data, etc.) that would naturally appear when training on real experimental LDOS data.
6. Figures are clear and informative.
7. Detailed model architecture included in Table I.


1. Not clear why the specific value of $\lambda = 0.1$ is chosen in the model.
2. Assumption of the model is that the lattice spacing between impurity states is exactly the superconducting coherence length $\xi$.
3. Shiba Lattice Model section lacks sufficient details for the reader. While many of these are buried in a methods section at the end of the paper, important physical quantities like the $V_0$, the strength of the disorder potential appear out of nowhere.
4. Is $24\times \xi$ a reasonable field of view in terms of a real experiment?
5. Claim in the discussion: "our protocol essentially overcomes the long-standing problem of identifying the topology of 2D superconductors." is likely overselling. Only a single model has been considered, in some restricted region of parameter space on mostly idealized data. The approach is interesting and promising, and will likely spur further work.
6. Data set and code are not included or made available with the manuscript. This will hinder reproducibility and potentially impact of the work.


The authors report on a supervised machine learning approach to classify topological states of 2D chiral superconductors and insulators using local density of states (LDOS) data. The method is tested on the Shiba lattice, an engineered quantum system formed by arranging magnetic impurities on a superconducting surface. Synthetic data is generated using an effective Hamiltonian that couples a simple BdG superconductor to magnetic impurities. By using an ensemble of networks to make predictions, good accuracy (over 90%) is achieved in classifying Chern numbers up to |C| = 3.

Overall the individual sections of the manuscript are clearly written with descriptive figures. The inclusion of detailed methods, only as a sort of appendix at the end reduces readability and make it more difficult to get a picture of the model and dataset before the results are presented. The choice of model parameters, and their connection to real engineered quantum systems is also lacking. However, the results are promising and will hopefully spur further work using real experimental data. As such, I believe this work opens a new pathway in an existing research direction, the identification of unambiguous signatures of topological superconductors. The lack of data or code with the manuscript is a drawback in this sense, and will make follow ups by the community more difficult.

Requested changes

1. Page 1: "... imperative to device ..." → "... imperative to devise"
2. Enhance the level of detail in the "Shiba Lattice Model and Test Data Set" section. For example, while a picture of a STM spectra is shown in Figure 1a), more discussion is needed on why this is the expectation from an experiment. The sentence: "The modulus of the Chern number |C| determines the number of chiral edge modes, and therefore the LDOS data contains information about |C| ..." should be expanded upon do better explain how STM spectra is sensitive to edge modes.
3. Appearance of disorder potential scale $V_0$ needs to be better explained earlier in the manuscript. The reader sees it 2 times before it is eventually explained on page 7.
4. Tone down claim in discussion regarding solution to the problem of identifying topology of 2D superconductors.
5. Extra space should be removed in spin kets in top paragraph in the RHS column of page 6.
6. What is "respectively" referring to in the following sentence "at the bulk gap is at least $3.5/N_x \approx 0.15$ and 0.15, respectively" on page 7?
7. Page 8, "CNNs [23] are build ..." → "CNNs [23] are built ..."

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: reasonable
  • grammar: perfect

Login to report or comment