# Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors

### Submission summary

 As Contributors: Ion Cosma Fulga · Max Geier Arxiv Link: https://arxiv.org/abs/2007.13781v3 (pdf) Code repository: https://zenodo.org/record/4526466 Date submitted: 2021-02-12 21:38 Submitted by: Geier, Max Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct $d$-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to $(d-2)$-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.

###### Current status:
Editor-in-charge assigned

We would like to thank all referees for their positive reports and for thoroughly studying our paper. We are very thankful for the constructive criticism and have modified our manuscript accordingly. We hope that the modifications provide further clarification, improve the readability, and make our results stand out more clearly. We believe that thanks to the referees suggestions we were able to further improve the quality of the manuscript.

We have included our responses to the individual referee reports as replies on the submissions page. For convenience, we provide in a separate email to the editor our manual "difference file" of our manuscript where all our new text insertions are marked in red and relevant deletions are in gray and crossed out.

Sincerely,
Max Geier, Ion Cosma Fulga, Alexander Lau

### List of changes

General:
* changed manuscript title slightly (crystalline -> rotation symmetric)
* expanded introduction and conclusion to emphasize the central results
* improved readability by including section summaries and introductions as well as minor adjustments and cross-references throughout the text
* included several footnotes throughout the text providing detailed information
* new Appendix A about holonomy equivalence classes of disclinations
* new subsection Appendix C.4 on the validity of the topological crystal construction for finite size samples
* other small changes and clarifications throughout the Appendix
* corrected typos throughout the text

Section 1:

* Added references 60,61,62 and sentence in the introduction in the paragraph on the literature
* Extended the summary of the manuscript to also contain the main results

Section 2:

* Changed section title to "Disclinations"
* added more precise definition of the lattice in terms of charge densities
* added a footnote in Sec. 2.1
* added Ref. 64 on definition of primitive unit cell
* clarified caption of Fig. 1
* added a reference to Appendix B.2 on disclination dipoles
* in Sec. 2.3, point to new Appendix A for details on holonomy classes
* added Refs. 35,50,53,65,66 in Sec. 2.3
* added footnote in Sec. 2.5 to clarify the notation in Eq.(2)
* in Sec. 2.6: changed text on geometric pi fluxes for clarification
* added footnote in Sec. 2.6 to support the discussion
* minor changes in the text and in the subsection titles to improve readability

Section 3:

* added "Strong" to the section title
* Changed section introduction
* clarification regarding strong topological phases in Sec. 3.1 and added footnote
* clarification regarding two-fold symmetric case in Sec. 3.4 and added footnote
* minor modifications to text and titles to improve readability

Section 4:

* Changed section title to "Disclinations in topological crystals"
* small changes in subsection titles
* small changes in the text to improve readability
* the section introduction points to Appendix C.4 which contains a discussion on the validity of the approach for finite size samples
* added a note in the caption of Fig. 7
* corrected a typo in Eqs. (7) to (10)
* additional footnote in Sec. 4.4 clarifying the definition of the topological invariants used in Eqs. (7) to (10)
* additional paragraph at the end of Sec. 4.4 elaborating on the translation holonomy vector T

Section 5:

* added section outline to the section introduction
* re-arranged the text into new subsections to improve clarity
* added footnote in Sec. 5.1 on flux quantization
* clarified interpretation of Cartan class BDI
* added footnote in Sec. 5.5 to clarify direct sums

Section 6:

* added reference to kwant software package
* minor clarification in the caption of Fig. 10
* added new Fig. 13 for 2pi/3 disclination and corresponding sentence in the text.

Section 7:
* split the Conclusions into "Results" and "Discussion"

Appendix

Appendix A:
* new appendix discussion the construction of the holonomy equivalence classes of disclinations

Appendix C.2:
* added footnote explaining the necessity of the block-offdiagonal coupling

Appendix C.4:
* new appendix discussing the validity of the topological crystal approach for finite size samples

### Submission & Refereeing History

Resubmission 2007.13781v3 on 12 February 2021
Submission 2007.13781v2 on 3 August 2020

## Reports on this Submission

### Report

In their resubmission, the authors have addressed all of my requests and comments in a satisfactory manner. I am therefore in favor of publication of the manuscript in its present form.

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