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Topological defects in a double-mirror quadrupole insulator displace diverging charge

by Isidora Araya Day, Anton R. Akhmerov, Daniel Varjas

Submission summary

As Contributors: Anton Akhmerov · Isidora Araya Day · Daniel Varjas
Arxiv Link: (pdf)
Code repository:
Date submitted: 2022-04-11 17:36
Submitted by: Araya Day, Isidora
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We show that topological defects in quadrupole insulators do not host quantized fractional charges, contrary to what their Wannier representation indicates. In particular, we test the charge quantization hypothesis based on the Wannier representation of a parametric defect and a disclination. Against the expectations, we find that the local charge density decays as $\sim 1/r^2$ with distance, leading to a diverging defect charge. We identify sublattice symmetry and not higher order topology as the origin of the previously reported charge quantization.

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Editor-in-charge assigned

Reports on this Submission

Anonymous Report 2 on 2022-6-14 (Invited Report)


1. investigation of a fundamental model that exhibits the corner charge
2. numerical results that the model with defect and without sublattice symmetry does not exhibit quantized defect charge


1. absence of discussions on a nontrivial topology indicated by Wannier representation.


In this work, the authors investigated a fundamental model with anticommuting double mirror symmetries, dubbed Benalcazar-Bernevig-Hughes (BBH) model, and discussed defect charge in the presence and absence of sublattice symmetry.

The authors showed that the defect charge is quantized in the presence of chiral symmetry but not in the absence. This implies that the quantization is not robust against symmetry breaking in contrast to ordinally topological phases. This finding matches the first item in expectations of acceptance criteria (

Although the presenting results are interesting, I would like to clarify the following things.

1. In Figure 3(a), the converged values q_{tot} of solid lines are almost equal to 0.5. Could the author provide evidence that it is not a numerical error? Or, when the \delta is larger, is the converged value far from 0.5?

2. In Figures 2 (b-d), it seems that the charge is distributed on the boundary of systems. Do the systems have nontrivial edge states or edge charge?

3. The authors said "We identify sublattice symmetry
and not higher order topology as the origin of the previously reported charge quantization." Then, I wonder if the sublattice symmetry is crucial in the fractional disclination charge of rotation symmetric systems reported in Ref. [10]. Could the authors show the importance of sublattice symmetry through the models in Ref. [10]?

Requested changes

1. The authors should add the converged value of solid lines in Figure 3(a).
2. The authors should add plot legends in Figure 2.

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

Author:  Isidora Araya Day  on 2022-06-15  [id 2583]

(in reply to Report 2 on 2022-06-14)
answer to question

Dear referee,

Thank you for your feedback. We would like to address your inquiry on the applicability of our results to Ref. 10. We would like to note that in addition to having a sublattice symmetry, Ref. 10 disregards the electron-lattice coupling. In other words, despite the dislocations are shown on a curved lattice, the authors consider all hoppings constant, unlike what would happen if one tried to create this Hamiltonian in a real material. Therefore, the charge quantization reported in Ref. 10 is to be expected. If one were to break the sublattice symmetry in the model of Ref 10 and in addition include the electron-lattice couplings, the scaling argument we present in our work (the paragraph containing Eq. 9) would apply in exactly the same way as to the BBH model.

We believe that adding a very similar model (that of Ref. 10) to the manuscript will not improve its readability, and the added value will not justify the extra work. Therefore, we would like to ask if the above explanation addresses your third question sufficiently.

Anonymous on 2022-06-27  [id 2611]

(in reply to Isidora Araya Day on 2022-06-15 [id 2583])
answer to question

Thank you for your reply.

Actually, I am not still convinced that the sublattice symmetry is essential for the models in Ref. [10]. The reasons are as follows.

  1. The point is whether the rotation symmetry can protect the quantized defect charge. If the sublattice symmetry is essential and the rotation symmetry is not, the rotation symmetry cannot solely quantize the defect charge. Here, although I think that such models have been realized in artificial systems as shown in Ref. [12], it does not matter if the models in Ref. [10] are realistic or not. I am concerned about which symmetries can protect the defect charge.

  2. In the manuscript, the authors emphasized that the sublattice symmetry is the origin of the quantized defect charge, as they said "We identify sublattice symmetry and not higher order topology as the origin of the previously reported charge quantization." In my understanding, "previously reported charge quantization" means the results of Ref. [10]. Since Ref. [10] discussed rotation symmetric systems, the authors should exclude the possibility of the quantization by the rotation symmetry.

  3. Although the authors claimed "We show that topological defects in quadrupole insulators do not host quantized fractional charges, contrary to what their Wannier representation indicates," they did not discuss the Wannier center arguments for the model the authors considered. While the rotation case is obvious, I am not sure that the Wannier center argument is applicable to the double mirror symmetry case.

Anonymous Report 1 on 2022-5-23 (Invited Report)


This paper provided adequate numerical evidence to demonstrate that the defect charge is not quantized in the double mirror quadrupole insulators. The logical flow is smooth and the argument is easy to follow.


It didn't directly explain the lack of defect charge quantization through the Wannier representation picture.


The present manuscript studied the quantization of charge in the double-mirror quadrupole insulator (Benalcazar-Bernevig-Hughes model) at the disclination defect. The author provided concrete numerical evidence to demonstrate that without the sublattice symmetry and the four-fold rotation symmetry, the total charge around the defect is not fractionally quantized in units of 1/2. Instead, the local charge density decays as $1/r^2$ when deviating from the defect origin.

The paper is interesting and reveals the importance of sublattice symmetry regarding the quantization of defect charge when additional $C_4$ symmetry is broken. These results are great additions to the previous works [Phys. Rev. B 101, 115115 (2020), Science 368(6495), 1114 (2020)] on bulk and defect correspondence in higher-order topological insulators when $C_n$ symmetry is required. Hence, I would recommend publication.

However, there are some points I would like to encourage the author to consider.

1, For the parametric Hamiltonian, although the bulk remains gapped along the loop, the edge gap closes, and Wannire representation changes. For Fig 2(b), what is the corresponding filling for the charge density? In that case, zero energy states are also localized at the edge and it's unclear if the summation on the right-hand side of eq (7) is quantized.

2, In Fig. 3(a), when sublattice symmetry is broken, the total charge is still converged to a finite value that is close to $1/2$ when $R\rightarrow \infty $. Is there a good physics understanding for that convergence? e.g. is it because the profile of Wannier orbitals loses the symmetry when $C_4$ is broken and portions of Wannier orbitals that fall into the integral region are therefore slightly different from the $1/2$? Would the deviation depend on how large the $C_4$ symmetry is broken?

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

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