Pith. sign in

REVIEW 3 major objections 6 minor 47 references

Reduced Wannier representation for topological bands

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read By using fewer trial orbitals than bands, exponentially localized Wannier functions can be built for most of a topologically obstructed manifold, leaving a small complement that carries the Chern or Z2 index.

desk verdict A credible and potentially useful method for partially Wannierizing Chern and Z2 bands, but the exponential-localization claim would be stronger with full-BZ singular-value checks for the larger supercells. read the letter →

arxiv 2412.17084 v3 pith:3CAHMUUF submitted 2024-12-22 cond-mat.mes-hall cond-mat.mtrl-sci

Trey Cole , David Vanderbilt This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords WannierfunctionstopologicalobstructionCherninsulatorHaldanemodelKane-MeleZ2indexsupercellprojectionmethod
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 claims that a topologically obstructed band manifold can be split into a large trivial subspace that admits exponentially localized Wannier functions and a small complementary subspace that carries the topological index. The construction uses fewer trial orbitals than there are bands, so the projection overlap matrix stays full rank everywhere in the Brillouin zone instead of becoming singular at the band-inversion momentum. The authors demonstrate this on Haldane and Kane-Mele models in supercells, reaching Wannier fractions of 3/4 and 6/8 in 2x2 cells and up to 24/25 and 16/18 in larger cells. If correct, the method gives a practical route to localized tight-binding-like descriptions of most of a Chern or Z2 band, with the obstructive sector cleanly separated.

What carries the argument

The key object is the rectangular inner-product matrix $A_{mn}(k)=\langle\psi_{mk}|g_n\rangle$ between Bloch eigenstates of the target manifold and a set of $J$ trial orbitals. When $J=M$ and the band is topological, $A_k$ drops rank at the inversion momentum and the $S_k^{-1/2}$ normalization diverges; when $J<M$, $A_k$ is $M\times J$ and full rank, so its singular value decomposition yields a semi-unitary projector $V\,1_{M\times J}\,W^\dagger$ defining a smooth $J$-dimensional subspace. That subspace is then refined by subspace selection, which minimizes the gauge-invariant spread $\Omega_I$, and maximal localization, which minimizes the gauge-dependent spread $\tilde\Omega$, producing the exponentially localized reduced Wannier functions whose complement carries the topology.

What would settle it

Compute the smallest singular value of $A_k$ on a dense full Brillouin-zone mesh for the $5\times5$ Haldane supercell at $f_W=24/25$ and the $3\times3$ Kane-Mele supercell at $f_W=16/18$; if it reaches zero at any point, the claimed exponential localization of the reduced Wannier functions fails.

Watch

Extended reading notes

Core claim

The central discovery is that the topological obstruction to exponential Wannier localization can be confined to a lower-dimensional subspace rather than poisoning the whole band manifold. In the projection method, the obstruction appears as a rank deficiency of the $M \times J$ inner-product matrix $A_k$ at momenta where the band character inverts. By omitting trial functions so that $A_k$ becomes rectangular with full rank at every $k$, the projected Bloch-like states remain smooth and periodic, and the resulting $J$ Wannier functions decay exponentially. The complement, obtained by subtracting the trivial projector from the occupied projector, inherits the Chern number or the Z2 oddness; for a $C=1$ Haldane band it is one band, and for a $Z_2=1$ Kane-Mele insulator it is one Kramers pair. The number of Wannier functions that can be constructed is bounded by $1 - N_{\max}/N_b$, where $N_{\max}$ is the maximal null-space dimension of $A_k$, and supercell folding raises the fraction toward one while breaking primitive translational symmetry.

Load-bearing premise

The method requires that after omitting enough trial functions, the rectangular overlap matrix $A_k$ has full rank with all singular values bounded away from zero at every $k$ in the Brillouin zone; the paper checks singular values along high-symmetry paths but does not report a full-zone verification for every supercell or an analytic proof.

Editorial extensions

If this is right

  • For a Chern insulator with $C=1$, all but one band of the occupied manifold can be Wannierized, with Wannier fraction bounded by $f_W \le 1 - 1/N_{sc}^2$ on an $N_{sc}\times N_{sc}$ supercell.
  • For a $Z_2$-odd insulator, all but one Kramers pair admit exponentially localized Wannier functions respecting time-reversal symmetry, with fraction bounded by $f_W \le 1 - 2/n_{occ}$.
  • The trivial subspace yields Wannier-interpolated bands that track the low-energy occupied bands, while the topological subspace reproduces the inverted character at the band-inversion point and carries the net Chern number.
  • The procedure works without borrowing opposite-topology bands, so it remains applicable when no nearby band of opposite Chern number exists, and it can be incorporated into first-principles Wannier workflows.
  • In flat-band or moiré systems, a supercell commensurate with a charge-density wave could isolate a Wannierizable trivial sector, potentially simplifying strongly correlated calculations.

Reading between the lines

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

  • The maximal null-space dimension $N_{\max}$ of $A_k$ is effectively a real-space measure of the obstruction; it might be possible to prove a general identity tying $N_{\max}$ to the Chern number or $Z_2$ index, making the Wannier-fraction formula exact rather than an upper bound.
  • The omitted trial site acts like a localized defect whose position is a free choice; treating that position as a variational parameter could reduce spreads further, and in magic-angle twisted bilayer graphene the 'fidget spinner' charge pattern suggests which three sites to keep.
  • The decomposition suggests a concrete strategy for interacting flat-band models: if a fractional filling occupies the topological sector, the trivial Wannier sector still supplies a local basis for the remaining degrees of freedom, an application the paper motivates but does not test.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a construction of exponentially localized Wannier functions that span a subspace of a topologically obstructed band manifold. The key idea is to use fewer trial orbitals than occupied bands in the projection method, so that the rectangular overlap matrix A_k can have full column rank everywhere in the Brillouin zone, yielding a smooth gauge on the selected subspace. The complement of this 'reduced Wannier subspace' inherits the topological index. The method is demonstrated on the Haldane model in its Chern-insulating phase and on the Kane-Mele model in its Z2-odd phase, using supercells to increase the fraction of the occupied manifold that can be Wannierized. A Wannier fraction fW is defined, with a bound derived from the maximal null-space dimension of A_k.

Significance. If the construction works as claimed, it offers a practical way to separate a large 'trivial' part of a topological band manifold from the small 'itinerant' part that carries the Chern or Z2 obstruction. This could be useful in flat-band and moiré contexts where one wants a localized Wannier basis for correlation calculations while keeping the topological subspace accessible. The paper is clearly written, the numerical spreads and Berry-curvature results are internally consistent, and the authors provide an explicit rank-nullity bound on the achievable Wannier fraction. The implementation in an extension of PythTB is a further strength, as it makes the procedure reproducible in principle.

major comments (3)
  1. [Secs. III.D and IV.D, Eq. (32), Figs. 2(b) and 11(b)] The central smoothness assumption is verified only for the initial projection in the small supercells, not for the final subspaces at the claimed maximum Wannier fractions. The bound fW <= 1 - Nmax/Nb is necessary but not sufficient: a fixed set of omitted columns can still fail to have full rank at some k, because the null space at rank-deficient points need not be contained in the omitted columns. The paper reports singular values only along a high-symmetry path for the 2x2 Haldane and Kane-Mele cases, and these are for the initial projection, before the subspace-selection and maximal-localization steps. No full-BZ singular-value or minimum-gap data are given for the 5x5 Haldane (fW=24/25) or 3x3 Kane-Mele (fW=16/18) supercells, nor is an analytic argument supplied. Since rank deficiency is a codimension-two condition that can occur at generic points missed by a path, this missing check is load-bearing for the claim that the reduced Wannier functions are exponentially localized.
  2. [Secs. III.B and IV.B, Figs. 3 and 12] The evidence for exponential localization is indirect. The figures plot bin-averaged density weights on a finite discrete-k supercell; they demonstrate decay of the density, not exponential localization of the Wannier amplitudes in the thermodynamic limit. Please either prove the existence of a smooth periodic gauge (for example, by establishing a uniform lower bound on the relevant singular values over the full BZ for the final optimized subspace), or provide a convergence study of the Wannier tails as the supercell and k-mesh sizes are increased. Without this, the central claim remains an assertion.
  3. [Sec. III.D, paragraph after Eq. (35)] The statement that 'The dimensionality of the null space at the singularities remains invariant through band folding' is asserted without proof. In a supercell, multiple primitive-cell k-points may fold to the same supercell k-point, and it is not obvious that the null-space dimension at those points is simply the primitive-cell value. Please provide a justification or a direct numerical check for the 5x5 and 3x3 supercells.
minor comments (6)
  1. [Eq. (23)] The object (A_K)_{4,n} with n varying is a row of the inner-product matrix, but the text calls it 'an entire column of zeros.' Please correct the row/column terminology.
  2. [Sec. II.B, discussion after Eq. (11)] The sentence 'If A is singular, then V W† becomes semi-unitary' is incorrect; for a square matrix A, V and W from the SVD are unitary even when A is singular, and the breakdown is the divergence of S^{-1/2}. The semi-unitary case is the rectangular one of Eq. (12). Please rephrase to avoid confusion about the mechanism.
  3. [Eq. (12)] The notation 1_{M x J} should be defined explicitly, for example as the first J columns of the M x M identity matrix or as the rectangular diagonal matrix with unit entries.
  4. [Sec. III.D, Fig. 8] The text says that trial wave functions are placed on 'randomly chosen subsets of low-energy sites' for the 5x5 supercell. Please specify the number of realizations, the random seed or distribution, and whether the plotted spreads are averaged over realizations, so that the results are reproducible.
  5. [Figs. 3 and 12, insets] The phrase 'the supercell conjugate to the discrete k-mesh' is unclear. Please specify the real-space supercell size and the k-mesh used for the plots, and clarify whether the plotted quantity is the Wannier function density or the site-projected weight.
  6. [References, [31]] Reference [31] is cited as the implementation but is not described. Please indicate whether this is a publicly available code repository and provide a URL or DOI if applicable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the reduced Wannier construction is a direct application of standard projection and disentanglement, and the Wannier-fraction bound follows from rank-nullity, not from fitted inputs.

full rationale

The central derivation is self-contained and not circular. The reduced Wannier construction follows the standard projection formalism of Refs. [28,29,32]: with J<M trial functions, the M×J overlap matrix A_k defines a J-dimensional smooth subspace whenever A_k has full rank and bounded singular values; Eq. (12) is then a semi-unitary projection, and Wannier functions are obtained by Fourier transform. The paper verifies the required rank/gap condition numerically for the 2x2 cells and checks exponential decay directly (Figs. 3 and 12). The Wannier-fraction bound of Eq. (32) is an elementary rank-nullity consequence of the definition of Nmax, not a fitted target. The split C_occ = C_triv + C_topo with C_triv=0 follows from the established theorem that exponentially localized Wannier functions imply vanishing Chern number; the paper then independently integrates Berry curvature to confirm C_topo=C_occ. No parameter is fit to the claimed fraction, and no load-bearing conclusion rests on an unverified self-citation. The only concerns are verification gaps (singular values for the 5x5 and 3x3 cells are not shown over the full BZ), which are correctness risks, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central construction rests on standard Wannier and MLWF theory plus a model-specific assumption that omitting Nmax trial functions removes the rank deficiency globally. The model parameters and supercell sizes are chosen inputs, not fitted quantities. No new physical entities are introduced.

free parameters (2)
  • Supercell size N_sc = 2 and 5 (Haldane), 3 (Kane-Mele)
    Chosen by hand to illustrate different Wannier fractions; it is a methodological control parameter, not a number fitted to data. The Wannier fraction bound depends on it.
  • Omitted trial orbital positions = One low-energy site omitted in 2x2 Haldane, one site and its Kramers partner in Kane-Mele, one C3-symmetric site in…
    The choice of which trial site to omit changes the Wannier spreads and center shifts. For the maximal 5x5 case the omitted site is singled out as a symmetry center, indicating a selection that improves the visual symmetry of the result.
assumptions (4)
  • domain assumption The topological obstruction manifests as a rank deficiency of the inner-product matrix A_k at isolated points in the BZ.
    Argued in Sec. II E and used to motivate the rectangular projection in Sec. III B; it is a property of the trial-function choice and model, not a general theorem.
  • ad hoc to paper After omitting Nmax trial functions, A_k is full rank with bounded singular values for every k.
    This is the load-bearing assumption for smoothness and exponential localization, verified only along high-symmetry paths for 2x2 cells and not checked over the full BZ for larger supercells.
  • standard math A smooth periodic frame for a J-dimensional subbundle of the occupied manifold yields exponentially localized Wannier functions spanning that subbundle.
    Standard Wannier theorem, see Refs. [6,20]; used throughout Secs. III and IV.
  • domain assumption Subspace selection keeps the reduced manifold inside the occupied manifold so that P_topo = P_occ - P_triv is a valid partition.
    Sec. II C and Eq. (27); if the disentanglement mixed in conduction-band states, the topological-subspace projector would no longer partition the occupied bands.
invented entities (1)
  • None (no new physical entities)
    purpose: The paper introduces only the methodological concept of a reduced Wannier representation.
    No new particle, force, dimension, or conserved quantity is postulated. The reduced Wannier representation is a construction, not an invented entity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reduced Wannier representation for topological bands." pith.science (2026). https://pith.science/paper/3CAHMUUF

@misc{pith2026241217084,
  author       = {Pith},
  title        = {Pith review of: Reduced Wannier representation for topological bands},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CAHMUUF}},
  note         = {Machine review of arXiv:2412.17084}
}
read the original abstract

Bands with non-trivial topological indices have a topological obstruction preventing them from being represented by exponentially localized Wannier states. Here, we propose a procedure to construct exponentially localized Wannier functions that span a subspace of topologically obstructed bands through the use of the projection method. These Wannier functions form what we refer to as a "reduced Wannier representation," indicating that the Wannier functions necessarily do not span the full topological manifold. By constructing supercells, we obtain reduced Wannier functions that break the primitive translational symmetry while capturing a substantial portion of the topologically obstructed manifold. This approach effectively decomposes the topological manifold into two subspaces: an itinerant subspace inheriting the topology and a localized trivial subspace represented by reduced Wannier functions. We consider the Haldane and Kane-Mele tight-binding models as the platforms for our investigation.

Figures

Figures reproduced from arXiv: 2412.17084 by the authors.

Figure 1
Figure 1. FIG. 1. Bands for the Haldane model on a 2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The centers of the three reduced Wannier functions [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Decay of the electron density (weights [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Folded bands of the topological occupied mani [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The interpolation of (a) the Berry curvature and (b) [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The dependence of the gauge-invariant (blue tri [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Bands for the Kane-Mele model on a 2 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Decay of the electron density (weights [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Wannier functions (blue circles) at Wannier fraction [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 24 canonical work pages

  1. [1]

    G. H. Wannier, The structure of electronic excitation lev- els in insulating crystals, Phys. Rev. 52, 191 (1937)

    work page 1937

  2. [2]

    Kohn, Analytic properties of bloch waves and wannier functions, Phys

    W. Kohn, Analytic properties of bloch waves and wannier functions, Phys. Rev. 115, 809 (1959)

    work page 1959

  3. [3]

    J. M. Foster and S. F. Boys, Canonical configurational interaction procedure, Rev. Mod. Phys. 32, 300 (1960)

    1960

  4. [4]

    S. F. Boys, Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another, Rev. Mod. Phys. 32, 296 (1960)

    1960

  5. [5]

    J. D. Cloizeaux, Energy bands and projection operators in a crystal: Analytic and asymptotic properties, Phys. Rev. 135, A685 (1964)

    1964

  6. [6]

    Strinati, Multipole wave functions for photoelectrons in crystals

    G. Strinati, Multipole wave functions for photoelectrons in crystals. iii. the role of singular points in the band structure and the tails of the wannier functions, Phys. Rev. B 18, 4104 (1978)

    work page 1978

  7. [7]

    I. V. Solovyev, First-principles wannier functions and effective lattice fermion models for narrow-band com- pounds, Phys. Rev. B 73, 155117 (2006)

    work page 2006

  8. [8]

    Jung and A

    J. Jung and A. H. MacDonald, Tight-binding model for graphene π-bands from maximally localized wannier functions, Phys. Rev. B 87, 195450 (2013)

    work page 2013

Show all 47 references
  1. [9]

    Gresch, Q

    D. Gresch, Q. Wu, G. W. Winkler, R. H¨ auselmann, M. Troyer, and A. A. Soluyanov, Automated construc- tion of symmetrized wannier-like tight-binding models from ab initio calculations, Phys. Rev. Mater. 2, 103805 (2018)

    2018

  2. [10]

    Lihm and C.-H

    J.-M. Lihm and C.-H. Park, Reliable methods for seam- less stitching of tight-binding models based on maximally localized wannier functions, Phys. Rev. B 99, 125117 (2019)

    2019

  3. [11]

    Q. Li, J. S. Smith, Y. Yin, C. Wang, M. V. Klymenko, J. H. Cole, and N. V. Medhekar, Localized wannier func- tion based tight-binding models for two-dimensional al- lotropes of bismuth, New Journal of Physics 23, 063042 (2021)

    2021

  4. [12]

    J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Spec- tral and fermi surface properties from wannier interpola- tion, Phys. Rev. B 75, 195121 (2007)

    2007

  5. [13]

    Calzolari, N

    A. Calzolari, N. Marzari, I. Souza, and M. Buon- giorno Nardelli, Ab initio transport properties of nanos- tructures from maximally localized wannier functions, Phys. Rev. B 69, 035108 (2004)

    2004

  6. [14]

    Kruthoff, J

    J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insu- lators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017)

    2017

  7. [15]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 10.1038/nature23268 (2017)

    2017 doi

  8. [16]

    H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99, 195455 (2019)

    2019

  9. [17]

    H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and wannier obstructions, Phys. Rev. Lett.121, 126402 (2018)

    2018

  10. [18]

    Thonhauser and D

    T. Thonhauser and D. Vanderbilt, Insulator/chern- insulator transition in the haldane model, Phys. Rev. B 74, 235111 (2006)

    2006

  11. [19]

    A. A. Soluyanov and D. Vanderbilt, Wannier representa- tion of Z2 topological insulators, Phys. Rev. B 83, 035108 (2011)

    2011

  12. [20]

    Vanderbilt, BERRY PHASES IN ELECTRONIC STRUCTURE THEORY: Electric Polarization, Orbital Magnetization and Topological Insulators(2018)

    D. Vanderbilt, BERRY PHASES IN ELECTRONIC STRUCTURE THEORY: Electric Polarization, Orbital Magnetization and Topological Insulators(2018)

    2018

  13. [21]

    J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Building blocks of topological quantum chemistry: Ele- mentary band representations, Phys. Rev. B 97, 035139 (2018). 14

    2018

  14. [22]

    Cano and B

    J. Cano and B. Bradlyn, Band representations and topo- logical quantum chemistry, Annual Review of Condensed Matter Physics 12, 225 (2021)

    2021

  15. [23]

    T. M. Gunawardana, A. M. Turner, and R. Barnett, Op- timally localized single-band wannier functions for two- dimensional chern insulators, Phys. Rev. Res. 6, 023046 (2024)

    2024

  16. [24]

    Bouhon, A

    A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wil- son loop approach to fragile topology of split elemen- tary band representations and topological crystalline in- sulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019)

    2019

  17. [25]

    Song and B

    Z.-D. Song and B. A. Bernevig, Magic-angle twisted bi- layer graphene as a topological heavy fermion problem, Phys. Rev. Lett. 129, 047601 (2022)

    2022

  18. [26]

    Hu and Q

    H. Hu and Q. Si, Coupled topological flat and wide bands: Quasiparticle formation and de- struction, Science Advances 9, eadg0028 (2023), https://www.science.org/doi/pdf/10.1126/sciadv.adg0028

    2023 doi

  19. [27]

    L. Chen, F. Xie, S. Sur, H. Hu, S. Paschen, J. Cano, and Q. Si, Metallic quantum criticality enabled by flat bands in a kagome lattice (2023), arXiv:2307.09431 [cond- mat.str-el]

    2023 arXiv

  20. [28]

    Souza, N

    I. Souza, N. Marzari, and D. Vanderbilt, Maximally local- ized wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001)

    2001

  21. [29]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012)

    2012

  22. [30]

    Coh and D

    S. Coh and D. Vanderbilt, Python tight binding (PythTB) (2022)

    2022

  23. [31]

    Cole and D

    T. Cole and D. Vanderbilt, Reduced wannier representa- tion for topological bands (2025)

    2025

  24. [32]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997)

    1997

  25. [33]

    Study, K¨ urzeste wege im komplexen gebiet, Mathema- tische Annalen 60, 321 (1905)

    E. Study, K¨ urzeste wege im komplexen gebiet, Mathema- tische Annalen 60, 321 (1905)

    1905

  26. [34]

    phases” and “distance

    A. K. Pati, Relation between “phases” and “distance” in quantum evolution, Physics Letters A 159, 105 (1991)

    1991

  27. [35]

    Q. Li, J. Dong, P. J. Ledwith, and E. Khalaf, Con- straints on real space representations of chern bands (2024), arXiv:2407.02561 [cond-mat.str-el]

    2024 arXiv

  28. [36]

    Shayegan, Wigner crystals in flat band 2d electron systems, Nature Reviews Physics 4, 212 (2022)

    M. Shayegan, Wigner crystals in flat band 2d electron systems, Nature Reviews Physics 4, 212 (2022)

    2022

  29. [37]

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, Proceed- ings. Physics 14, 816 (2013)

    2013

  30. [38]

    W. Yang, D. Zhai, T. Tan, F.-R. Fan, Z. Lin, and W. Yao, Fractional quantum anomalous hall effect in a singular flat band (2024), arXiv:2405.01829 [cond-mat.mes-hall]

    2024 arXiv

  31. [39]

    Zhao and S.-Q

    A. Zhao and S.-Q. Shen, Quantum anomalous hall ef- fect in a flat band ferromagnet, Phys. Rev. B 85, 085209 (2012)

    2012

  32. [40]

    Y. D. Kato, Y. Okamura, S. Minami, R. Fujimura, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, R. Arita, Y. Tokura, and Y. Takahashi, Optical anomalous hall effect enhanced by flat bands in ferromagnetic van der waals semimetal, npj Quantum Materials 7, 73 (2022)

    2022

  33. [41]

    C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Flat bands and wigner crystallization in the honeycomb opti- cal lattice, Phys. Rev. Lett. 99, 070401 (2007)

    2007

  34. [42]

    Jaworowski, A

    B. Jaworowski, A. D. G¨ u¸ cl¨ u, P. Kaczmarkiewicz, M. Kupczy´ nski, P. Potasz, and A. W´ ojs, Wigner crystal- lization in topological flat bands, New Journal of Physics 20, 063023 (2018)

    2018

  35. [43]

    D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum hall effect in the absence of landau levels, Na- ture Communications 2, 389 (2011)

    2011

  36. [44]

    Kang and O

    J. Kang and O. Vafek, Symmetry, maximally localized wannier states, and a low-energy model for twisted bi- layer graphene narrow bands, Phys. Rev. X 8, 031088 (2018)

    2018

  37. [45]

    J. Yu, M. Xie, B. A. Bernevig, and S. Das Sarma, Magic- angle twisted symmetric trilayer graphene as a topolog- ical heavy-fermion problem, Phys. Rev. B 108, 035129 (2023)

    2023

  38. [46]

    A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised wannier functions, Computer Physics Communications 185, 2309 (2014)

    2014

  39. [47]

    Marrazzo, S

    A. Marrazzo, S. Beck, E. R. Margine, N. Marzari, A. A. Mostofi, J. Qiao, I. Souza, S. S. Tsirkin, J. R. Yates, and G. Pizzi, The wannier-functions software ecosystem for materials simulations (2023)

    2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.