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Bulk-boundary-defect correspondence at disclinations in crystalline topological insulators and superconductors
by Max Geier, Ion Cosma Fulga, Alexander Lau
This is not the latest submitted version.
This Submission thread is now published as SciPost Phys. 10, 092 (2021)
|As Contributors:||Ion Cosma Fulga · Max Geier · Alexander Lau|
|Arxiv Link:||https://arxiv.org/abs/2007.13781v2 (pdf)|
|Date submitted:||2020-08-03 11:30|
|Submitted by:||Geier, Max|
|Submitted to:||SciPost Physics|
We establish 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.
Submission & Refereeing History
Published as SciPost Phys. 10, 092 (2021)
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Reports on this Submission
Anonymous Report 3 on 2020-10-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2007.13781v2, delivered 2020-10-28, doi: 10.21468/SciPost.Report.2124
1- novel and of immediate relevance
2- exhaustive classification results
3- clear illustrations
4- simple explicit models
2- should better discriminate original results from prior ones
3- some argument not clear
In this work, the authors establish a correspondence between bulk band topology and anomalous states bound to disclination defects. The disclinations are constructed via the Volterra processes, and both the role of the Franck angle and of the translational holonomy are investigated. The appearance of disclination-bound states is considered both for strong topological insulators possibly breaking translational symmetry as well as for crystalline topological insulators characterized using the technique of topological crystals. The theoretical arguments of this work are supported by clear and very helpful illustrations. The conclusions obtained by this work provide an important extension of prior works on the closely related topic of bound states at dislocations, and their appearance is very timely. The results of the analysis appear very exhaustive, original, and of immediate scientific value.
In summary, I am strongly inclined to recommend this work for publication.
On the other hand, I have to criticize the overall organization of the manuscript, which has resulted in the manuscript being a slow and cumbersome read. Studying this work costs an unexpectedly long time – not because of the length or difficulty, but rather because of the often sloppy structure. The authors have appropriately moved some technical parts of the argumentation into appendices, but I found their overall structure rather disorganized – e.g. when certain section of the main text defers detailed discussion to an appendix, whereas the corresponding appendix retrieves results derived in later sections of the text. At other occasions, a paragraph of discussion is included in the text, which does not bear a clear relation to the prior/previous text and lacks a clear motivation (few examples listed further below).
Concerning the structure, I think the manuscript would be readily improved if the authors could do the following two straightforward amendments:
1.) Begin each section with a short summary of its role: How does it connect to the previous sections? How is it structured into subsections? How are the results obtained here needed in the following sections, and how are they relevant for the overarching goal of the paper? How is this section supplemented by Appendices? Similar (perhaps shorter) summary could be placed at the beginning of most subsections (especially the longer ones). Although such summaries would admittedly further increase the length of the manuscript, it would at the same time greatly help the readers navigate through the arguments and discussions.
2.) Place a summary of the main results somewhere: either just after introduction, or in the beginning of the conclusions. It seems to me that the main results are scattered on pages 18—20, especially in Eqs. (7—11) (Remark: Note that the notation “$\mu$” for topological invariants is not even introduced until one gets to read these equations), in the following characterization of systems with unitary rotation symmetry whose representation commutes/does not commute with all internal unitary symmetries/antisymmetries + in Tables 1 and 2. In the present version of the manuscript, it is difficult to locate the results of the analysis are without immersing deep into reading.
Note that I am not asking for new results or for restructuring of the manuscript, so this should not be difficult to achieve.
Furthermore, I think the authors should do a better job distinguishing in their presentation the original results from a review of prior works. At a few occasions, I found the presented arguments not very rigorous (some specific cases are listed below), although I understand that the authors had to find a balance between rigor and clarity. Finally, while the explicit model examples discussed in the last Section are clear and helpful, I was a bit disappointed not to find any motivating example even mentioned until finally getting to apply the derived machinery in the last Section.
I think the authors should keep in mind the above-listed points in mind when preparing their manuscript for resubmission.
Besides these general remarks and comments on the structure, I also have the following more specific comments that the authors should consider:
1.) The work provides a great review of the classification of disclination defects, part of which I have not known before. In contrast, the notion $Hol(\Omega)$ is introduced without a clear mathematical formulation. Do I understand correctly that one (1) constructs an “order-parameter space” M=E/G (E = Euclidean group, G = space group), (2) uses fundamental group $\pi_1(M)$ to describe codimension-2 defects, and finally (3) partitions the fundamental group into equivalence classes $Hol(\Omega)$? A brief construction of the holonomy classes would be helpful (even if only as a footnote).
2.) Related to the previous comment: Do I understand correctly that the listed holonomies in Eq. (1) apply only to Bravais lattices? This is not clearly specified, and it seems to me that more complicated space groups decorated with multiple sites and/or with nonsymmorphic symmetries are not captured by this equation. (Or are these more complicated cases somehow covered by the discussion in Appendices A.1 and A.2?) In either case, the authors should make sure the assumptions for this equation are stated clearly.
3.) In Fig. 1(d—e), shouldn’t the blue dashed Volterra cut go through mid-bonds and mid-cells, rather than through the vertices?
4.) Section 2.5 about spinful fermions begins with an argument that I did not understand, even after several attempts. It says that “when transporting a half-integer spinful particle around a $2\pi$ disclination, there are two effects contributing a $\pi$ phase to its wavefunction: (i) the rotation of the real space coordinate system, and (ii) the basis rotation of the local degrees of freedom”. But (i) and (ii) just look like the passive vs. the active description of the same thing (rotating the coordinates in which to express the degrees of freedom, vs. rotating the degrees of freedom with respect to the coordinates). What do I miss? Could the authors make this argument more explicit?
5.) When building up the topological crystal for space group p2 in Fig. 5(a) and in Appendix B.1, the quarter unit cell (i.e. the 2-cell) is decorated by three 1-cells. The fourth one, which would be (x,a/2) with x in range [0,a/2], is missing. Therefore, it is not true, as written in the Appendix, that the three 1-cells “together cover the complete boundary of the 2-cell upon translating [+rotating?] them with the crystalline symmetry”. Am I right? Then why is this fourth 1-cell dropped? Is it because one thinks about the cells in homological terms, i.e. the boundary of such a cell would simply be the boundary of the sum of the listed three 1-cells, thus making it redundant?
6.) It is not completely clear how the results in Eqs. (7—10) should be read. For example for the pi/2-disclination in Eq. (9), the holonomy has been claimed to be $Z_2$-valued. Then what are the possible values of the two-component vector T in (9)? Do T-vectors (1,0) and (0,1) (which should be equivalent to each other after local rearrangement of atoms) lead to the same number of bound states? It might be worth elaborating a bit on this result, e.g. how the bound state is moved if a disclination with non-zero translational holonomy is split into a pure disclination and a pure dislocation.
7.) When discussing the classification result on page 20, the authors write “(ii) In the remaining subset of symmetry classes where strong, rotation-symmetry protected second-order topological phases are forbidden, the anomaly at the disclination may still be determined from the bulk topology alone.” However, assuming I read Tables 1 and 2 correctly, rather then this “might” being the case, it actually “always is” the case. Or do I misunderstand how to read the tables?
8.) In Fig. 12 for the sixfold-symmetric class D second-order TSC, it might be illuminating to also explicitly show the results for a 2pi/3 (i.e. “doubled”) disclination, when the theory predicts the absence of a disclination-bound state.
9.) Why do the authors speak of “bulk-boundary-defect correspondence” instead of just “bulk-defect correspondence”?
10.) The authors add a short paragraph on disclination dipole in App. A.2, but I didn’t understand the purpose. It seems to relate to nothing else in the rest of the manuscript. Similarly, there is a paragraph on first-order topology near the end of App. B.2 which appears to be misplaced, and probably should be a part of App. C. I also didn’t understand how the paragraph on the relation to Ref.  on page 17 fits in the text. (There were a few more such paragraphs, but these three I found particularly confusing.)
11.) Could the authors more clearly explain why Fig. B.4(d) corresponds to a block-off-diagonal coupling with the x-Pauli matrix (cf. text on p.48)?
12.) In the discussion of fluxes bound to defects under Eq. (11), the charge “2e” is set in the magnetic flux quantum. Is this because this discussion only applies to superconductors? Or does this discussion also apply to Gedankenexperiments with topological insulators?
• In Table 2, class AII: The column for phase phi should be empty(?)
• Caption to Fig. A.3 should refer to Eq. (19) [instead of (20)]
• Extra “the” on p. 47 in “upon translating the them with”
• On page (50), there should be “The topological crystal shown in Fig. B.4 (c)” [rather than (d)].
• Two paragraphs further, in sentence “that (a) is a weak topological phase and (b) is a strong” replace by (d) and (e).
Assuming all the listed issues are properly resolved, I would most likely recommend the work for publications after the authors’ resubmission.
Report 2 by Frank Schindler on 2020-10-3 (Invited Report)
- Cite as: Frank Schindler, Report on arXiv:2007.13781v2, delivered 2020-10-02, doi: 10.21468/SciPost.Report.2038
4- Great figures
1- Often not rigorous
2- Presentation is somewhat disorganized
In this manuscript, the authors derive a correspondence between bulk crystal topology and disclination bound states. They focus on second-order topological insulators with protected corner states. While the disclination response of such phases has already been partially discussed in the literature, a systematic treatment of all symmetry classes has so far been missing. The present manuscript fills this gap in a comprehensive manner by classifying the disclination response for all internal and point group symmetries. It also provides useful topological indicators that predict bound states.
I am therefore in favor of publication, but not before the following concerns are adequately addressed (see also additional requested changes below):
1- I do not understand the derivation of bound states for twofold symmetry. It is a general problem of the paper that the arguments made are often of a very qualitative character. This can be good -- it is pedagogical and physically intuitive -- but already in the case of twofold rotation symmetry, the dangers of oversimplification become evident: In the "Twofold rotation symmetry" part of section 3.4, it is stated that "In the resulting sample, the bulk and all boundaries are gapped by construction". Firstly, this is incorrect, as there is one remaining corner state at the boundary as required by anomaly cancellation [it is in fact shown in Fig 4(a)]. Secondly, what is the precise argument for why the corner state cannot remain on the boundary (while its partner does stay there)? The authors should argue this case more compellingly.
2- The derivation of disclination bound states based on topological crystals is neat. However, the manuscript glosses over the distinction between two very different length scales: (a) The extent $a$ of the physical unit cell. (b) The extent $A$ of the topological crystal unit cell, which has to satisfy $A \gg a$ in order for each of the topological crystal unit cells to accommodate subdimensional topological phases of its own. The derivation in section 4.4 implicitly assumes that the translational part of disclinations is of size $A$, but in realistic materials it will rather be of size $a$. It should be argued why ignoring this mismatch nevertheless produces physically meaningful results.
3- In section 5.3 it is stated that "In symmetry classes for which the bulk-boundary-defect correspondence holds, the direct sum of a first-order topological phase with itself cannot lead to a second-order topological phase". What about, for instance, class AII? Combining two copies of a time-reversal symmetric 3D topological insulator in a C2/C4/C6-symmetric fashion gives a second-order topological insulator, see e.g. section VIIB in the supplementary material of Science Advances Vol. 4, no. 6, eaat0346.
1- There's a consistent miss-spelling: "Franck" should really be the last name of Frederick Charles Frank
2- Typo in the caption of Fig. 2: "The unit cells of three- and sixfold symmetric lattice are parallelograms" -- isn't this only true for the threefold case?
3- inconsistency in and around Eq (2): the vector r sometimes has an index i and sometimes not
4- In section 2.5, the difference between "quantized magnetic flux" and "fluxoid quantization" should be explained, if any
5- In Fig. 4(a), why is it necessary to split the system in half in the first step? The top half seems to not play any role in the cutting-and-gluing procedure that follows (this may be related to me not understanding the argument for bound states here, see point 1 above)
6- In Fig. 7, some disclination bound states seem to be unpaired (see e.g. 2nd row, 2nd column). This is impossible by the anomaly cancellation requirement. Indeed, upon closer inspection, in these cases the boundary always hosts an odd number of gapless states and so necessarily remains gapless even when translational symmetry is relaxed. This consistency should be pointed out explicitly, right now it is confusing that only one state is circled in green without further comment.
7- At the bottom of page 17, the $\nu$ invariants need to be defined before they are used (or their definition referenced in case I missed it)
8- The notion of a "geometric $\pi$-flux quantum" needs to be defined precisely. What's the difference to a magnetic $\pi$-flux?
9- In Eq. (11), there's a "mod 2" missing on the right-hand side
10- typo in Table 2, row "AII": the column belonging to $\phi$ should be empty
11- Caption of Fig. 10: "lowest eigenstates with $E \geq 0$" -- these should be at exactly $E = 0$ due to the particle hole symmetry
12- Typo in the conclusion "as construced"
Anonymous Report 1 on 2020-9-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2007.13781v2, delivered 2020-09-27, doi: 10.21468/SciPost.Report.2026
1) Adequately written
2) Timely subject
3) Exposed theory matched by quite some intuitive examples
Retrieves quite some known results affecting the originality/novelty a bit
In this work, Geier et al consider the effect of lattice defects, in particular disinclinations, in rotationally symmetric free fermion topological phases. This is an old subject. Indeed, for the SSH model -although one would call this now a symmetry obstructed band topology- zero modes were directly tied via an index theorem to the underlying topology. This directly links to well known Jackiw-Rebbi solutions. More generally, such index theorems have been considered in the context of various systems that have the possibility of hosting Dirac Fermions. Defects, when assumed to act locally, can have the same type of monodromy effect, mimicking closely the Callan-Harvey mechanism used to describe the “conventional” bulk boundary correspondence in free-fermion topological phases of matter. This is a timely subject due to the active research into crystalline protected topological band theory. In this regard the authors also consider e.g. higher order topological phases.
The paper is nicely written and interesting. Although the authors retrieve some previous results, the paper is self-contained and also has new discussions and examples. Hence, the paper may meet the criteria for publication. Before deciding on this matter, I like to discuss a few points nonetheless.
a)Although I value the freedom of authors to phrase their text and title to their liking, there are a two aspects I like to discuss in this case;
1] In the first sentence of the abstract the authors say they “establish” a link. As witnessed by the the main text, the authors refer to many papers on the subject. Hence the word establish, having the connotation of being first, is not completely fair in my eyes, affecting especially readership interested only in the abstract. Hence I suggest to use e.g, consider.
2]Related, I urge the authors to change their title. Given the many works and the fact that this focusses on C_n rotation characterized phases, this should be reflected better in the title. As it stands now it appears a general correspondence is found.
b)In sec 2.1 I like to point out that monodromy defects, apart from ref 54 are known to characterize, SPT states generally, see also e.g. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.86.115109
c)In Eq. 1 the holonomy quotes the results of Eq. 81 in ref 54, it is important to note that the translations have been modded out. The authors refer to rotation anomaly above, but considering only Eq.1 and it notation this can be tricky.
d)A main point is the terminology on strong topology. In sec 2.1 the authors state “A strong topological phase remains unaffected when translation symmetries are broken.” In principle one can define of course whatever one wants, but in case of TRS strong means that one does not need a translational symmetry axes to define weak indices. As class AII has a Z_2 invariant in 2D (*and*) 3D, a layered construction makes sense, i.e. the product over the pfaffian of the sewing matrix is gauge invariant in each plane. But translational symmetry is tricky. For topological (*band theory*) a lattice is implicitly assumed, c.f. also first sentence of sec 4.1- although here it is clear that its role has not been used. And this implicit presence is even more so assumed for crystalline invariants. Hence strong seems to mean that the topological protection comes from a 2D subgroup in e.g. a plane in 3D, the translations the authors refer to denote the preserved perpendicular direction. I think formulating it from this angle is clearer than stating that translations can be broken as implicitly it can only be perturbatively broken or the crystalline symmetries have to preserved on average. In any case there is an implicit sense of the underlying lattice and hence translation symmetry. In this regard, one should also note that for a weak phase e.g. doubling the unit cell may trivialize the phase, then the translations are altered but there is translational symmetry in every direction. Hence, my request to formulate this slightly different.
e)Purely out of interest. There is isomorphism between class A and AIII, which is particular reflected in the K-theory and according constraints of rotation invariants; https://journals.aps.org/prx/pdf/10.1103/PhysRevX.7.041069.
Specifically it relates a piece in class A to a piece in class A \oplus a piece in class AIII. Maybe this is reflected in the relation in adding a dimension and stacking. My question is whether the authors saw some of these relations specifically, see also Appendix A.
f)Sec 5.3 can be improved a bit to my taste. Basically, in the appendix and the examples of sec 6 the relation to the presence of extra symmetries is hinted upon. But apart from enumerating the results in the table, the discussion should be expanded. Relating the different symmetry classes is generally not so straightforward. Indeed spinless 1D E irreps of C_4 or C_6 can e.g. glue together into 2D reps that can give some form of topology, showing the crucial role of spinless TRS in this case and more generally.