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To minimize a band's quantum geometry, place its orbitals at the symmetric positions the tight-binding model allows.

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2026-08-01 07:02 UTC pith:CQGJLWD3

load-bearing objection Solid fixed-lattice theorem for the integrated quantum metric, but the advertised 'must obey' claim for the Berry-curvature variance is false as stated—and the paper's own Appendix B2 is the counterexample. the 1 major comments →

arxiv 2607.21581 v1 pith:CQGJLWD3 submitted 2026-07-23 cond-mat.str-el

Symmetry and Quantum Geometry in Bloch Bands

Ziwei Wang , Charlie Raca , Steven H. Simon This is my paper
classification cond-mat.str-el
keywords quantum geometryintegrated quantum metricBerry curvature varianceorbital embeddingspatial symmetrymagnetic translation symmetryideal bandsHofstadter model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Quantum geometric quantities such as the integrated quantum metric and the variance of Berry curvature depend on where orbitals sit inside the unit cell, not just on the hopping parameters of a tight-binding model. This paper establishes that if a tight-binding model is compatible with a spatial symmetry—possibly combined with a gauge transformation or time-reversal—then the orbital embeddings that minimize these two quantities must themselves be symmetric, up to an overall translation and with a fine-tuned caveat that at least one global minimum is symmetric. The argument works because both quantities are low-degree polynomials in the orbital positions, so a symmetry operation maps one minimum to another, and uniqueness or group averaging then forces symmetry. A sympathetic reader would care because this makes geometry-independent predictions of tight-binding models meaningful and gives a concrete construction rule for designing flat-band, fractional Chern insulator, and ideal-band models.

Core claim

The paper's central claim is that, in order to minimize the integral of the trace of the quantum metric tensor or the variance of the Berry curvature, the real-space realization of a tight-binding model must obey all the spatial symmetries that the tight-binding model is compatible with, up to gauge transformation and possible time-reversal operation. For generic non-degenerate cases the minimizing embedding is itself symmetric; in fine-tuned cases with multiple minima, at least one global minimum must be symmetric. The proof relies on the fact that the quantum geometric quantities are polynomials of the orbital positions up to second order, that spatial operations preserve these quantities,

What carries the argument

The central object is the 'orbital embedding': the assignment of real-space positions to the orbitals of a tight-binding model, including both the lattice vectors and the intra-unit-cell positions. The proof's key mechanism is the transformation rule for Bloch wavefunctions under a change of embedding, \tilde u_a(k) = e^{-ik\cdot r_a} u_a(k), which makes the quantum metric tensor a quadratic polynomial and the Berry curvature a linear polynomial of the orbital positions. Because both integrated quantities are bounded below, their minima can be studied via quadratic forms; applying a spatial symmetry to a minimizing embedding generates another minimizer, and the uniqueness of the integrated-q

Load-bearing premise

The proof that the optimal relative orbital positions do not change when the unit cell is reshaped assumes the diagonal components of the integrated quantum metric can be minimized independently; for oblique lattices, where the off-diagonal metric component enters the trace, this independence can fail, and the symmetric-shape conclusion leans on that step.

What would settle it

Take a tight-binding model with a spatial symmetry, such as a C3-symmetric model, and numerically minimize the integrated quantum metric over all lattice vectors and internal orbital positions; if a global minimum is found whose orbital set is not invariant under the symmetry up to a common translation, the central claim is false. For the unit-cell shape part, minimize I on an oblique lattice with a mirror-symmetric model and check whether a sheared lattice with non-symmetric internal coordinates beats every rectangular symmetric embedding.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For the Hofstadter model with uniform flux, the standard placement of orbitals on a regular grid inside the magnetic unit cell is a global minimizer of both the integrated quantum metric and the variance of Berry curvature; no rearrangement of orbitals within the unit cell can beat it.
  • Any ideal band—one saturating the lower bound I = 2π|C|—must be realized with an orbital embedding that obeys all symmetries the underlying tight-binding model supports, which explains why lattice Landau-level parent models are naturally symmetric.
  • If a symmetry does not uniquely fix orbital positions, the allowed symmetric embeddings form a connected manifold, so tuning among symmetric embeddings can vary quantum geometric properties while preserving optimality.
  • The same symmetry-minimization logic applies to the spread of maximally localized Wannier functions: after optimizing over Bloch phases, the most localized Wannier orbitals also sit at symmetric embeddings.
  • The result extends to gauge-composed symmetries, including magnetic translations, so changing magnetic unit cell conventions or gauge choices that preserve periodicity leaves the integrated quantum metric and the variance of Berry curvature unchanged, provided mild divisibility conditions hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unit-cell-shape optimization relies on the assumption that the diagonal components of the integrated quantum metric can be minimized independently for any lattice; for oblique lattices the trace includes the off-diagonal metric component through G = AᵀA / det A, so the optimal relative embedding may depend on the lattice shape. A numerical scan over sheared lattices would map exactly where the
  • The fine-tuned degeneracy caveat for the variance of Berry curvature means that models built from decoupled or doubled copies can have non-symmetric global minima; physical models constructed by stacking independent sublattices may evade the symmetry rule for Berry-curvature flatness.
  • The connected-manifold property suggests a practical design strategy: for any model with a spatial symmetry but no unique Wyckoff position, one can restrict optimization of quantum geometric quantities to the symmetric-submanifold, and this is sufficient to find the global minimum.
  • The paper implicitly legitimizes computing quantum geometry at the physically symmetric embedding as the canonical geometry-independent choice, but it also warns that when symmetry does not fix the embedding uniquely, the physical orbital positions may differ from the geometry-minimizing ones, which could matter for comparing tight-binding predictions with ab initio or experimental systems.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the dependence of two quantum-geometric quantities — the integrated quantum metric I and the variance of the Berry curvature σ²_Ω — on the real-space embedding of a tight-binding model, i.e., on the lattice vectors and intra-cell orbital positions. The central claim is that any embedding minimizing these quantities must obey all spatial symmetries with which the tight-binding model is compatible, up to gauge transformations and time reversal. For I, the paper proves that the minimizer is unique up to a uniform translation, which forces symmetry. For σ²_Ω, it proves only that at least one global minimizer is symmetric when the quadratic part has extra null vectors, and it constructs a doubled Haldane model in which non-symmetric embeddings are also global minima. The paper extends the argument to magnetic translations and other composite symmetries, and applies it to the Hofstadter model, ideal bands, and maximally localized Wannier functions.

Significance. The paper is a self-contained analytic contribution to a currently active problem: which orbital embedding should be used when computing geometry-independent quantum-geometric quantities. The fixed-lattice proof for I is clean and rigorous, and it goes beyond Ref. [30] by treating cases where symmetry does not uniquely fix the orbital positions and by proving uniqueness up to translation. The treatment of gauge-transformed spatial symmetries, especially magnetic translations, is original and gives a non-trivial justification of the conventional regular-grid embedding of the Hofstadter model. The paper is also honest about the limitations of the σ²_Ω statement, providing an explicit counterexample in Appendix B2 and a footnote that narrows the claim. The main weakness is not the mathematics but a mismatch between the unqualified abstract/theorem statement and the actual result for σ²_Ω.

major comments (1)
  1. [Abstract and Section II.B] The abstract and the bold statement in Section II.B claim that the real-space embedding minimizing the variance of Berry curvature or the integrated quantum metric 'must obey all the spatial symmetries' the model is compatible with. This is proved for I: Appendix B1 shows the minimizer is unique up to a uniform translation, so applying any symmetry to a minimizer yields the same embedding. For σ²_Ω, however, the proof in Section III only establishes that at least one global minimum is symmetric when extra null vectors of the quadratic part exist. Appendix B2 gives a concrete counterexample: the doubled Haldane model, in which shifting one honeycomb copy by an arbitrary offset leaves σ²_Ω exactly unchanged, so non-symmetric embeddings are also global minima. Footnote 2 concedes exactly this, but the abstract, the theorem statement, and the conclusion do not carry the qualification. Becaus
minor comments (5)
  1. [Introduction] Typo: 'an two-dimensional insulator' should be 'a two-dimensional insulator'.
  2. [Section IV] Typo: 'shownshown' in the first paragraph should be 'shown'.
  3. [Appendix C] The simultaneous-minimization step after Eq. (C5)–(C8) is correct but too terse. The reason one can minimize \(\bar g_{xx}\) and \(\bar g_{yy}\) simultaneously is that they depend on disjoint sets of Cartesian orbital coordinates, not on the relative coordinates \(\tilde x_a,\tilde y_a\) separately for oblique lattices. Please state this explicitly, and clarify that the 'same relative embedding' means the same physical Cartesian embedding expressed in the new basis. As written, the reader may worry that off-diagonal metric components under lattice deformations invalidate the argument; they do not, but the text should say why.
  4. [Appendix D] In Eq. (D12), the notation \(\tilde\Omega(k)=q\Omega(k)\) is ambiguous: it should be stated explicitly that \(\tilde\Omega\) is the sum of Berry curvatures of the folded descendant bands in the supercell gauge. Otherwise the factor \(q\) appears unmotivated. The point that the supercell construction does not directly preserve the variance of the folded sum is made, but the subsequent use of \(\tilde\Omega\) to extract \(\Omega'\) needs this clarification to be fully rigorous.
  5. [Section VI, corollary (1)] The sentence 'the embedding of an ideal band must obey all symmetries of the underlying tight-binding model' is safe because the ideal-band condition fixes I at its minimum, but it should be phrased as a consequence of the uniqueness result for I. Without that qualifier, the sentence can be misread as inheriting the unqualified σ²_Ω claim.

Circularity Check

0 steps flagged

No circularity: the central theorem is proved from the model definitions, and the only soft spot (unqualified variance-minimizer statement) is expressly qualified by the paper itself.

full rationale

The paper's derivation chain is self-contained. The central claim is established by direct computation: orbital-position dependence enters through the explicit transformation u_a(k)→e^{-ik·r_a}u_a(k) (Eqs. 13–16), which makes I and σ²_Ω polynomials of degree ≤2 in the embedding. Existence of a minimum follows from boundedness; uniqueness for I (up to translation) is proved in Appendix B1 via a Cauchy–Schwarz saturation argument; the group-averaging step for degenerate σ²_Ω minima is standard convexity over the symmetry orbit; and Appendix C is a separate variational argument over lattice vectors. Gauge transformations are handled by derivation in Appendix D, not by assumption. Ref. [30] is cited only as a recovered special case, and the only author-overlapping citation (Ref. [37], Simon & Rudner) supplies terminology/context, not a load-bearing premise. Footnote 2 and Appendix B2 explicitly qualify the unqualified 'must obey' wording for σ²_Ω and exhibit non-symmetric global minima; this concerns the truth-value of the advertised claim rather than a circular reduction. No fitted parameter is relabelled as a prediction, and no result is imported from a self-citation chain, so there is no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No numerical parameters are fitted to data; the paper is a proof. The non-trivial input is a collection of domain assumptions about smoothness of wavefunctions, the invariance of quantum geometry under symmetry operations, and the validity of the gauge-transformation and lattice-vector lemmas. The most fragile entry is the final lattice-vector lemma, which is also identified as the weakest assumption.

axioms (6)
  • standard math A bounded-below quadratic polynomial in real variables has a global minimum modulo null vectors; group averaging of a convex invariant function maps a minimizer to a symmetric minimizer.
    Used in Section III to establish existence of minima and to symmetrize minimizers in the degenerate variance-of-Berry-curvature case.
  • domain assumption The Bloch wavefunctions of the band are smooth in k and every orbital has non-vanishing weight somewhere in the BZ (no trivially removable orbitals).
    Appendix B1 uses this to prove the integrated quantum metric has no flat directions beyond uniform translation. If violated, e.g. for multi-band sets with disjoint orbital support, non-symmetric minimizers exist and the 'must obey' claim fails.
  • domain assumption Spatial symmetry operations are Euclidean isometries acting linearly (up to translation) on orbital positions, so applying them to any embedding and relabeling orbitals leaves the integrated quantum metric and Berry-curvature variance unchanged.
    Used in Section III and Appendix A to map a minimizer to another minimizer; the whole symmetry argument depends on this invariance.
  • domain assumption Gauge transformations that participate in symmetries either take the form \hat c_{R,a} -> \hat c_{R,a} e^{i(k0·R+φ_a)}, or, if they alter the magnetic unit cell, the unit-cell dimensions satisfy m|q and n|q for the variance invariance to hold.
    Appendix D proves invariance of I and σ² only under these conditions; arbitrary gauge transformations beyond these forms are not covered.
  • domain assumption The symmetry group used in the averaging construction is finite (or the averaging integral is well-defined).
    Section III uses a finite-group average 1/|G| to construct a symmetric global minimum in the degenerate σ² case.
  • domain assumption For any unimodular change of lattice vectors, the same relative embedding minimizes the integrated quantum metric (the simultaneous-minimization claim of Appendix C).
    This is required to extend the symmetry result to optimization over unit-cell shape. It is asserted in Appendix C but not proved for oblique lattices, where the off-diagonal metric component can enter the trace through the lattice matrix G = A^T A / det A.

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read the original abstract

Quantum geometric quantities have featured heavily in the discussion of the properties of quantum systems in recent years. Among quantities most commonly discussed is the variance of Berry curvature and the integral of the trace of the quantum metric tensor. Despite their usefulness, it is known that they suffer from one significant complication: for a tight-binding model, the quantum geometric quantities depend not only on the parameters of the tight-binding model itself, but also the real space geometry of the tight-binding model, the so-called "orbital embedding". One explicitly geometry-independent quantity is therefore the minimal value of the quantum geometric quantity out of all possible real space geometries. In this work, we demonstrate that, if the tight-binding model is compatible with certain spatial symmetries, then the real space geometry that minimizes the variance of the Berry curvature or the integral of the trace of the quantum metric tensor must obey all those spatial symmetries. We further show that the statement is applicable to systems with magnetic translation symmetries and other composite symmetries, with implications for the quantum geometry of the Hofstadter model.

Figures

Figures reproduced from arXiv: 2607.21581 by Charlie Raca, Steven H. Simon, Ziwei Wang.

Figure 1
Figure 1. Figure 1: FIG. 1: The honeycomb lattice (a) and a decorated [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Two honeycomb lattices with an offset and [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Reference graph

Works this paper leans on

45 extracted references · 1 linked inside Pith

  1. [1]

    equivalent

    Integrated quantum metric In this section, we show that, excluding trivially re- movable orbitals, the only (two) flat directions (i.e. com- binationsof{r a,i}thatarenotallzeroandkeepthequan- tum geometric quantity unchanged) for the integrated quantum metric are the uniform shifts in the orbital po- sitions. The diagonal components of the quantum metric ...

  2. [2]

    doubled” version of the Hamiltonian with some “external field

    V ariance of Berry curvature Now we turn our attention to the variance of Berry curvature. Under a change of embedding, the Berry cur- vature transforms as Ω(k)→Ω(k)− X a ϵijra,i∂kj |ua(k)|2.(B7) For a certain combination ofra,i to be a flat direction, the necessary and sufficient condition is X a ϵijra,i∂kj |ua(k)|2 = 0for allk.(B8) Clearly, a uniform sh...

  3. [3]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two- Dimensional Periodic Potential, Physical Review Letters 49, 405 (1982)

    1982

  4. [4]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of Modern Physics82, 1959 (2010)

    1959

  5. [5]

    Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    2018

  6. [6]

    S.A.Parameswaran, R.Roy,andS.L.Sondhi,Fractional Chern insulators and theW∞ algebra, Physical Review B85, 241308 (2012)

    2012

  7. [7]

    M. O. Goerbig, From fractional Chern insulators to a fractional quantum spin hall effect, The European Phys- ical Journal B85, 15 (2012)

    2012

  8. [8]

    Y.-L. Wu, B. A. Bernevig, and N. Regnault, Zoology of fractional Chern insulators, Physical Review B85, 075116 (2012)

    2012

  9. [9]

    Murthy and R

    G. Murthy and R. Shankar, Hamiltonian theory of frac- tionally filled chern bands, Phys. Rev. B86, 195146 (2012)

    2012

  10. [10]

    Dobardžić, M

    E. Dobardžić, M. V. Milovanović, and N. Regnault, Geo- metrical description of fractional Chern insulators based on static structure factor calculations, Physical Review B88, 115117 (2013)

    2013

  11. [11]

    Claassen, C

    M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux, Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators, Physical Re- 16 view Letters114, 236802 (2015)

    2015

  12. [12]

    T. S. Jackson, G. Möller, and R. Roy, Geometric stability of topological lattice phases, Nature Communications6, 8629 (2015)

    2015

  13. [13]

    Bauer, T

    D. Bauer, T. S. Jackson, and R. Roy, Quantum geometry and stability of the fractional quantum hall effect in the hofstadter model, Phys. Rev. B93, 235133 (2016)

    2016

  14. [14]

    C. H. Lee, M. Claassen, and R. Thomale, Band structure engineering of ideal fractional Chern insulators, Physical Review B96, 165150 (2017)

    2017

  15. [15]

    Ozawa and B

    T. Ozawa and B. Mera, Relations between topology and the quantum metric for Chern insulators, Physical Re- view B104, 045103 (2021)

    2021

  16. [16]

    Mera and T

    B. Mera and T. Ozawa, Uniqueness of Landau levels and their analogs with higher Chern numbers, Physical Re- view Research6, 033238 (2024)

    2024

  17. [17]

    J. Shi, J. Cano, and N. Morales-Durán, Effects of Berry curvature on ideal fractional Chern insulator many-body gaps, Physical Review Research8, L022045 (2026)

    2026

  18. [18]

    Roy, Band geometry of fractional topological insula- tors, Physical Review B90, 165139 (2014)

    R. Roy, Band geometry of fractional topological insula- tors, Physical Review B90, 165139 (2014)

    2014

  19. [19]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact Landau Level Description of Geometry and Interaction in a Flatband, Physical Review Letters127, 246403 (2021)

    2021

  20. [20]

    Mera and T

    B. Mera and T. Ozawa, Kähler geometry and Chern in- sulators: Relations between topology and the quantum metric, Physical Review B104, 045104 (2021)

    2021

  21. [21]

    Wang and Z

    J. Wang and Z. Liu, Hierarchy of Ideal Flatbands in Chi- ral Twisted Multilayer Graphene Models, Physical Re- view Letters128, 176403 (2022)

    2022

  22. [22]

    P. J. Ledwith, A. Vishwanath, and E. Khalaf, Family of Ideal Chern Flatbands with Arbitrary Chern Number in Chiral Twisted Graphene Multilayers, Physical Review Letters128, 176404 (2022)

    2022

  23. [23]

    Mera and T

    B. Mera and T. Ozawa, Engineering geometrically flat Chern bands with Fubini-Study Kähler structure, Phys- ical Review B104, 115160 (2021)

    2021

  24. [24]

    P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vor- texability: A unifying criterion for ideal fractional Chern insulators, Physical Review B108, 205144 (2023)

    2023

  25. [25]

    J. Wang, S. Klevtsov, and Z. Liu, Origin of model frac- tional Chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra, Physical Review Research5, 023167 (2023)

    2023

  26. [26]

    Z. Liu, B. Mera, M. Fujimoto, T. Ozawa, and J. Wang, Theory of Generalized Landau Levels and Its Implica- tions for Non-Abelian States, Physical Review X15, 031019 (2025)

    2025

  27. [27]

    Fujimoto, D

    M. Fujimoto, D. E. Parker, J. Dong, E. Khalaf, A. Vish- wanath, and P. Ledwith, Higher Vortexability: Zero- Field Realization of Higher Landau Levels, Physical Re- view Letters134, 106502 (2025)

    2025

  28. [28]

    Peotta and P

    S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nature Communications6, 8944 (2015)

    2015

  29. [29]

    Julku, S

    A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. Törmä, Geometric Origin of Superfluidity in the Lieb- Lattice Flat Band, Physical Review Letters117, 045303 (2016)

    2016

  30. [30]

    Liang, T

    L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and P. Törmä, Band geometry, Berry curvature, and super- fluid weight, Physical Review B95, 024515 (2017)

    2017

  31. [31]

    Herzog-Arbeitman, V

    J. Herzog-Arbeitman, V. Peri, F. Schindler, S. D. Hu- ber, and B. A. Bernevig, Superfluid Weight Bounds from Symmetry and Quantum Geometry in Flat Bands, Phys- ical Review Letters128, 087002 (2022)

    2022

  32. [32]

    Huhtinen, J

    K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, and P. Törmä, Revisiting flat band supercon- ductivity: Dependence on minimal quantum metric and band touchings, Physical Review B106, 014518 (2022)

    2022

  33. [33]

    Tam and S

    M. Tam and S. Peotta, Geometry-independent superfluid weight in multiorbital lattices from the generalized ran- dom phase approximation, Physical Review Research6, 013256 (2024)

    2024

  34. [34]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized Wannier functions for composite energy bands, Physical Review B56, 12847 (1997)

    1997

  35. [35]

    Fruchart, D

    M. Fruchart, D. Carpentier, and K. Gawędzki, Parallel transport and band theory in crystals, Europhysics Let- ters106, 60002 (2014)

    2014

  36. [36]

    Dobardžić, M

    E. Dobardžić, M. Dimitrijević, and M. V. Milovanović, Effective description of Chern insulators, Physical Re- view B89, 235424 (2014)

    2014

  37. [37]

    Lim, J.-N

    L.-K. Lim, J.-N. Fuchs, and G. Montambaux, Geome- try of bloch states probed by stückelberg interferometry, Physical Review A92, 063627 (2015)

    2015

  38. [38]

    N. R. Cooper, J. Dalibard, and I. B. Spielman, Topologi- calbandsforultracoldatoms,ReviewsofModernPhysics 91, 015005 (2019)

    2019

  39. [39]

    S. H. Simon and M. S. Rudner, Contrasting lattice ge- ometry dependent versus independent quantities: Rami- fications for Berry curvature, energy gaps, and dynamics, Physical Review B102, 165148 (2020)

    2020

  40. [40]

    Fuchs and F

    J.-N. Fuchs and F. Piéchon, Orbital embedding and topology of one-dimensional two-band insulators, Physi- cal Review B104, 235428 (2021)

    2021

  41. [41]

    Parity Anomaly

    F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly", Physical Review Letters61, 2015 (1988)

    2015

  42. [42]

    Kapit and E

    E. Kapit and E. Mueller, Exact Parent Hamiltonian for the Quantum Hall States in a Lattice, Physical Review Letters105, 215303 (2010)

    2010

  43. [43]

    Qi, Y.-S

    X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quan- tization of the spin Hall effect in two-dimensional para- magnetic semiconductors, Physical Review B74, 085308 (2006)

    2006

  44. [44]

    Li and F

    B. Li and F. Wu, Quantum-Geometric Design of Lat- tice Generalized Landau Levels (2026), arXiv:2607.08702 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2026

  45. [45]

    L. Zhao, Y. Ge, S. Ryu, and J. Yu, Ideal bands in tight- binding models (2026), to appear

    2026