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$q$th-root non-Hermitian Floquet topological insulators
by Longwen Zhou, Raditya Weda Bomantara, and Shenlin Wu
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Submission summary
| Authors (as registered SciPost users): | Longwen Zhou |
| Submission information | |
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| Preprint Link: | scipost_202204_00036v1 (pdf) |
| Date submitted: | 2022-04-21 05:27 |
| Submitted by: | Zhou, Longwen |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Floquet phases of matter have attracted great attention due to their dynamical and topological nature that are unique to nonequilibrium settings. In this work, we introduce a generic way of taking any integer $q$th-root of the evolution operator $U$ that describes Floquet topological matter. We further apply our $q$th-rooting procedure to obtain $2^n$th- and $3^n$th-root first- and second-order non-Hermitian Floquet topological insulators (FTIs). There, we explicitly demonstrate the presence of multiple edge and corner modes at fractional quasienergies $±(0, 1,...2^n)π/2^n$ and $±(0, 1,..., 3^n)π/3^n$, whose numbers are highly controllable and capturable by the topological invariants of their parent systems. Notably, we observe non-Hermiticity induced fractional-quasienergy corner modes and the coexistence of non-Hermitian skin effect with fractional-quasienergy edge states. Our findings thus establish a framework of constructing an intriguing class of topological matter in Floquet open systems.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022-6-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202204_00036v1, delivered 2022-06-01, doi: 10.21468/SciPost.Report.5161
Strengths
1 Clear presentation
2 Convenient notation transferable to different fields in which floquet phases are discussed
3 original results regarding the conceptualization of exotic non-Hermitian topological states
4 possible realization of Z3 parafermions
Weaknesses
1 Too technical
2 No feasiblity / stability analysis of the proposed topological states
Report
The manuscript discusses an nth root extension of non-Hermitian topological states through Flouqet engineering. Some of the results do explicitly build up on previous work, but there are also original aspects in the manuscript. In principle I appreciate this work and tend to recommend it for publication.
Report 1 by A.M. Marques on 2022-5-20 (Invited Report)
- Cite as: A.M. Marques, Report on arXiv:scipost_202204_00036v1, delivered 2022-05-20, doi: 10.21468/SciPost.Report.5101
Strengths
1- Originality of the method to generate qth-root Floquet topological insulators.
2- The method covers any arbitrary qth-root, with q a positive integer.
3- These models allow for a controlled proliferation of topological edge states at any rational fractional quasienergies.
Weaknesses
1- Nothing major. Perhaps the construction of the qth-root generating procedure could be introduced in a more pedagogical way when it first appears in the paper, as I indicate below. This can be easily addressed by the authors.
Report
The authors developed a method for constructing qth-root versions of Floquet topological insulators (FTIs), generalizing the previous work of Ref. [45], which was limited to $2^n$th-root systems. In particular, the method allows one to generate topological states with any rational fractional quasienergy (in units of $\pi$), which is a key novelty of these models. Ultimately, the qth-root Floquet operator results from the multiplication of a diagonal matrix of U’s with a generalized shift operator, such that a block diagonal matrix is obtained when raising this operator to the qth power, with the blocks related to the parent model from which the topological properties are inherited. By taking the models of Refs.[62,66] as the parent models, several examples of qth-root FTIs are presented. I find these results novel and insightful, as well as relevant, given the recent surge in interest in high-root topology. Therefore, I recommend publication of this paper, provided the comments below are properly addressed first.
Requested changes
(1) I don’t know the current status of Ref. [45]. If it was already submitted, I believe the publication of this paper should only occur after the publication of Ref. [45], since this paper is a generalization of it.
(2) The general method outlined in eqs. (9)-(13) is not obvious at a first reading. Only after one explicitly works out an example can one understand it. I recommend the authors contextualize better this part by explaining in a couple of sentences, e.g. below eq. (10), the insights for introducing eqs. (11) and (12) such that eq. (13) is obtained. Also, immediately below eq. (13) it would be helpful if the authors worked out a specific example (say q=3), showing the form of the different $P, \tilde{H}, U$, etc., of the qth-root model. This would greatly clarify the method for the reader.
(3) The issue of the protecting symmetry of the edge states, discussed in the paragraphs above Sec. 3 and other places throughout, is a delicate point. The authors invariably relate it to the chiral symmetry of the parent model, which is true in a sense. However, at the level of the qth-root model, I would say that the 0 and $\pi$ edge states are protected by its chiral symmetry (which is built from copies of the $C$ symmetry of the parent model), while the fractional quasienergy edge states are not, since they are paired in symmetric quasienergies. For instance, in the square-root version of the SSH model we've studied in PRB 100, 041104(R) (2019), each of the finite energy edge states was shown to be protected by a “subchiral” symmetry $C_{1/2}$ (which incorporates the $C$ symmetry of the SSH model in one of its blocks, but not in the other), rather than by the global chiral symmetry $C_1$. I would expect something similar here, namely that each fractional quasienergy edge state should be protected by its own chiral-type symmetry, which locks that state at its specific gap. In other words, the protecting symmetries of the qth-root model, while closely connected to the parent model, should be derived directly from the starting qth-root model. I would like to hear the authors’ views on this, and eventually the inclusion of some extra clarifications on these points above Sec. 3.
(4) In the penultimate paragraph of Sec. 4, the authors classify the extra 1D edge states appearing in Figs. 4 and 5 as “trivial edge states”. Are the authors sure of this? Since, for instance, the parent model consists of dimerized SSH chains in the x direction and in the topological phase, I would expect these 1D states to be topological edge states coming from the weak topology of the system (as is known to occur, e.g., in the 2D SSH model).
Minor typos/comments on technical details:
i) I could be mistaken, but there is a general feeling that the $U_1$ and $U_2$ order, as defined in eq. (1), is not respected everywhere in the calculations below it. E. g., substituting (1) in (3) leads me to (4) with tau_+ and tau_- switched. Please recheck these calculations.
ii) In eq. (12), check if the$P_j\to P_{q-j}$ (with $P_q=P_0$) substitution should be made for the diagonal terms to have the correct order $U_1,U_2,U_3$,…
iii) In eq. (11) and below eq. (13), correct the index of the product.
iv) Below eq. (15): “characterized integer” $\to$ “characterized by integer”
v) Above eq. (20), the authors identify eq. (14) with $H(t)$ and eq. (15) with $H(t+1/2)$. In connection with point i), check if it should not be the converse instead, given the $U_1$ and $U_2$ definition in eq. (1). The same above eq. (26).
vi) Above (21) and again above eq. (27), $H_1$ and $H_2$ are switched.
vii) In the second paragraph below eq. (21): “generate states with E=±0,±2π/3 (E=±π/3)” $\to$ “generate states with E=±0,±2π/3 (E=±π/3,±π)”.
viii) Sec. A, 5th line from the end: the second $ν_0$ should be $ν_π$.