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2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation
by Miguel Alvarado and Alfredo Levy Yeyati
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Submission summary
| Authors (as registered SciPost users): | Miguel Alvarado |
| Submission information | |
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| Preprint Link: | scipost_202109_00011v2 (pdf) |
| Code repository: | https://doi.org/10.5281/zenodo.5593119 |
| Date submitted: | 2021-12-14 11:02 |
| Submitted by: | Alvarado, Miguel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Since the breakthrough of twistronics a plethora of topological phenomena in correlated systems has appeared. These devices can be typically analyzed in terms of lattice models using Green's function techniques. In this work we introduce a general method to obtain the boundary Green's function of such models taking advantage of the numerical Faddeev-LeVerrier algorithm to circumvent some analytical constraints of previous works. We illustrate our formalism analyzing the edge features of a Chern insulator, the Kitaev square lattice model for a topological superconductor and the Checkerboard lattice hosting topological flat bands. The efficiency and accuracy of the method is demonstrated by comparison to standard recursive Green's function calculations and direct diagonalizations.
List of changes
Dear editor,
Thank you for the referee reports on our manuscript. We have answered them separately and modified the manuscript in accordance to their comments. The list of changes in the manuscript is provided below.
1. We have uploaded to the Zenodo repository the Matlab codes needed to accomplish the Faddeev-Leverrier algorithm for the Chern Insulator as in Fig. 4 in the manuscript.
2. We include in the introductory section further explanations on the kind of systems that can be studied from the boundary Green’s function perspective and the general properties that can be computed.
3. Also in the introduction we mention the possibility to treat interactions diagrammatically using Green’s functions, giving a reference to a particular example treating the electron-phonon interaction.
4. We include an explanatory paragraph in section II to properly define the boundary Green’s function for systems with an arbitrary number of neighbours.
5. In the conclusions we add a comment to clarify that our method can be combined with recursive approaches to accomplish finite regions without translational symmetry (e.g. disordered heterostructures).
6. We have included in the manuscript the references suggested by both referees in connection to recursive Green’s function techniques.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2022-2-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202109_00011v2, delivered 2022-02-09, doi: 10.21468/SciPost.Report.4375
Report
In my first report I stated:
> 1. The manuscript needs to define "bGF". Section 2 begins with the words "To obtain the bGF we start" but it is never defined. This is, after all, the central concept of the text.
In their response to this report, the authors claim that a definition was already present in the manuscript:
“the definition of bGF appears at the third paragraph of the introductory section”. They refer to the following paragraph of version 1:
> In this work we focus on the boundary Green’s function (bGF) method, which is specifically suited to obtain transport properties in heterostructures [10–15]. The bGF approach allows also to explore electronic spectral properties such as the local density of states (LDOS) or checking out the bulk-boundary correspondence of topological phases and computing topological invariants [16,17]. Furthermore, the Green’s function formalism allows to incorporate in a natural way electron-phonon and/or electron-electron interaction effects. Even more, from bGFs it is possible to deduce effective Hamiltonians including all of these effects and obtain their topological properties [18–20].
I read this paragraph carefully multiple times before writing my first report. This is not a definition, not even a hand-waving one. There is not even a precise pointer to where such a definition could be found. Instead, the authors cite a series of six articles by themselves (Refs. 10 to 15). Are interested readers to look up all these six articles to even see what the current one is about?
I maintain that an article on “boundary Green’s functions” should contain an appropriate introduction of this concept. In an substantial article of 20 pages, half a page of text including a few equations could be certainly devoted to an introduction of the central concept. Such an introduction should have an appropriate level of mathematical rigor and contain specific pointers to more in-depth information.
In version 2 of the manuscript the paragraph has been expanded:
> In this work we focus on the boundary Green’s function (bGF) method, which is specifically suited to obtain transport properties in heterostructures based on non-equilibrium Green’s function techniques [10–15]. The bGFs encode information on the local spectral properties of semi-infinite regions. In addition to transport, such information is of special interest, for instance, in the case of topological phases where the boundary local density of states (LDOS)s can reveal the presence of edge states or other type of localized excitations, thus it is possible to check out the bulk-boundary correspondence of topological phases and computing topological invariants [16, 17]. Furthermore, Green’s function formalism allows to incorporate in a natural way electron-phonon and/or electron-electron interaction effects for example by means of diagrammatic techniques [18]. Even more, from bGFs it is possible to deduce effective Hamiltonians including all of these effects and obtain their topological properties [19–21].
However, in my opinion, this still does not define what “bGFs” are, it merely lists some properties and applications.
Looking at the examples and the provided references I have the impression that the “bGF” is about computing the Green’s function of a 2d bulk that is terminated by a 1d boundary, in other words a half-bulk. But is this true? A clear problem statement and perhaps even a helpful figure would go a long way to show immediately what this work is about.
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I find that my question number 3 has not yet been addressed. As I tried to point out in my first report, at least within the quantum transport community, the term “recursive Green’s function method” is used for the technique pioneered by A. MacKinnon in Zeit. f. Phys. B 59, 385 (1985). The purpose of this technique is the computation of the retarded Green’s function within a finite scattering region to which semi-infinite quasi-1d leads have been attached.
However when the authors refer to “standard recursive Green’s function calculations” (see abstract) they seem to mean a different recursive Green's function method that is a “well established tool to compute bGFs” (see section 5). Clarifying the terminology unambiguously would make the article easier to understand for researchers with a background in quantum transport.
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It’s laudable that the authors agreed to share the source code associated with the article. Unfortunately, the provided material consists of four matlab files without any further explanation. A short readme file that gives an overview of the different files and functions would be very useful if the interested reader is not supposed to reverse-engineer the code.
I tried to run the code with Octave (a free clone of matlab), but while the computation seems to finish (the progress indicator reaches 100%), the script fails subsequently with an error. ( Clearly, since the program is meant to be run with Matlab, this is not a fault, but the authors might be interested about this.)
Requested changes
1. Properly define “bGF”, the central concept of this article.
2. Clarify the relation to “standard recursive Green’s function calculations”.
3. Provide a problem statement that is clear to people who are new to “bGF” formalism: what is the purpose and current status of the “bGF” formalism? How does it compare to other approaches, for example the one introduced in Phys. Rev. Research 1, 033188 (2019)? What is the improvement brought about by the current article? (The introduction mentions “diverse disadvantages of the semi-analytical calculations” but this is very vague.)
In my opinion addressing these issues requires substantially reworking the introductory section.
Author: Miguel Alvarado on 2022-02-21 [id 2231]
(in reply to Report 1 on 2022-02-09)We thank the referee for his/her disposition to review our manuscript. We provide below the requested information.
In our last revision of the manuscript, we have tried to give further information on the bGF method. Now we understand that the Referee is asking to provide a direct definition of the bGF concept. As the mathematical definition is given in Section 2, what we have done is to add a qualitative definition in the introduction as the quantity which “encodes the local excitation spectrum at an open boundary” of semi-infinite system. We have furthermore restructured the introduction following the recommendation to focus on the purpose and status of the method, and pointing out the advantages with respect to other approaches.
The bGF can be computed using recursive methods, as described in Section 5 of the manuscript. As there exist a large variety of such methods which have been used for many decades within the quantum transport community we agree that there could be some ambiguity regarding what we mean by “standard recursive techniques”. We have thus removed the term “standard” in the abstract and refer the reader to the explicit definition of the recursive method, which is given in Section 5.
As mentioned in 1., we have rewritten the introduction to made it more clear not only the bGF concept but also what we mean by the “bGF method” for transport calculations. We hope that in this new writing we made clear the distinction between the method that we develop for the bGF calculation and the “bGF method” for transport calculations, which is something independent on how one obtains the bGF. We also explicitly mention the relation of our method to that described in Phys. Rev. Research 1, 033188 (2019). This work also uses the residue theorem to obtain the Fourier transform of the GF but takes a different path to obtain its poles GF in momentum space, by solving a generalized eigenvalue problem. Another important difference between the approaches is that in our method a non-zero broadening term (eta) that enters as an imaginary part of the energy is needed in order to simplify the integration by residues. In the PRR (2019) they must modify the residue integration by adding extra terms depending on the existence or not of boundary modes, and consequently, it’s mandatory to use a different routine to determine the existence or not of these localized modes. Even though our approach is less robust and works in the limit eta ≠ 0, it is quite transparent and easy to implement with a straightforward integration routine independent of the particularities of the Hamiltonian. We also explain the advantage of our method with respect previous semi-analytical approaches that demand the analytical expression of the coefficients of the characteristic polynomial. These other approaches are limited to small Hilbert spaces since a typical symbolic Laplace expansion to evaluate the characteristic polynomial is highly inefficient with computational complexity of O(N!).
We have included a README file in the code repository. In relation to the execution of our code in Octave we have found that the problem mentioned by the Referee is related to the visualization part of the subroutine. We have rewritten the plotting subroutine to facilitate the execution of the code by open-source users.