SciPost Submission Page
Topological chiral modes in random scattering networks
by Pierre A. L. Delplace
|As Contributors:||Pierre Delplace|
|Submitted by:||Delplace, Pierre|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
Using graph theory, we show the existence of interface chiral modes in random oriented scattering networks and discuss their topological nature. For particular regular networks (e.g. L-lattice, Kagome and triangular networks), an explicit mapping with time-periodically driven (Floquet) tight-binding models is found. In that case, the interface chiral modes are identified as the celebrated anomalous edge states of Floquet topological insulators and their existence is enforced by a symmetry imposed by the associated network.
Submission & Refereeing History
Reports on this Submission
Anonymous Report 2 on 2019-8-31 Invited Report
The paper deals with the topological nature of chiral modes appearing at interfaces in random oriented scattering networks. For certain cases, an exact mapping is found to Floquet topological insulators within the setting of non-interacting fermions. The paper presents an interesting perspective on the physics of topological edge/interface modes using graph theory, and their geometrical interpretation as such is undoubtedly of value. The paper is, for most part, clearly written and the exposition of the calculations is detailed well.
I recommend publication of the paper. However, I would like the author to address the following questions or discuss how they could be possibly answered.
1. The analysis of the results depends quite crucially on the strong phase rotation symmetry and then the topological nature of the Chern numbers. It will be useful to have some idea about what happens as one tunes away from the symmetric point, even if the system stays in the same topological phase. An an example, in non-interacting femionic systems, the gap in the bulk spectrum can be related to the localisation of the chiral edge modes within the topological phase. How does something like this show up in the network picture?
2. One of the strengths of the work presented here is that the presence of the chiral modes can be understood in non-periodic systems. Section 2 has a very clear description of how the cyclic oriented networks are obtained from the tight-binding models on regular lattices. I think the paper would benefit massively from a somewhat more detailed discussion of what kind of physical Hamiltonians and graphs should one think about when interpreting the results of the arbitrary disordered scattering networks.
3. What changes in terms of the network picture between the anomalous Floquet topological phase (edge modes with zero Chern number) and the topological phase which is not anomalous?
4. While it might be a topic of future research and outside the scope of this paper, it might be useful to have some directions about how the scattering network calculations need to be modified to take interactions into account.
Other than this, the paper, while generally clearly written, has quite a few typographical errors and would benefit from a thorough proof reading.
Anonymous Report 1 on 2019-8-2 Invited Report
Clear and well-written; elucidates and generalizes anomalous Floquet topological phases in a satisfying way.
No serious weaknesses; a few small suggestions below in the Report.
In this paper, the author maps the two-dimensional anomalous Floquet Anderson insulator (and related Floquet models) onto unitary scattering networks. Using graph theory and a “phase rotation symmetry” that such graphs obey, he gives a simple picture of when an anomalous phase (having a topological edge mode despite a vanishing Chern number) occurs in these models. This perspective allows him to generalize to non-Floquet evolution and unifies independent known results on scattering networks with known results in the Floquet setting. It also allows him to determine when an edge mode will occur even in cases where there is no known bulk invariant.
This is a very nice paper, based around an insightful idea. I think it’s essentially publishable as-is, but I have a couple of minor points for the author:
1. It would be good to comment on what happens to this picture with interactions. It seems at first glance that if the interactions are local, the scattering network map would still apply. Does this imply that the author’s graph-theoretic results also hold for the interacting case? Specifically does the argument in section 5.2 go through, and if not, why?
2. On p8 the author writes, “According to the mapping established above, one can now interpret the topological transition between the Floquet anomalous and trivial regimes in discrete-time tight-binding models as a percolation transition.” The implications of this claim should be expanded. What are the universal features of the transition, if it maps onto percolation? Is this known in the literature? If not, it seems like an important result.
3. There are a few small typos scattered throughout, so the paper should be given another close proofing. For instance, the labels (a) and (b) are missing in Fig 8.
I recommend the paper for publication.