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Isolated flat bands in 2D lattices based on a novel path-exchange symmetry

by Jun-Hyung Bae, Tigran Sedrakyan, Saurabh Maiti

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Submission summary

Authors (as registered SciPost users): Jun Hyung Bae · Saurabh Maiti · Tigran Sedrakyan
Submission information
Preprint Link: scipost_202301_00038v1  (pdf)
Date submitted: 2023-01-27 21:03
Submitted by: Maiti, Saurabh
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
Approach: Theoretical


The increased ability to engineer two-dimensional (2D) systems, either using materials, photonic lattices, or cold atoms, has led to the search for 2D structures with interesting properties. One such property is the presence of flat bands. Typically, the presence of these requires long-ranged hoppings, fine-tuning of nearest neighbor hoppings, or breaking time-reversal symmetry by using a staggered flux distribution in the unit cell. We provide a prescription based on carrying out projections from a parent system to generate different flat band systems. We identify the conditions for maintaining the flatness and identify a path-exchange symmetry in such systems that cause the flat band to be degenerate with the other dispersive ones. Breaking this symmetry leads to lifting the degeneracy while still preserving the flatness of the band. This technique does not require changing the topology nor breaking time-reversal symmetry as was suggested earlier in the literature. The prescription also eliminates the need for any fine-tuning. Moreover, it is shown that the subsequent projected systems inherit the precise fine-tuning conditions that were discussed in the literature for similar systems, in order to have and isolate a flat band. As examples, we demonstrate the use of our prescription to arrive at the flat band conditions for popular systems like the Kagome, the Lieb, and the Dice lattices. Finally, we are also able to show that a flat band exists in a recently proposed chiral spin-liquid state of the Kagome lattice only if it is associated with a gauge field that produces a flux modulation of the Chern-Simons type.

Current status:
Has been resubmitted

Reports on this Submission

Report 2 by Alexei Andreanov on 2023-5-31 (Invited Report)

  • Cite as: Alexei Andreanov, Report on arXiv:scipost_202301_00038v1, delivered 2023-05-31, doi: 10.21468/SciPost.Report.7281


1- Impact: classification and construction of flatbands is an important problem and any new approached are more than welcome. Even more important is establishing a connection between flatbands and symmetries.
2- Relevance: flat bands is an active topic of research motivated by their sensitivity to perturbations, in particular interactions which makes flatband models natural testbeds for novel and exotic phases of matter.


1- I think the presentation of this symmetry could be further improved. For instance in Fig. 10 it is not immediately obvious why the symmetry is present in (c) but not in (d)?


The authors propose a novel method to construct tight-binding Hamiltonians with a flat band -- dispersionless energy band, and derive a criterion for the flat band to be gapped away from the other (dispersive bands). The construction proceeds through projecting out the minority subsystem of the parent chiral sytem. The mismatch between the sizes of the majority and minority sublattices and the resulting rectangular matrix guarantees the presence of the flatband. This argument persists also in cases where the chiral symmetry is partially broken and hoppings within the minority subsystem are allowed. Whether the flatband is gapped or gappless is determined by the path-exchange symmetry related to how the minority and majority sublattices are connected.

This is an interesting and important work on flatbands that is definitely worth being published at SciPost Physics.

Requested changes

1- The deformation of the kagome lattice considered in Sec. 3 is known as breathing lattices in frustrated magnetism, and could be a convenient name for the deformation.
2- Within the Loedwin method flatband is always the GS (or the highest state in the spectrum) since effective Hamiltonian is positive definite. Is there a possibility to adapt/modify the method to flatbands in the middle of the spectrum?
3- Is there any connection to the following work: PHYSICAL REVIEW A 102, 053305 (2020) - Building flat-band lattice models from Gram matrices?
4- Does the Loedwin projection method extend to the case of other symmetries, which enforce the E -> -E symmetry of the spectrum, like (anti-)PT, etc?
5- The authors illustrate their findings with the Lieb and dice lattices which both have chiral symmetry. I wonder if the authors tried to take any Hamiltonian with a flatband as a ground state and reconstruct the parent lattice? The simplest example perhaps would be the sawtooth chain. Naively that seems to be possible, since the Hamiltonian can be made positive definite, and can be decomposed into a "square" of a rectangular matrix. However the hoppings might not be short-range again. If the answer is positive and there is a way to reconstruct the parent lattices, this would imply the hidden symmetry behind at least a subclass of flatbands.
6- Sec. 4.1: going beyond bipartiteness -- does this preserve short range hopping in general? or the long-range hoppings could be generated in this case?
7- It is important to compare/relate your path-exchange symmetry mechanism for the band touchings to the previously developed criterion for critical flat bands, e.g. flat bands with a touching: see "Singular flat bands", W Rhim, BJ Yang - Advances in Physics: X, 2021 - Taylor & Francis.

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

Author:  Saurabh Maiti  on 2023-07-14  [id 3812]

(in reply to Report 2 by Alexei Andreanov on 2023-05-31)

Thank you for your evaluation and the many useful suggestion. You can see our detailed response in the resubmission letter. I mostly wanted to touch on suggestions #4 and #5 which are relevant "next-step" questions and won't be addressed directly in our manuscript.
#4: This is a very relevant direction to think along. A good way to rephrase your question is to ask how do these symmetries/conditions emerge in the projected system from the parent system. Do they correspond to something simple? I hop you will agree with this that this question merits its own consideration separate from this work.
#5: We suspect that the parent system may not be unique, in general. We can construct two subsystems to connect to the subsystem of our interest and project both of them out. This is what stopped us from attempting this inverse problem. It certainly is an interesting idea to explore and again we hope that not including this in the current manuscript is reasonable.

Report 1 by Philippe St-Jean on 2023-5-18 (Invited Report)

  • Cite as: Philippe St-Jean, Report on arXiv:scipost_202301_00038v1, delivered 2023-05-18, doi: 10.21468/SciPost.Report.7211


1- Impact: the work is a very interesting perspective on a general method for engineering flat energy bands. There have been many proposals and demonstrations for implementing crystalline structures that could lead to such flat bands but very few have presented generic methods for realizing them. For example, Ref. [52] from the Bernevig group in 2022, is a recent important example, and the current manuscript provides an interesting alternative approach. Although one might find important similarities between these two works, the two approaches are sufficiently distinct to gain intuition and/or insightful perspectives.

2- Relevance: the subject of flat bands is a timely and important topic. A lot of effort is currently devoted to understand the emergence of these dispersive-less bands, notably because the interaction energy dominates and is expected to give rise to interesting many-body excitations.

2- Clarity: the paper is very well-written and allows for a clear understanding of the benefit of the method developed.


1- I find that the paper lacks quantitative descriptions of its main results. For example, it is hard to appreciate directly the flatness of the bands obtained as only figures of band structures are presented. I would have appreciated to see figures presenting directly the flatness of the bands as a function of the relevant parameters.

2- The paper also lacks discussion on the limitations of the model developed. The entire work is limited to systems that are well described in the tight-binding approximation. However, this is only an approximation and real systems will deviate from it; in that context, what would be the influence of these deviations from tight-binding?


This manuscript reports a theoretical work where a generic model for engineering flat bands in crystalline systems (solid-state, photonics, atomic...) is presented. I think this is a timely and and important topic, hence this paper provides insightful and interesting perspectives on this field. I think this manuscript is worth publishing in SciPost, provided that the authors address the minor points I raised above.

Requested changes

1- Include dedicated figures presenting a quantitative metric describing the flatness of the bands (e.g. E_max - E_min for a band as a function of the relevant parameters like r). I would also like to see figures presenting the evolution of the gap energy as a function of these parameters to better see the lift of degeneracy between the flat and dispersive bands.

2- Include a discussion (even a brief one) on the impact of deviating from the tight-binding approximation. This could be done, e.g., by including small next-nearest-neighbor couplings or other orbitals, and discussing how this impact the flatness and the gaps. This will be crucial to better understand the robustness of their scheme.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Saurabh Maiti  on 2023-07-14  [id 3813]

(in reply to Report 1 by Philippe St-Jean on 2023-05-18)

Thank you for your time and evaluation of our work. You can see our detailed response in the resubmission letter. I mostly wanted to address your concern about including the next-nearest-neighbor(nnn) terms. We think you are completely justified in wondering what happens when these imperfections are included. To address this we included a section in the resubmitted version. But I just wanted to emphasize two points: 1. The formulation is exact. But to future readers I wanted to clarify that your question has to do not with the formalism but the applicability of it to real systems where there are nnn effects (which is a very relevant point). 2. The formalism still handles nnn well if the bipartite conditions is maintained. But in space respecting Euclidian geometry, the type of nnn usually breaks the bipartite nature. And this is harmful: it disperses rather quickly. We quantify this in a plot. We also quantify the gaps of the isolated flat bands in parent systems and projected systems. The effect size in projected systems usually gets smaller.

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