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Edge mode locality in perturbed symmetry protected topological order

by Marcel Goihl, Christian Krumnow, Marek Gluza, Jens Eisert, Nicolas Tarantino

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Marek Gluza · Marcel Goihl
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-06-06
Date submitted: 2019-05-10 02:00
Submitted by: Goihl, Marcel
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


Spin chains with symmetry-protected edge zero modes can be seen as prototypical systems for exploring topological signatures in quantum systems. These are useful for robustly encoding quantum information. However in an experimental realization of such a system, spurious interactions may cause the edge zero modes to delocalize. To stabilize against this influence beyond simply increasing the bulk gap, it has been proposed to harness suitable notions of disorder. Equipped with numerical tools for constructing locally conserved operators that we introduce, we comprehensively explore the interplay of local interactions and disorder on localized edge modes in the XZX cluster Hamiltonian. This puts us in a position to challenge the narrative that disorder necessarily stabilizes topological order. Contrary to heuristic reasoning, we find that disorder has no effect on the edge modes in the Anderson localized regime. Moreover, disorder helps localize only a subset of edge modes in the many-body interacting regime. We identify one edge mode operator that behaves as if subjected to a non-interacting perturbation, i.e., shows no disorder dependence. This implies that in finite systems, edge mode operators effectively delocalize at distinct interaction strengths. In essence, our findings suggest that the ability to identify and control the best localized edge mode trumps any gains from introducing disorder.

Author comments upon resubmission

Dear readers,

we have included the latest feedback of the referee and included a discussion of our fitting errors.
As pointed out in our comment of version 2, we caused some misunderstanding by using the term 'stability'. We have now included a more detailed discussion on that subtle issue and went on describing the phenomenon we want to study as 'localization' since 'stability' is common to describe phases. Moreover it occurred to us, that we did not explain the error bars we obtain from our fitting procedure. These are least-squares errors of the fitting algorithm and hence strengthen our claim that the data for the interacting system is meaningful, even though the amount of available data points is rather small.

We look forward to hearing from you,
the authors

List of changes

- added a discussion in the introduction on why localization of the edge modes is crucial for employing these systems for quantum information tasks.
- removed claims about a 'challenged narrative' altogether.
- added a discussion of the fitting errors in the captions of the plots as well as in the result section.

Published as SciPost Phys. 6, 072 (2019)

Reports on this Submission

Anonymous Report 2 on 2019-6-4 (Invited Report)


I think that after the further revisions of the manuscript, and in view of the convincing answer of the authors, the paper now deserves publication. Hence i recommend to accept the paper.


I recommend publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2019-5-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1901.02891v3, delivered 2019-05-19, doi: 10.21468/SciPost.Report.960


The authors significantly expanded the text and I find the discussion of the algorithm now much more clear and accessible.
They also added a discussion of the decay of the operator support, explaining the plateaus visible in the numerical data. The discussion of the numerical procedure, system sizes and statistical analysis was improved.

In summary, I find that the referee comments have been addressed in
a satisfactory manner and I recommend the article for publication in SciPost Physics.

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

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