Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

Gourab Paul; Kuntal Bhattacharyya; Saurabh Basu; Srijata Lahiri

REVIEW 4 major objections 6 minor 51 references

Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Twisted bilayer α-T3 aligned with h-BN is claimed to turn a degenerate flat band into an isolated topological band with Chern number -1, switching to -2 and back under tuning of the hopping ratio and twist angle.

desk verdict Competent TBG-style calculation producing a plausible Chern phase diagram for twisted bilayer α-T3 with h-BN; the main gaps are the missing monolayer-plus-mass baseline and the absence of any finite-w1/w3 robustness check. read the letter →

arxiv 2508.18657 v2 pith:KVFEGCNQ submitted 2025-08-26 cond-mat.mes-hall

Gourab Paul , Srijata Lahiri , Kuntal Bhattacharyya , Saurabh Basu This is my paper

classification cond-mat.mes-hall

keywords twistedbilayerα-T3latticeflatbandsmoirésuperlatticeChernnumbertopologicalphasetransitionh-BNsubstrateWannierchargecenters

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that twisting two sheets of the α-T3 lattice—a honeycomb lattice whose hopping ratio α interpolates between graphene and the dice lattice—can turn a degenerate, topologically trivial flat band into a single non-degenerate topological band. The authors argue that, once the degeneracy is lifted by aligning each sheet with a hexagonal boron nitride layer (modelled as a staggered potential on two sublattices), a nearly flat sub-band away from charge neutrality carries Chern number -1. As the twist angle θ and hopping ratio α are varied, this band undergoes gap closings at the M point and the Γ point, switching to Chern number -2 and back to -1. The central interpretive claim is that the topology is generated by band folding from the twist, not inherited from the monolayer, which is topologically trivial. A reader would care because a tunable isolated flat band with nonzero Chern number is precisely the kind of platform proposed for interaction-driven and topological phases.

What carries the argument

The central object is the continuum-model Hamiltonian of a twisted bilayer α-T3 lattice built from two monolayer Dirac Hamiltonians rotated by ±θ/2 and coupled by interlayer tunnelling matrices at the three moiré momentum transfers. The argument is carried by two ingredients: the generalized chiral limit w1=w3=0, which makes the middle band exactly flat and isolates it, and the staggered h-BN mass term M Sz acting on the A and C sublattices, which lifts the degeneracy into sub-bands. Topology is diagnosed by the evolution of hybrid Wannier charge centers and by the numerically computed Chern number on the discretized moiré Brillouin zone; the bandwidth of the nearly flat band functions as th

What would settle it

Compute the Chern number of the lower nearly flat band with the physical interlayer couplings w1 and w3 restored (and with the h-BN moiré potential included) at θ=1.08°, α=1; if the band is no longer isolated or C differs from -1, the chiral-limit result is not representative. Also compute the monolayer with the staggered mass M; a nonzero Chern number there would undercut the claim that twist-induced band folding creates the topology.

Watch

Extended reading notes

Core claim

At the magic angle θ=1.08° and dice limit α=1, in the chiral limit where only one interlayer hopping amplitude survives, the twisted bilayer α-T3 lattice has an exactly flat, highly degenerate band at zero energy. Adding h-BN layers as a staggered mass term M on the A and C sublattices broadens this middle band and produces two non-degenerate nearly flat sub-bands at E=±M, at the edges of the spectrum. The paper's discovery is that the lower nearly flat band, away from charge neutrality, is topologically non-trivial with Chern number C=-1, while the ten-band bundle near charge neutrality remains trivial. Tracking this band in the α-θ plane, the Chern number switches from -1 to -2 through a g

Load-bearing premise

The isolated topological band is demonstrated only in a simplified model: the chiral limit (w1=w3=0), a staggered-mass-only h-BN substrate with the moiré potential neglected, and no check that the staggered mass alone leaves the monolayer band trivial—so if any of these neglected pieces closes the gap or already produces the topology, the central claim would need revision.

Editorial extensions

If this is right

If the central claim holds, a twisted bilayer α-T3 device aligned with h-BN should show an isolated band with Chern number -1 over a substantial range of twist angles and hopping ratios, making it a candidate for a Chern insulator at appropriate filling.
Varying α or θ across the M-point or Γ-point gap closings should switch the Hall conductance, observable as a change in the transverse conductivity in steps of e²/h, provided the nearly flat band can be isolated.
The same mechanism—band folding plus a sublattice-selective substrate mass—may produce topological flat bands in other twisted lattices whose monolayer flat bands are trivial.
Since the sub-bands near charge neutrality are shown to be trivial via multi-band Wannier charge center evolution, transport or edge-state probes should show conducting edge channels only at energies of the nearly flat band, not near the band center.
The bandwidth analysis implies that twist angle has a stronger influence on flatness than the hopping ratio, so experiments aimed at maximizing flatness should prioritize θ over α.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: The paper leaves implicit that the C=-2 region, if realized, could support two chiral edge modes and possibly higher-Chern correlated states; this is an extension beyond their explicit single-band analysis.
Editorial: The h-BN moiré potential and the w1,w3 interlayer terms are neglected; including them could shift the phase boundaries or close the gap. A natural testable extension is to map the Chern number with these terms included.
Editorial: The authors do not compute the monolayer-plus-staggered-mass baseline, so the 'topology from band folding' interpretation would be sharpened by showing that a single α-T3 layer with the staggered mass stays trivial; if it does not, the origin claim would need re-attribution.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper is a straightforward, well-executed application of the Bistritzer–MacDonald machinery to twisted bilayer α-T3, with an h-BN staggered mass added to lift the flat-band degeneracy. What's genuinely new is the Chern number phase diagram across the α–θ plane: the nearly flat band takes C = −1, passes through C = −2, and returns to C = −1, with gap closings at the M and Γ points. The calculations look internally consistent, the WCC plots match the Chern numbers, and the paper is honest about the region where the gap is too small to define a Chern number. That is a solid subfield contribution.

The paper does something worth crediting: it uses standard machinery without inventing new parameters. The values for w2, M, and vF are taken from existing TBG literature, and the chiral limit w1 = w3 = 0 is stated explicitly. The bandwidth analysis as a function of α and θ is a nice touch, even if it is mostly what one would expect.

Where it softens: the central interpretive claim—that the topology arises from band folding rather than from the h-BN mass—rests on an untested premise. The paper says the monolayer α-T3 flat band is non-topological, but that statement is about a monolayer without the h-BN mass. Since the bilayer calculation always includes M Sz, the authors should have shown a monolayer-plus-M baseline so we can see that the mass alone does not produce a nonzero Chern number. In the dice limit, the flat band lives entirely on the B sublattice and the A/C mass does not affect it, so the claim is plausible, but it is not demonstrated.

The second soft spot is that every result sits in the chiral limit w1 = w3 = 0. Finite w1 or w3 is not forbidden by the geometry; it is set to zero to maintain generalized chiral symmetry. The paper never tests whether a small w1 or w3 closes the gap between the nearly flat band and its neighbors or shifts the phase-transition lines. Given the abstract's claim that the topological band is isolated over a "substantial area" of the α–θ plane, that robustness check really should be there. Without it, the phase diagram is conditional on a specific idealization.

Minor: the paper says results are converged about k-point sampling but gives no numbers, and no code or data is provided. That makes independent replication more work than it should be.

Overall, the core phase diagram is probably right within the stated model, but the origin story and the robustness beyond the chiral limit are under-supported. This deserves a serious referee—someone should look at it—but the referee should require the monolayer+M baseline and at least one finite-w1/w3 calculation before it is published as-is.

Recommendation: send to peer review with the expectation of revision.

Referee Report

4 major / 6 minor

Summary. The paper studies a continuum model of twisted bilayer alpha-T3 with Bernal stacking, in the chiral limit w1=w3=0, and with an h-BN-induced staggered mass M acting on the A and C sublattices. It reports that, in the dice limit and at the magic angle, the highly degenerate middle band splits under h-BN alignment, producing two nearly flat sub-bands at E=±M. The sub-band near charge neutrality is argued to be topologically trivial via multi-band hybrid Wannier center evolution, while the sub-band at E=-M is claimed to be topologically non-trivial with Chern number C=-1. The paper maps Chern numbers of this nearly flat band across the alpha-theta plane, finding regions with C=-1, C=-2, and a return to C=-1, with gap-closing transitions at M and Gamma points. The bandwidth of the NFB is also studied as a function of alpha and theta, showing that flatness is maximal near the dice limit and the magic angle but degrades for intermediate alpha.

Significance. If the central claims hold, this work extends the flat-band topology program from twisted bilayer graphene to the alpha-T3 family, introducing the hopping ratio alpha as an additional tunable parameter alongside the twist angle. The Chern numbers and gap-closing transitions are mutually consistent, the model is stated transparently, and the parameters M, w2, and vF are taken from prior TBG literature rather than fitted. The paper also honestly notes several modeling simplifications. However, because the main phase diagram and the topological interpretation rest on three untested assumptions—the chiral limit, the neglect of the monolayer-plus-h-BN baseline, and the valley-projected treatment—the significance of the result for a physical alpha-T3 sample is not yet established. The findings are valuable as a model study if these gaps are addressed.

major comments (4)

[Making Moiré dice lattice topological; Fig. 6] All Chern numbers and the phase diagram are computed in the generalized chiral limit w1=w3=0. These couplings are not forbidden by the alpha-T3 geometry; setting them to zero is a model choice, not a symmetry of the lattice. A finite w1 or w3 will generically broaden the flat bands and can close the gap that isolates the NFB, making the Chern number ill-defined. The paper should compute at least representative finite w1 and/or w3 cases near the dice limit and magic angle, and show whether the C=-1/-2 phase regions persist. Without this, the 'substantial area' claim in the abstract and the NFB phase diagram are only statements about a specific idealized model.
[Topological characterization of NFBs] The text states that because the monolayer alpha-T3 flat band is topologically trivial, the emergent topology in the bilayer is due to band folding induced by twist. This comparison is not controlled: the bilayer calculation includes the h-BN staggered mass M, while the monolayer baseline is computed without M. A monolayer alpha-T3 with a sublattice mass M on A and C can acquire nonzero Berry curvature in each valley, so the twist-induced band folding may not be the sole origin of the reported topology. The authors should compute the monolayer-plus-M model (with the same M and, ideally, the same valley projection) and compare the Berry curvature or Chern number before attributing the topology to band folding.
[Eq. (1) and model; K vs K' valley] The continuum model explicitly neglects intervalley scattering and the authors state they focus on the K valley. The Hamiltonian used is thus a single-valley model. Since the h-BN staggered mass M Sz is time-reversal symmetric in the full lattice, the K' valley will generically contribute opposite Berry curvature, so the total Chern number over the full moiré Brillouin zone may vanish. The manuscript should clarify whether the reported C=-1 and C=-2 are valley-projected Chern numbers or total Chern numbers. If they are valley-projected, the system is more properly described as a valley-Chern/valley-Hall system, and the abstract's 'topologically non-degenerate' claim should be qualified accordingly.
[Fig. 7(a) and text: 'magic angle limit' for all alpha] The magic angle is determined for the dice limit (alpha=1) at theta=1.08 degrees, but the paper then fixes theta=1.08 degrees for all alpha when discussing the 'magic angle limit' and the alpha-dependence of the bandwidth. The magic-angle condition for twisted bilayer alpha-T3 likely depends on alpha, so theta=1.08 degrees may not be the flat-band condition for intermediate alpha. The authors should either derive the alpha-dependent magic-angle curve or explicitly state that they are fixing theta, not the magic angle. This affects the flatness claims in Fig. 7 and the interpretation of the phase diagram near the nominal magic-angle line.

minor comments (6)

[Eq. (2)] There appears to be a typographical error in the phase factor: 'e−iθq−θ)' should likely read 'e^{-i(θ_q-θ)}' or similar. Please correct the parentheses and check the sign conventions.
[Model Hamiltonian and Band Structure] The h-BN moiré potential is neglected with a citation to TBG. Since the alpha-T3 lattice has three sublattices and a different symmetry, the authors should provide a short justification for why this neglect is valid here, or at least acknowledge that it is an additional uncontrolled approximation.
[Fig. 6] The abstract claims the topological band is isolated over a 'substantial area' of the alpha-theta plane, but the phase diagram shows regions and Chern numbers without quantifying the gap to neighboring bands. A separate map of the isolation gap (e.g., min gap above/below the NFB) would make the 'isolated' claim concrete.
[Conclusion] The conclusion states that results are converged with respect to the number of reciprocal-space lattice points, but no convergence data are shown. Please provide a brief convergence test in the main text or supplementary material.
[Throughout] The phrase 'singular sub-band' and 'topologically non-degenerate singular sub-band' is used without definition. If this refers to the singular flat band of alpha-T3 or to the single-band nature of the NFB, please define it in the text.
[Fig. 5 caption] The text refers to 'Fig. [5(a)-5(e)]' with bracket formatting; please remove the brackets and format references consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Chern phase diagram is computed from an explicitly specified Hamiltonian with parameters taken from external TBG literature; self-citations are motivational or methodological and do not carry the derivation.

full rationale

The paper's central claims—the emergence of a topological nearly flat band (NFB) and the C = -1/-2 phase diagram in the α-θ plane—are obtained by direct numerical diagonalization of a Bistritzer-MacDonald continuum Hamiltonian (Eqs. 1-4) followed by Fukui's Chern-number evaluation and hybrid Wannier-charge-center evolution. No target Chern number is fitted, and no prediction is constructed from its own output. The parameters w2 = 110.7 meV and M = 17 meV are explicitly borrowed from prior TBG studies (Refs. [34,38,41-44]), not fitted to the quantity being predicted. The chiral limit w1 = w3 = 0 is stated as an explicit modeling assumption in 'Making Moiré dice lattice topological' and is taken from external TBG chiral-limit literature (Ref. [40]), not from the authors' own prior work. The monolayer α-T3 flat band's trivial topology is supported by an external reference (Ref. [26]), and the paper's own supplementary calculations are not used to substitute for that fact. The self-citations that do appear (Refs. [36,37] for α-dependent monolayer topology, Ref. [48] for WCC methodology) are not load-bearing: the bilayer derivation does not reduce to them, and removing them would not change the computed phase diagram. The paper also states clear limitations—neglect of the h-BN moiré potential, restriction to the chiral limit, and no finite w1/w3 test—but these are robustness concerns about the model's physical completeness, not circular steps in the derivation. Therefore no circularity is identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model inputs are borrowed material parameters and the physical knobs α and θ; no parameters were fitted to the target Chern numbers, so circularity burden is low. The real burden is the package of modeling assumptions: chiral limit, valley conservation, and the simplified h-BN mass term. No new particles, forces, dimensions, or conserved quantities are introduced.

free parameters (3)

w2 (interlayer tunneling amplitude) = 110.7 meV
Borrowed from TBG literature (Refs [34,38]); required for the chiral-limit Hamiltonian and the existence of the flat band.
M (h-BN staggered sublattice mass) = 17 meV
Borrowed from TBG/h-BN literature (Refs [41-44]); it lifts the middle-band degeneracy and fixes the NFB energy scale at E = ±M.
vF (Fermi velocity) = 6326.1 meV·Å
Taken from graphene; sets the kinetic scale of the continuum model but is not fitted to any target of this paper.

assumptions (5)

domain assumption Continuum Bistritzer-MacDonald approximation with momentum conservation at the rotated valleys is valid for small twist angles.
Used in Eq. (1) and Eq. (3) to construct the bilayer Hamiltonian.
domain assumption Interlayer scattering conserves valley, so intervalley scattering can be neglected and the K and K' sectors are independent.
Stated in the interlayer Hamiltonian section after Eq. (2).
domain assumption The chiral limit w1 = w3 = 0, with only w2 interlayer tunneling, is representative of the system.
Adopted from TBG chiral-limit studies (Ref [40]); it enables an exactly flat middle band and the isolated NFB, but this choice is not independently justified for α-T3.
domain assumption h-BN alignment is described only as a staggered mass M Sz acting on A and C sublattices, with the B sublattice unaffected and the h-BN moiré potential neglected.
Introduced in the section 'Making Moiré dice lattice topological'; this is a strong simplification borrowed from TBG treatments (Refs [41,42,45]).
domain assumption Bernal (A-B) stacking with zero relative in-plane translation τ0 = 0 and spinless fermions.
Defined in the model Hamiltonian section; the topological results depend on this stacking choice.

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Cite this review

Pith. "Pith review of Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice." pith.science (2026). https://pith.science/paper/KVFEGCNQ

@misc{pith2026250818657,
  author       = {Pith},
  title        = {Pith review of: Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KVFEGCNQ}},
  note         = {Machine review of arXiv:2508.18657}
}

abstract

We investigate an interesting interplay of destructive interference due to lattice geometry and band folding due to enlargement of the Brillouin zone in generating and subsequently modifying the band topology in a twisted bilayer $\alpha$-$T_3$ system. The pronounced degeneracy of the emergent flat band in the dice limit of the $\alpha$-$T_3$ lattice is removed on alignment with h-BN layers, resulting in the formation of sub-bands with varying topological characteristics. Remarkably, while the sub-band near charge neutrality exhibits a trivial behavior, a topologically non-degenerate singular sub-band emerges away from charge neutrality. The topological band remains isolated from the rest of the bands for a substantial area of the $\alpha - \theta$ plane (where $\alpha$ and $\theta$ correspond to the hopping ratio and twist angle respectively) while exhibiting multiple phase transitions as a function of the aforementioned parameters via hybridization with its nearest bands. We study the evolution of the hybrid Wannier charge center and the Chern number to characterize the different emergent topological phases. Finally, the degree of flatness of the topological band is studied as a function of both $\alpha$ and $\theta$ to explicitly show the influence of quantum interference and band folding on the width of the topological band.

Figures

Figures reproduced from arXiv: 2508.18657 by the authors.

**Figure 2.** FIG. 2. (a) Brillouin zones of the top and bottom lay [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Band structure of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Evolution of the hybrid WCC along the direction [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. The band structures of the NFB in the dice limit are shown in panels (a)–(e) for increasing twist angles [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. The phase plot of the Chern number for the NFB [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. The bandwidth of the NFB is shown (a) as a function [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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Works this paper leans on

51 extracted references · 40 canonical work pages

[1]

Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies- Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, Phys. Rev. Lett. 124, 076801 (2020)

work page 2020
[2]

Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

are denoted by a1, a2 (˜a1, ˜a2), and the corresponding sublattice position vectors are indicated by τX (τ ˜X ), with X = A, B, C and ˜X = ˜A, ˜B, ˜C. The nearest-neighbor hopping amplitudes between A–B and B–C sites are labeled as t and αt, respectively. conduction bands of the pristine nearest-neighbor dice lattice resembling graphene, the triply degene...

work page Pith review arXiv 2025
[3]

U U † H0(− θ 2)) (1) where H0 corresponds to the intralayer Hamiltonian of the individual α-T3 layers and U denotes the interlayer tunneling matrix. It is to be noted that the Hamiltonian for a monolayer α-T3 lattice when twisted by an angle θ (denoted by H0(θ)), and expanded around the rotated 3 valley point Kθ, can be written as HKθ(qθ)= vF qθ ⎛ ⎜ ⎝ 0 e...

work page
[4]

Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Nature 583, 215 (2020)

work page 2020
[5]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature 556, 80 (2018). 7

work page 2018
[6]

Tanaka, J

M. Tanaka, J. I. j. Wang, T. H. Dinh, D. Rodan-Legrain, S. Zaman, M. Hays, A. Almanakly, B. Kannan, D. K. Kim, B. M. Niedzielski, K. Serniak, M. E. Schwartz, K. Watanabe, T. Taniguchi, T. P. Orlando, S. Gustavs- son, J. A. Grover, P. Jarillo-Herrero, and W. D. Oliver, Nature 638, 99 (2025)

work page 2025
[7]

P. J. Ledwith, A. Vishwanath, and E. Khalaf, Exotic car- riers from concentrated topology: Dirac trions as the origin of the missing spectral weight in twisted bilayer graphene (2025), arXiv:2505.08779 [cond-mat.str-el]

arXiv 2025
[8]

P. J. Ledwith, J. Dong, A. Vishwanath, and E. Khalaf, Phys. Rev. X 15, 021087 (2025)

2025

Show all 51 references

[9]

Khalaf and A

E. Khalaf and A. Vishwanath, Nature Communications 13, 6245 (2022)

2022
[10]

Z. D. Song and B. A. Bernevig, Phys. Rev. Lett. 129, 047601 (2022)

2022
[11]

B. A. Bernevig, Z. D. Song, N. Regnault, and B. Lian, Phys. Rev. B 103, 205411 (2021)

2021
[12]

Z. D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Phys. Rev. B 103, 205412 (2021)

2021
[13]

B. A. Bernevig, Z. D. Song, N. Regnault, and B. Lian, Phys. Rev. B 103, 205413 (2021)

2021
[14]

B. Lian, Z. D. Song, N. Regnault, D. K. Efetov, A. Yaz- dani, and B. A. Bernevig, Phys. Rev. B 103, 205414 (2021)

2021
[15]

Bistritzer and A

R. Bistritzer and A. H. MacDonald, Proceedings of the National Academy of Sciences 108, 12233 (2011)

2011
[16]

H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Phys. Rev. X 8, 031089 (2018)

2018
[17]

Serlin, C

M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Ba- lents, and A. F. Young, Science 367, 900 (2020), https://www.science.org/doi/pdf/10.1126/science.aay5533

2020 doi
[18]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018)

2018
[19]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science 365, 605 (2019), https://www.science.org/doi/pdf/10.1126/science.aaw3780

2019 doi
[20]

M. Kang, L. Ye, S. Fang, J. S. You, A. Levitan, M. Han, J. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg, M. K. Chan, R. D. McDonald, D. Graf, K. Kaznatcheev, E. Vescovo, D. C. Bell, E. Kaxiras, J. van den Brink, M. Richter, M. Prasad Ghimire, J. G. Checkelsky, and R. Comin,...

2020
[21]

Jiang, M

W. Jiang, M. Kang, H. Huang, H. Xu, T. Low, and F. Liu, Phys. Rev. B 99, 125131 (2019)

2019
[22]

Taguchi, M

Y. Taguchi, M. Hirayama, and T. Miyake, Phys. Rev. B 111, 115105 (2025)

2025
[23]

Mukherjee, A

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. ¨Ohberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015)

2015
[24]

Jiang, S

W. Jiang, S. Zhang, Z. Wang, F. Liu, and T. Low, Nano Letters 20, 1959 (2020)

1959
[25]

C. Y. Chen, E. Li, H. Xie, J. Zhang, J. W. Y. Lam, B. Z. Tang, and N. Lin, Communications Materials 5, 54 (2024)

2024
[26]

Vidal, R

J. Vidal, R. Mosseri, and B. Dou¸ cot, Phys. Rev. Lett. 81, 5888 (1998)

1998
[27]

Wang and Y

F. Wang and Y. Ran, Phys. Rev. B 84, 241103 (2011)

2011
[28]

Dey and T

B. Dey and T. K. Ghosh, Phys. Rev. B98, 075422 (2018)

2018
[29]

K. W. Lee, P. H. Fu, and Y. S. Ang, Phys. Rev. B 109, 235105 (2024)

2024
[30]

Dey and T

B. Dey and T. K. Ghosh, Phys. Rev. B99, 205429 (2019)

2019
[31]

F. K. Kunst, G. van Miert, and E. J. Bergholtz, Phys. Rev. B 97, 241405 (2018)

2018
[32]

L. L. Ye, C. Z. Wang, and Y.-C. Lai, Phys. Rev. B 110, 075108 (2024)

2024
[33]

Tamang and T

L. Tamang and T. Biswas, Phys. Rev. B 107, 085408 (2023)

2023
[34]

Illes, J

E. Illes, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B 92, 245410 (2015)

2015
[35]

F. Li, Q. Zhang, and K. S. Chan, Scientific Reports 12, 12987 (2022)

2022
[36]

D. Ma, Y. G. Chen, Y. Yu, and X. Luo, Phys. Rev. B 109, 155159 (2024)

2024
[37]

X. Zhou, Y. C. Hung, B. Wang, and A. Bansil, Phys. Rev. Lett. 133, 236401 (2024)

2024
[38]

Islam, K

M. Islam, K. Bhattacharyya, and S. Basu, Phys. Rev. B 110, 045426 (2024)

2024
[39]

Bhattacharyya, S

K. Bhattacharyya, S. Lahiri, M. Islam, and S. Basu, Phys. Rev. B 110, 235432 (2024)

2024
[40]

Koshino, New Journal of Physics 17, 015014 (2015)

M. Koshino, New Journal of Physics 17, 015014 (2015)

2015
[41]

J. M. B. L. dos Santos, N. M. R. Peres, and A. H. C. Neto, Phys. Rev. Lett. 99, 256802 (2007)

2007
[42]

Tarnopolsky, A

G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Phys. Rev. Lett. 122, 106405 (2019)

2019
[43]

Y. H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Phys. Rev. B 99, 075127 (2019)

2019
[44]

T. Cea, P. A. Pantale´ on, and F. Guinea, Phys. Rev. B 102, 155136 (2020)

2020
[45]

Liu and X

J. Liu and X. Dai, Phys. Rev. B 103, 035427 (2021)

2021
[46]

X. Sun, S. Zhang, Z. Liu, H. Zhu, J. Huang, K. Yuan, Z. Wang, K. Watanabe, T. Taniguchi, X. Li, M. Zhu, J. Mao, T. Yang, J. Kang, J. Liu, Y. Ye, Z. V. Han, and Z. Zhang, Nature Communications 12, 7196 (2021)

2021
[47]

Zhang, D

Y.-H. Zhang, D. Mao, and T. Senthil, Phys. Rev. Res. 1, 033126 (2019)

2019
[48]

Taherinejad, K

M. Taherinejad, K. F. Garrity, and D. Vanderbilt, Phys. Rev. B 89, 115102 (2014)

2014
[49]

A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B 83, 035108 (2011)

2011
[50]

Lahiri and S

S. Lahiri and S. Basu, Phys. Rev. B 109, 115424 (2024)

2024
[51]

Fukui, Y

T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Physical Society of Japan 74, 1674 (2005)

2005

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies- Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, Phys. Rev. Lett. 124, 076801 (2020)

work page 2020

[2] [2]

Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

are denoted by a1, a2 (˜a1, ˜a2), and the corresponding sublattice position vectors are indicated by τX (τ ˜X ), with X = A, B, C and ˜X = ˜A, ˜B, ˜C. The nearest-neighbor hopping amplitudes between A–B and B–C sites are labeled as t and αt, respectively. conduction bands of the pristine nearest-neighbor dice lattice resembling graphene, the triply degene...

work page Pith review arXiv 2025

[3] [3]

U U † H0(− θ 2)) (1) where H0 corresponds to the intralayer Hamiltonian of the individual α-T3 layers and U denotes the interlayer tunneling matrix. It is to be noted that the Hamiltonian for a monolayer α-T3 lattice when twisted by an angle θ (denoted by H0(θ)), and expanded around the rotated 3 valley point Kθ, can be written as HKθ(qθ)= vF qθ ⎛ ⎜ ⎝ 0 e...

work page

[4] [4]

Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Nature 583, 215 (2020)

work page 2020

[5] [5]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature 556, 80 (2018). 7

work page 2018

[6] [6]

Tanaka, J

M. Tanaka, J. I. j. Wang, T. H. Dinh, D. Rodan-Legrain, S. Zaman, M. Hays, A. Almanakly, B. Kannan, D. K. Kim, B. M. Niedzielski, K. Serniak, M. E. Schwartz, K. Watanabe, T. Taniguchi, T. P. Orlando, S. Gustavs- son, J. A. Grover, P. Jarillo-Herrero, and W. D. Oliver, Nature 638, 99 (2025)

work page 2025

[7] [7]

P. J. Ledwith, A. Vishwanath, and E. Khalaf, Exotic car- riers from concentrated topology: Dirac trions as the origin of the missing spectral weight in twisted bilayer graphene (2025), arXiv:2505.08779 [cond-mat.str-el]

arXiv 2025

[8] [8]

P. J. Ledwith, J. Dong, A. Vishwanath, and E. Khalaf, Phys. Rev. X 15, 021087 (2025)

2025