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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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Submission summary
Authors (as registered SciPost users):  Ahmed Abouelkomsan · Emil Bergholtz · Daniel Varjas 
Submission information  

Preprint Link:  scipost_202107_00047v1 (pdf) 
Code repository:  https://zenodo.org/record/5102818#.YPk3hRMza3I 
Date submitted:  20210722 14:31 
Submitted by:  Varjas, Daniel 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
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 twoband 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.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202228 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00047v1, delivered 20220208, doi: 10.21468/SciPost.Report.3879
Report
This paper address the question of finding lattice models possessing bands with exactly constant (nonvanishing) 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 nogo theorem for the case of twoband models stating that in any twoband 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 constantBerrycurvature 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 KapitMueller model with constant Berry curvature. Finally, they show by numerical inspections that although the modified KapitMueller model still possesses the ground state degeneracy, it is less stable than the original KapitMueller 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 nonexpert 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 KapitMueller model.
Report #1 by Anonymous (Referee 1) on 20211013 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00047v1, delivered 20211013, doi: 10.21468/SciPost.Report.3659
Strengths
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.
Weaknesses
The paper is not wellwritten. The order of section is not chosen properly. The results seem contradictory (because of poor presentation).
Report
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 KapitMuller 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 selfcontained.
Requested changes
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.
Author: Daniel Varjas on 20220225 [id 2246]
(in reply to Report 1 on 20211013)
We thank the Referee for reviewing our manuscript, and finding that we studied an interesting research question. We regret that the Referee found the presentation of the results hard to follow. We made several changes, also highlighted in the attached manuscript.
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.
We changed the order of the sections in order to clarify the message of the paper, and provided an overview of the paper’s structure in the last paragraph of the introduction. Now we first present the positive result about the existence of constant curvature lattice models, and only after this present the nogo theorem about constant curvature in 2band models. We made sure to clarify that there are two nogo theorems in the manuscript, the second one shows that the ideal droplet condition (which we find to govern the FCI physics instead of the fluctuations of the Berry curvature) cannot be satisfied simultaneously with constant curvature in any lattice model.
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.
We clarified our arguments as to why certain models benefit from flattening the Berry curvature, and others do not. We added an analysis in sec VI. where we quantify the deviation from the strictest ideal flatband condition (11), and we find that this is the relevant quantity for the interacting physics. This quantity is improved by the flattening in the Hofstadter model, but is made worse in the KM model. This is clear intuitively, as the KM model has wavefunctions that are already ideal by construction, and cannot be improved. We added a new panel to fig. 6, showing that this quantity is a good predictor of both the third highest 2body energy’s deviation from zero, and of the size of the finitesize groundstate splitting.
Attachment:
Constant_curvature_diff_rvutAfQ.pdf
Anonymous on 20220325 [id 2323]
(in reply to Daniel Varjas on 20220225 [id 2246])The authors have properly addressed my questions and concerns and I support its publication.
Author: Daniel Varjas on 20220225 [id 2245]
(in reply to Report 2 on 20220208)We thank the Referee for the positive review of our manuscript and the constructive criticism. We answer the two points below, and attach the updated manuscript with changes highlighted.
Following the referee’s suggestion, we rephrased the third and fourth paragraphs of the introduction to make it more accessible, and to better highlight the structure of the manuscript.
We thank the Referee for pointing out that the analysis of the numerical results was not sufficiently clear. We added an analysis in sec VI. where we quantify the deviation from the strictest ideal flatband condition (11), and we find that this is the relevant quantity for the interacting physics. This quantity is improved by the flattening in the Hofstadter model, but is made worse in the KM model. This is clear intuitively, as the KM model has wavefunctions that are already ideal by construction, and cannot be improved. We added a new panel to fig. 6, showing that this quantity is a good predictor of both the third highest 2body energy’s deviation from zero, and of the size of the finitesize groundstate splitting.
Attachment:
Constant_curvature_diff.pdf