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Topological Lattice Models with Constant Berry Curvature
by Daniel Varjas, Ahmed Abouelkomsan, Kang Yang, Emil J. Bergholtz
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|Authors (as Contributors):||Ahmed Abouelkomsan · Emil Bergholtz · Daniel Varjas|
|Date submitted:||2021-07-22 14:31|
|Submitted by:||Varjas, Daniel|
|Submitted to:||SciPost Physics|
Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of bosonic fractional Chern insulator states. In fact, we show that an "ideal flatband" cannot have constant Berry curvature.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2022-2-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00047v1, delivered 2022-02-08, doi: 10.21468/SciPost.Report.3879
This paper address the question of finding lattice models possessing bands with exactly constant (non-vanishing) Berry curvature. The findings of the paper regarding this question can be divided into two major results. First, they show that roughly speaking it is possible to have exactly flat Berry curvature in models with more than two internal degrees of freedom per unit cell (three or more bands). As a complement to this finding, the authors prove a no-go theorem for the case of two-band models stating that in any two-band model, the berry curvature does vary within the Brillouin zone. Second, they show that albeit the possibility of constructing lattice models with constant Berry curvature, this property does not generally improve the topological robustness of the system, namely in a fractional Chern insulator model. This second finding is, in fact, against the initial motivation in pursuit of such constant-Berry-curvature models.
The aforementioned findings are given in detailed analytical proofs for the absence (existence) of the possibility of finding constant berry curvature in two (three or more) band models. Also, considering certain models, it is shown how to obtain the constant Berry curvature besides exploring the stability of FCI states under such conditions. In particular, the authors start with lattice models, introduced by Kapit and Mueller, whose ground states can be tailored to be exactly identical with Laughlin states. Then they apply their Berry curvature flattening recipe to obtain a modified Kapit-Mueller model with constant Berry curvature. Finally, they show by numerical inspections that although the modified Kapit-Mueller model still possesses the ground state degeneracy, it is less stable than the original Kapit-Mueller model with varying Berry curvature.
My overall assessment, based on the findings and presentation of the paper is positive, and after addressing my following concerns, I can suggest its publication.
1. I find the introduction a bit technical which for a general audience of SciPost Phys. So I suggest extending/modifying the introduction to make it more accessible for non-expert readers.
2. The second finding of the stability of FCI states under the constant Berry curvature condition is mainly supported by numerical calculations. It would be much more helpful if the authors can provide an intuitive physical picture for these findings as well. For instance, provide some explanation why in the case of the Hofstadter model with flux φ = 1/4 per plaquette, the flattening does act in favor of a more ideal model, but not in the case of φ = 1/3 Kapit-Mueller model.
Anonymous Report 1 on 2021-10-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00047v1, delivered 2021-10-13, doi: 10.21468/SciPost.Report.3659
This paper has studied the possibility of flatenning the Berry curvature and the energy bad in a topological band structure and its impact on the stability of fractional Chern insulators. This is definitely an interesting question.
The paper is not well-written. The order of section is not chosen properly. The results seem contradictory (because of poor presentation).
This paper has studied the possibility of flatenning the Berry curvature and the energy bad in a topological band structure and its impact on the stability of fractional Chern insulators (FCIs). They show that a uniform Berry curvature in the 1st Brillouin zone does not necessarily lead to a more stable FCI. They have also shown that an local lattice model with perfectly flat bands and exactly uniform Berry curvature cannot exist. However, if we compromise and allow a small variation in the Berry curvature or flatness of the energy band the model can indeed be realized on a lattice.
This main subject definitely of great ineterest for the researchers in the field of fractional quantum Hall effect (FQHE). However, the paper lacks a consistent flow and is not well written. The order of section is not chosen properly. The results seem contradictory (because of poor presentation).
For example, the authors discuss extensively the interplay between flat band structure and a flat Berry curvature and their impact on the stability of the FCIs. Later on, they argue such band structures cannot exist. This is quite confusing for the readers. Moreover, they discuss that such models are not as good as the regular Kapit-Muller model of 1/3 FQH state, and more stable than its 1/4 counterpart. A proper explanation is lacked in my opinion.
Finally, a lot of equations have been used without elucidating their origin and importance. Since their is no page limit in this journal, they need to expand those sections to make the article more accessible and self-contained.
The order of some section must change. The author should discuss their no go theorem first and more clearly emphasize that relaxing the exactly flat energy or exactly uniform Berry curvature is the loophole to their no go theorem.
They should also provide a better and more convincing explanation of their rather contradictory results on the stability of the FCIs in their nearly inform model.
The paper clearly needs a substantial revision and become more available and more appealing to meet the criteria and standards required by Scipost.