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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

This is not the latest submitted version.

Submission summary

As Contributors: Miguel Alvarado
Preprint link: scipost_202109_00011v1
Date submitted: 2021-09-08 13:44
Submitted by: Alvarado, Miguel
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


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.

Current status:
Has been resubmitted

Reports on this Submission

Report 2 by Adrian Feiguin on 2021-11-22 (Invited Report)


1- This work builds upon previous papers by the authors and collaborators where the boundary Green's function is introduced in 1D, by extending the formalism to arbitrary dimensions and using the Faddeev-LeVerrier algorithm to solve for the Green's function.

2- The authors claim to show a considerable improvement compared to recursive methods in terms of stability and computational cost.

3- They demonstrate the method with several applications to 2D (quadratic) topological Hamiltonians.

4- The method is described in great details, including pseudo-code.

5- The formalism is beautiful and elegant. It can become a standard tool in the study of heterostructures within ab-initio frameworks. I recommend the authors cite relevant references in this context such as:
- Haydock R. The recursive solution of the Schrödinger equation. Comput Phys Commun. (1980) 20:11–6.
-Viswanath VS, Müller G. The Recursion Method: Application to Many Body Dynamics. Berlin; Heidelberg: Springer (1994).


1- The authors briefly mention in the introduction that the method can be extended to problems with electron-electron interactions. The paper is self-contained and I would not request an example of such a calculation, but it would be useful to understand how the formalism would be extended to a case in which one has quartic terms. Maybe a descriptive paragraph in the conclusions would suffice.


I recommend it for publication in SciPost after my aforementioned comments are addressed.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

Author:  Miguel Alvarado  on 2021-11-26  [id 1973]

(in reply to Report 2 by Adrian Feiguin on 2021-11-22)

We thank the referee for his/her disposition to review our manuscript. We provide below the requested information which would hopefully allow him/her to complete a report.

In the main text we quote some references regarding the use of GF to compute magnetic interaction or to deduce effective Hamiltonians for such systems [1-3]. On the other hand, Green’s functions provide the appropriate starting point to include interactions by means of diagrammatic techniques. We shall include a comment along these lines in the revised manuscript.

[1] Z. Wang and S.-C. Zhang, Simplified topological invariants for interacting insulators, Phys Rev. X2, 031008 (2012), doi:10.1103/PhysRevX.2.031008
[2] M. Iraola, N. Heinsdorf, A. Tiwari, D. Lessnich, T. Mertz, F. Ferrari, M. H. Fischer, S. M. Winter, F. Pollmann, T. Neupert, R. Valentí and M. G. Vergniory, Towards a topological quantum chemistry description of correlated systems: the case of the hubbard diamond chain, arXiv preprint arXiv:2101.04135 (2021),2101.04135.
[3] D. Lessnich, S. M. Winter, M. Iraola, M. G. Vergniory and R. Valentí, Elementary band representations for the single-particle green’s function of interacting topological insulators, Physical Review B104(8) (2021), doi:10.1103/physrevb.104.085116

Anonymous Report 1 on 2021-10-20 (Invited Report)


The manuscript under consideration introduces an improved variant of the so-called boundary Green’s function (bGF) method, which is presented as specifically suited for (numerically?) obtaining transport properties in heterostructures.

I am familiar with several of the established approaches that are mentioned (e.g. recursive Green's function method, wave matching). However, I find myself unable to assess this manuscript due to the problems listed under below under "requested changes".

In addition to the requested changes, I would also like to rise the following point: Is an example implementation of the new method available to interested readers? Ideally, in the spirit of open access, a working example implementation would accompany the manuscript.

Once the "requested changes" have been addressed I will be hopefully able to properly review this manuscript.

Requested changes

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.

2. More generally, I think that it should be clearer what this article is about to a person who has research experience in the field but might be not already familiar with the existing bGF literature by the authors. What kind of systems does the method apply to? What kind of properties can be computed? Is the new method suitable for computing mundane transport properties like the conductance of a disordered heterostructure?

3. The authors claim that their new method of computing transport properties is in some ways superior to established methods. This should be demonstrated clearly and quantitatively. Section 5 contains a comparison to "standard recursive Green’s function calculations" but I struggle to recognize therein what is commonly known as the recursive Green's function approach [1] in the quantum transport community.

[1] A. MacKinnon, Zeit. f. Phys. B 59, 385 (1985).

  • validity: -
  • significance: -
  • originality: -
  • clarity: low
  • formatting: perfect
  • grammar: good

Author:  Miguel Alvarado  on 2021-10-22  [id 1872]

(in reply to Report 1 on 2021-10-20)

We thank the Referee for his/her disposition to review our manuscript. We provide below the requested information which would hopefully allow him/her to complete a report.

In the first place, we agree that an example implementation available to interested readers would be desirable. We have thus 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 (see This link will be provided in the manuscript when revision be allowed by the editor, after this first round of referees.

Regarding the requested further information on the boundary Green’s function method, we plan to revise our manuscript in order to give further explanations on it and on its relationship with other methods to obtain transport properties based on Green’s function techniques. Let us try to summarize here what we believe could be the origin of some possible misunderstanding.

Boundary Green’s functions (the definition of bGF appears at the third paragraph of the introductory section) encode information on the local spectral properties of semi-infinite systems. Such information is of special interest, for instance, in the case of topological phases where the boundary spectral densities can reveal the presence of edge states or other type of localized excitations. Additionally, bGFs are the basic input to compute general transport properties of different type of junctions within the so-called “boundary Green’s function method”, which is based on non-equilibrium Green’s function techniques as explained in Refs. [10-15] of the manuscript. From the Referee comments we realize that the idea on this method might not be clearly stated within the manuscript. In particular, we have realized that we should emphasize that this method is not appropriate to calculate transport properties of extended disordered system as the one described in the reference pointed out by the Referee, but is rather better suited for the case of short junctions.

Extended disordered systems could be, of course, studied using recursive Green’s function techniques which do not require any kind of translational invariance. On the other hand, recursive Green’s function techniques can be applied to obtain the bGF of semi-infinite translational invariant systems and that is what the comparison by the end of the manuscript is about.

Finally let us comment that our FLA method and the recursive methods could be combined to describe systems where two semi-infinite translational invariant regions are coupled by a disordered region.

We plan to revise our manuscript in order to clarify the above mentioned issues and also add a reference to the manuscript pointed out by the referee in connection to recursive Green’s functions calculations. We are now fully open to consider any other possible suggestion by the Referee.

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