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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 is not the current version.

Submission summary

As Contributors: Marcel Goihl
Arxiv Link:
Date submitted: 2019-01-14
Submitted by: Goihl, Marcel
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Quantum Physics


Spin chains with a symmetry-protected edge zero modes can be seen as prototypical systems for exploring topological signatures in quantum systems. 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 such systems. 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 non-interacting regime. Moreover, disorder helps localize only a subset of edge modes in the truly 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 despite the presence of disorder. In essence, our findings suggest that the ability to identify and control the best localized edge mode trumps any gains from introducing disorder.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2019-3-15 Invited Report


1. Providing interesting results concerning the question how disorder, interactions and their mutual interplay affect localized edge modes


1. While the results are obtained for a specific (class of) spin system(s) , the statements in the abstract are made in a possibly too general way.


I read the manuscript with interest and think that, on the whole, it is an interesting contribution to the subfield of topologically-ordered systems and in particular to the question whether perturbations (through disorder and/or interactions) affect the localizing behavior of edge modes.
The authors show numerically for the specific example of a disordered XZX cluster hamiltonian (based on a spin chain) that an additional perturbation J tend to delocalize the edge modes, while the character of J turns out to be important: If J leaves the system non-interacting, this delocalization is essentially disorder independent, while the interacting counterpart shows an additional disorder dependence.

However, the conclusions drawn from the results based on their specific spin model are presented, both in the abstract and in parts of the text, in a rather general way. I think one should be a bit more careful with the level of generalization.

Also, in the abstract and several times in the text, it is stressed rather generally that "the narrative that disorder necessarily stabilizes topological order" is challenged. While this is the case in certain instances (see Ref. [16-18]) the way this opinion is put forward in this paper implies that this is the general believe for topological systems. On the other hand, too strong disorder in weakly gapped systems has also been shown to spoil topological behavour. As another example one could mention the physics of the topolgical Anderson insulator (see eg Groth et al, Phys Rev Lett 2009): While it also shows a disorder-induced transistion into a topological phase, the overall phase diagram is much more complicated.

Requested changes

1. Referring to both, the prefactor J in front of the perturbation and the parameter $\eta$, as an interaction is a bit unfortunate and might be confusing: For instance, in the caption to Fig. 2c) (last sentence in figure caption) the authors talk about "non-interacting results" (meaning $\eta=0$) presented as a function of interaction strength $J$ in Panel c.

2. I did not understand how the time-dependent quantity $B_0(t)$ enters, see Eq. (15), and what is its meaning. Also replacing the time average there by a basis expansion looks like invoking an ergodicity argument, e.g. equilibration. But is this justified within this study of localization effects?

3. In Fig. 1c the color coding is nearly not distinguishable in the symbols forming the straight line.

4. Fig. 3c): Color code given in upper right box does not coincide with the red and blue symbols used in the panel.

5. Fig. 2a,b): Where does the symmetry with respect to site number comes from that does not exist for $\eta=0$, see Fig. 1?

6. Page 6, second paragraph: "red dashed-dotted" should be "grey dashed dotted"?

7. First sentence in abstract: ".. with a .. modes"

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

Anonymous Report 2 on 2019-3-3 Invited Report


1- interesting investigation of stabilization of topological order by disorder


1- very hard to read, intricate notation
2- no finite size scaling
3- insufficient numerical quality to back claims


The authors investigate a topological one dimensional quantum system in the presence of interactions and disorder. This system can in the absence of disorder be diagonalized exactly using a Jordan-Wigner transformation to Majorana operators and it is found in this limit that there exist edge modes due to the topological nature of the problem. The problem can still be diagonalized exactly in the presence of XX and YY interactions, but is promoted to a full quantum many-body problem in the presence of additional ZZ interactions.

The central problem investigated in this article is the locality of edge modes in the presence of interactions and disorder, and previous claims that disorder stabilizes topological order are challenged. For this, a method introduced in a previous paper by the authors to obtain local integrals of motion of the many-body localization problem is adapted to find conserved edge operators and their locality is studied by introducing a measure of locality, which compares the obtained operator to a truncation to a subset of the system.

The authors claim that the support of the edge modes acquires exponential tails into the bulk of the system, leading to no disorder dependence in the noninteracting case, but to stronger localization of some edge modes in the presence of disorder

In general, I find the paper interesting, since it seems to be a fresh approach to the problem of edge modes.
Unfortunately, the extensive use of mathematical notation and jargon makes it quite hard to read and I strongly suggest to explain the main ideas and the method in plain english before casting them into equations. Also, the notation should be explained this way, examples are the set complements, first featuring in Eq (16), as well as what stands behind Eq. (14), which is unclear to me. Why is there a floor function (?) appearing here?

Claims about exponential (in what?) corrections of the vanishing commutator of the edge mode operators with the Hamiltonian were made, and I would find it useful to see a numerical verification of this claim.

The description of the algorithm for the construction of the edge modes is really not clear and I think it should be extended considerably and explained in more detail, not relying on the previous work by the authors (one particularly unclear point is why the order of eigenvectors should matter).

In the final part of the paper, numerical calculations of edge mode operators where performed and the localization is checked by considering the supp(B,S) support quantifier. It is constructed such that it yields one if the operator is entirely supported on the subsystem S and is smaller than one otherwise. The decay of the support with decreasing subsystem size is used to quantify the decreasing operator weight at long distances. It would be useful to add an illustration of the subsystems used here.

The support does not decrease smoothly, there are considerable jumps and plateaus visible. Is it clear where they stem from?

The authors analyze the decay of the support in terms of exponential functions which are fitted to the numerical data. While in Fig. 1 this appears plausible (minus the jumps in the support function), I have a hard time believing this analysis in Fig 2 in the presence of interactions. Here, essentially three points are used to analyze the exponential decay, which are hardly sufficient to get a trustworthy result, in particular since they often don't even seem to represent a good fit.
I think what is dearly missing in this analysis is a careful study of the dependence of the results on system size. It is clear that there is considerable leakage of certain edge modes in the bulk of the system and it is therefore crucial to increase the size as much as possible, given the state of the art of full diagonalization, I believe that at least system sizes up to L=15 should be feasible for this 2^L Hilbert space. This would add at least another data point for the analysis of the exponential decay and make it more plausible. In addition, the stability with system size should be analyzed, comparing results for different chain lengths.

In Fig. 1 right there appears to be an artifact in the errorbars (?), which vanish suddenly at 10^-4. What is the reason for this?

In conclusion, I cannot recommend the publication of this paper in SciPost Physics in its current form for the following reasons: The introduced method for finding conserved operators is not new; the readability of the work could be improved; the numerical results are not convincing and should be extended in terms of system size and in particular by a careful finite size scaling analysis due to the severe limitations in system sizes. I suggest that the authors implement these recommendations in their changes to the manuscript before resubmission.

Requested changes

see report

  • validity: ok
  • significance: ok
  • originality: good
  • clarity: low
  • formatting: excellent
  • grammar: excellent

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