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Quantum Geometry in the NbSe$_2$ Family I: Obstructed Compact Wannier Function and New Perturbation Theory

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the Fermi-level band of monolayer 1H-NbSe2 is an obstructed atomic band whose Wannier function is nearly identical to a strictly compact three-site orbital, and that a new two-energy perturbation theory…

desk verdict The compact Wannier approximation and the NNN>NN cancellation are the real results; the 'new perturbation theory' is under-tested because its wavefunction overlaps with DFT lack a baseline. read the letter →

arxiv 2507.02047 v1 pith:VSG6KH7Z submitted 2025-07-02 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el

Jiabin Yu , Yi Jiang , Yuanfeng Xu , Dumitru Călugăru , Haoyu Hu , Haojie Guo , Sandra Sajan , Yongsong Wang
show 3 more authors
Miguel M. Ugeda Fernando De Juan B. Andrei Bernevig
This is my paper
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.str-el
keywords 1H-NbSe2obstructedatomicbandcompactWannierfunctionquantumgeometrytransitionmetaldichalcogenidesflatperturbationtheorytight-bindingmodel
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 argues that the Fermi-level band of monolayer 1H-NbSe2, and of five related 1H transition-metal dichalcogenides, is not an ordinary atomic band but an obstructed atomic band: its Wannier charge center sits at the empty 1c site rather than on any atom. In the three-band model obtained by Wannierizing the DFT bands, this obstructed Wannier function is almost exactly the most compact three-site orbital allowed by symmetry, with squared overlap 0.9379 for NbSe2 and above 0.90 for all six compounds. Projecting to a single band then produces a notable hopping hierarchy: the next-nearest-neighbor hopping (0.0955 eV) is much larger than the nearest-neighbor hopping (0.0178 eV), because the onsite and nearest-neighbor contributions cancel. For NbSe2 the paper also develops a two-energy perturbation theory, built on a simplified six-band Hamiltonian, that reproduces the top three DFT bands and gives a perturbative wavefunction with reported 0.98 overlap with the ab initio obstructed Wannier function. The point of these results is that the obstructed, compact Wannier structure is a concrete handle on the quantum geometry that may feed into the charge-density-wave and superconducting orders in this material family.

What carries the argument

The load-bearing object is the maximally compact obstructed Wannier function $w^\dagger_{\mathrm{compact},R}$, a three-site orbital defined in the $sp^2$-like rotated basis $(\tilde{c}^\dagger_{R,1},\tilde{c}^\dagger_{R,2},\tilde{c}^\dagger_{R,3})$ obtained by rotating the Nb $d_{z^2}$, $d_{xy}$, $d_{x^2-y^2}$ orbitals. Because the rotation makes the three-fold rotation act by permutations, the local three-site cluster decouples from its translations, and its symmetric eigenvector $(1,1,1)/\sqrt{3}$ is exactly the $A_1$ Wannier state at the empty 1c site. This object carries the argument: its nearly perfect overlap with the numerical Wannier function converts "obstructed atomic" from a symmetry label into an explicit, short-ranged wavefunction. The second mechanism is the new two-energy perturbation theory, which replaces the resolvent $1/(E-H_1)$ by $a+bE$ chosen to be exact at the two flat-band energies $E'_0=0$ and $E_1=2.417$ eV, turning a nonlinear eigenvalue problem into a linear effective Hamiltonian with a metric factor $1-SbS^\dagger$; this is what makes the reduction from six bands to three bands tractable even though the conventional gap and bandwidth conditions fail.

What would settle it

Recompute the perturbative wavefunction from the full six-band DFT Hamiltonian with the selenium in-plane hopping kept at its realistic value and with the upper bands not flattened, then check the squared overlap with the ab initio Fermi-level Wannier function; if it falls well below the reported 0.98, the success of the perturbation theory depends on the removed terms.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the isolated $m_z$-even band crossing the Fermi level of monolayer 1H-NbSe2 realizes the obstructed atomic band representation A1@1c, and that its Wannier function is nearly identical to the strictly compact three-site orbital $w^\dagger_{\mathrm{compact},R}=\frac{1}{\sqrt{3}}(e^\dagger_{R+a_1+a_2,1}+e^\dagger_{R,2}+e^\dagger_{R+a_1,3})$ written in the $sp^2$-rotated basis of the three-band model; the squared overlap is 0.9379, with root-mean-square overlap 0.970. The same compact orbital describes the Fermi-level band with more than 90% accuracy in MoS2, NbS2, TaS2, TaSe2 and WS2. In the one-band projection, the nearest-neighbor hopping $t_w(a_1)=0.0178$ eV is much smaller than the next-nearest-neighbor hopping $t_w(2a_1+a_2)=0.0955$ eV, and the paper traces this to a near cancellation between three-band onsite and nearest-neighbor terms in the compact Wannier basis. For NbSe2 the paper further claims a new perturbation theory: starting from a simplified six-band Hamiltonian in which the selenium sites have only onsite terms, it approximates the upper three bands by flat bands at 0 and 2.417 eV and obtains an effective three-band Hamiltonian whose bands match the simplified six-band model and whose Bloch state overlaps the DFT obstructed Wannier function with reported probability 0.980.

Load-bearing premise

The whole perturbative construction assumes that a simplified six-band Hamiltonian, with most selenium-to-selenium hopping removed and the upper three bands idealized as exactly flat at 0 and 2.417 eV, still represents the true hybridization of the DFT bands; if it does not, the reported 95-98% wavefunction overlaps are comparisons to the simplified model rather than to NbSe2.

Editorial extensions

If this is right

  • For all six compounds studied, the isolated Fermi-level band is obstructed atomic (A1@1c), so quantum-geometric effects should be included in any model of their correlated orders.
  • The one-band NNN model with the reported hoppings reproduces the DFT Fermi-level band, so a single-orbital model on the obstructed Wannier lattice captures the flat-band physics.
  • The NN/NNN hopping hierarchy follows from a cancellation between onsite and nearest-neighbor terms in the compact basis, making it a generic feature of this material family rather than a fitting accident.
  • The new perturbation theory yields an effective three-band Hamiltonian and an analytic wavefunction for the obstructed band, enabling parameter-free model Hamiltonians for correlated studies.
  • If the compact Wannier picture is correct, the obstructed Wannier charge center at the empty 1c site should be visible in real-space imaging, as the simulated STM charge density in the paper shows.

Reading between the lines

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

  • One consequence the paper leaves implicit is that the ~0.94 compact-orbital overlap means the single-band model's quantum geometry is largely pinned by symmetry, so modest changes in hopping parameters should not drastically change the band's geometric properties.
  • A testable extension is to build an interacting Hubbard model on the triangular lattice of 1c Wannier centers using the compact orbital and compare its superfluid weight or charge-density-wave susceptibility with a full DFT-based calculation.
  • The two-energy flat-band perturbation scheme should generalize to other materials with a quasi-flat Fermi-level band and a weakly dispersive upper manifold, even where the conventional condition that the gap exceed the hybridization is violated.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper revisits the low-energy electronic structure of monolayer 1H-NbSe2 and five related TMDs. From Wannier90-derived tight-binding models (6-band, 3-band, and 1-band), the authors identify the isolated mz-even Fermi-level band as an obstructed atomic band representation A1@1c. In the 3-band model, a three-site compact Wannier state w_compact,R = (e_{R+a1+a2,1}+e_{R,2}+e_{R+a1,3})/sqrt(3) reproduces the Wannier90 state with squared overlap 0.9379 for NbSe2 and above 0.90 for the other compounds. The 1-band model shows NNN hopping (0.0955 eV) much larger than NN hopping (0.0178 eV), traced to cancellation between onsite and NN contributions in the 3-band model. The paper then develops a 'new perturbation theory' that replaces (E-HSe)^{-1} by a+bE and applies it to a simplified Se-onsite NNN 6-band model with z=0, reporting effective 3-band dispersions and wavefunction overlaps with the DFT band of 0.980 and 0.977 in the main text, or 0.9638 and 0.9538 in the Appendix.

Significance. The compact Wannier and hopping-cancellation results are a solid contribution if they hold: they are parameter-free in the sense of direct Wannier90 output, the compact ansatz is fixed by symmetry and the dominant hoppings rather than fitted, and the BZ-averaged overlap is computed independently of the Wannier90 state. The NNN>NN explanation via explicit decomposition into onsite, NN, and NNN contributions (Eqs. S2.53-S2.58, Tables S1-S2) is crisp and falsifiable. The new perturbation theory is potentially interesting but currently under-validated: it is built on an ad hoc z=0 Hamiltonian and flat-band inputs, and its wavefunction overlaps are not benchmarked against the exact eigenstates of that same simplified model. The paper would be strengthened considerably by adding that baseline and by quantifying the z and flat-band errors.

major comments (3)
  1. [Sec. IV.B, Eq. (41); App. C.1, Eq. (S3.18)] The wavefunction overlaps reported in Eq. (41) (0.980) and Eq. (53) (0.977), and in Eqs. (S3.18), (S3.41), (S3.49), are quoted against the DFT-precise Bloch state w_k for the Fermi-level band. However, no baseline is given: the exact Fermi-level eigenstate of the simplified Se-onsite NNN model H6,Se-Onsite,NNN in Eq. (S3.9) is never compared with w_k. Because H6,Se-Onsite,NNN was chosen precisely because it 'maintains the shape' of the relevant band (Sec. IV.B), a high overlap of its exact eigenstate with DFT may already be present, in which case the reported accuracy is not attributable to the new perturbation method. If, instead, the exact eigenstate of the simplified model has low overlap with DFT, then the flat-band replacement appears to do the work and the method is not validated. The authors should report the BZ-averaged overlap between the exact Fermi-level eigenstate of H6,Se-Onsite,NNN and the DFT-precise state, and preferably the same overlap for a conventional energy-independent downfolding of the same 6-band model.
  2. [Sec. IV.B, Eqs. (30)-(31)] The replacement (E - HSe(k))^{-1} -> a + bE is exact at the two energies E=0 and E=E1, but the actual Fermi-level band of H6,Se-Onsite,NNN spans -0.3555 to 0.8016 eV and the upper two bands span 1.997 to 2.937 eV (Sec. IV.B). No bound or estimate is given for the interpolation error over these intervals, and the validation in Fig. 2(a) is a visual comparison of the effective-model bands with the bands of the same simplified model used to read off E0'=0 and E1=2.417 eV. To make the claim that the effective 3-band model captures the three DFT bands quantitative, the paper should report a mean absolute error or similar metric for the effective-model bands against both H6,Se-Onsite,NNN and the full DFT bands, and the BZ-averaged wavefunction overlap of the effective-model eigenstates with the exact eigenstates of H6,Se-Onsite,NNN.
  3. [Eqs. (S3.7)-(S3.9)] The Se-onsite NNN model is obtained by setting to zero all Se-Se a1 hoppings except the px-px term and then multiplying that term by a factor z which is set to 0, although the realistic Wannier90 value is z=1. The paper motivates z=0 by the band-shape insensitivity shown in Fig. S11, but all wavefunction overlaps in Eqs. (41), (53), (S3.18), (S3.41), and (S3.49) are computed at z=0, and no wavefunction overlap is reported for z>0. Since the abstract and Sec. IV claim these overlaps with the ab initio (DFT) state, the representativeness of z=0 should be quantified: report the exact and perturbative overlaps for z=0.2, 0.4, and 1, or equivalently the overlap between the z=0 and z=1 Fermi-level eigenstates of the simplified model.
minor comments (4)
  1. [Eq. (41) vs Eq. (S3.18); Eq. (53) vs Eq. (S3.41)] The main text reports sqrt(1/N sum |...|^2) values (0.980 and 0.977) while the Appendix reports 1/N sum |...|^2 values (0.9638 and 0.9538) for the same quantities. The text labels both as 'probability overlap'; please use one definition throughout, or explicitly state that the main-text numbers are RMS overlaps.
  2. [Sec. III.B, Eq. (19) and Eq. (S2.50)] The sentence says the probability overlap is 0.94 and then gives an expression equal to 0.970. Please clarify that 0.94 is the squared overlap of Eq. (S2.50) and 0.970 is its square root.
  3. [Tables S1-S2] The estimated tw(a1) values in Table S2 differ from the Wannier90 values in Table S1 by up to roughly 0.04 eV for MoS2 and WS2. The text's wording that the estimates capture the qualitative difference is accurate, but a sentence in the caption acknowledging the quantitative discrepancy would prevent readers from expecting closer agreement.
  4. [Sec. V] The word 'Qantum' in the first sentence of the Discussion should be 'Quantum'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: compact Wannier overlap and perturbation wavefunctions are computed against external DFT/Wannier90 benchmarks; the NNN>NN explanation is a cross-model consistency check, not a fitted prediction.

full rationale

The paper's central derivations are self-contained against external DFT data. The compact Wannier state w†_compact (Eq. 16) is fixed by the p3m1 EBR A1@1c and the dominant local 3×3 block M (Eqs. 13–14); its squared overlap 0.9379 with the Wannier90 1-band Wannier state (Eq. S2.50) is a computed quantity, not an imposed fit. The NNN>NN single-band hoppings are obtained independently from Wannier90 (Eq. 23), and the 3-band-based formulas (Eqs. S2.53–S2.58) provide an approximate accounting of those hoppings with the same qualitative ordering; this is a post-hoc explanation of one model by another derived from the same DFT, not a prediction forced by construction. The new perturbation theory (Sec. IV) uses a simplified Se-onsite NNN 6-band Hamiltonian chosen for invertibility of E−HSe (Eq. S3.9); the effective 3-band model (Eq. 39) is validated against that simplified Hamiltonian, and the wavefunction overlap with the DFT-precise Bloch state (Eqs. 41, S3.41) is an external benchmark. A missing baseline—the overlap of the exact Fermi-level eigenstate of H6,Se-Onsite,NNN with the DFT state—leaves the attribution of the 0.980 overlap underdetermined, but this is a correctness/interpretation concern, not a circularity: no equation in the chain defines its target in terms of itself, and no fitted parameter is renamed as a prediction. No load-bearing self-citation or imported uniqueness theorem appears; references to prior work (e.g., [90], [97], [123]) are standard attributions, not the argument's basis.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central compact Wannier claim rests on the standard Wannier90/DFT pipeline and symmetry analysis, which are domain assumptions rather than free fits. The perturbation theory sections introduce several hand-set parameters (z, E0', E1, and the simplification parameters) that are needed for the analytic wavefunctions and effective models. No new physical entities are postulated.

free parameters (4)
  • z scaling factor for Se-Se a1 px-px hopping = 0 (set to 0 in H6,Se-Onsite,NNN)
    The simplified 6-band model Eq. (S3.7)-(S3.9) includes a scaling factor z for the Se-Se a1 px-px hopping; z=0 is chosen to make Ek-HSe(k) invertible for the Fermi-level band, while the realistic value is 1.
  • Flat-band energies E0' and E1 = 0 eV and 2.417 eV
    The new perturbation theory replaces the energy denominator by a+bE fixed at E0'=0 and E1=2.417 eV, the mean energies of the lower and upper groups of the simplified 6-band model (Sec. IV.B). These are inputs, not derived.
  • Simplification 1 parameters tA, tB, Ez, Ex = tA=-0.6300 eV, tB=0.7700, Ez=-12/5, Ex=-8/5
    Closed-form wavefunctions use approximated hopping matrices with about 10% deviation from Wannier90 values (Eq. S3.43); these are hand-chosen to yield compact analytical expressions.
  • Simplification 2 parameters ESe, tSeNb, ENb = ESe=-1.8755 eV, tSeNb=-2ESe/3, ENb=0.5313 eV
    Further simplified model with about 30% deviation from Wannier90 values (Eq. S3.50-S3.51); used for analytical wavefunctions with 88-93% overlap.
assumptions (6)
  • domain assumption DFT-PBE band structure of monolayer 1H-NbSe2 and related TMDs is an accurate reference.
    All models and wavefunction overlaps are validated against DFT bands; if the DFT-PBE electronic structure is wrong, the conclusions change.
  • domain assumption Wannier90 disentanglement and maximal localization yield trustworthy Wannier functions and tight-binding parameters.
    The 6-band, 3-band, and 1-band models are constructed from Wannier90 output with no independent check of gauge dependence or disentanglement ambiguity.
  • domain assumption The mz-even and mz-odd sectors decouple in the monolayer, so the 6 mz-even bands can be treated alone.
    Used throughout from Sec. II; justified by symmetry and the roughly 1 eV gap to the closest mz-odd band.
  • ad hoc to paper The Se-onsite NNN 6-band model with z=0 remains representative of the DFT Fermi-level band.
    The simplification drops most Se-Se a1 hoppings to make Ek-HSe(k) invertible; the paper shows the band shape is preserved but gives no rigorous error bound.
  • ad hoc to paper The upper three bands can be approximated by flat bands at 0 eV and 2.417 eV for the perturbation.
    The new perturbation theory in Sec. IV.A assumes this two-energy flat-band structure; the dispersion around these energies is treated as a perturbation.
  • ad hoc to paper Lambda_k has non-negative diagonal elements for the simplified model.
    Required to define sqrt(Lambda) in Eq. (39); verified numerically for the Se-onsite NNN model but not proven in general.

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Cite this review

Pith. "Pith review of Quantum Geometry in the NbSe$_2$ Family I: Obstructed Compact Wannier Function and New Perturbation Theory." pith.science (2026). https://pith.science/paper/VSG6KH7Z

@misc{pith2026250702047,
  author       = {Pith},
  title        = {Pith review of: Quantum Geometry in the NbSe$_2$ Family I: Obstructed Compact Wannier Function and New Perturbation Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VSG6KH7Z}},
  note         = {Machine review of arXiv:2507.02047}
}
abstract

We revisit the electronic structure and band topology of monolayer 1H-NbSe$_2$, which hosts both superconductivity and charge density wave, and its related compounds 1H-MoS$_2$, NbS$_2$, TaS$_2$, TaSe$_2$ and WS$_2$. We construct a 6-band, a 3-band, and - simplest of all - a single-band model for this material family, by directly Wannierizing the ab initio bands. All host obstructed atomic isolated bands away from the atomic positions near the Fermi energy. We find that in the 3-band model, the obstructed atomic Wannier function can be well approximated by an optimally compact Wannier function with more than 90% accuracy for all the compounds, rising to a remarkable 94% accuracy in NbSe$_2$. Interestingly, the simplest single-band model has next nearest-neighboring hopping larger than the nearest-neighboring hopping (by nearly an order of magnitude for MoS$_2$, NbSe$_2$, TaSe$_2$ and WS$_2$), which comes from the cancellation between the atomic onsite terms and the atomic nearest-neighboring hopping after projecting to the obstructed atomic Wannier functions. Furthermore for NbSe$_2$, we employ a novel approximation scheme to obtain an effective Hamiltonian that captures the 3 bands originating mainly from the Nb atom. We also use conventional perturbation theory to derive the ab initio obstructed Wannier function with 95% accuracy. Our results pave the way for future study of the effect of quantum geometry on the correlated phases in this family of materials.

Figures

Figures reproduced from arXiv: 2507.02047 by the authors.

Figure 1
Figure 1. FIG. 1. (a)NbSe [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The band structures of DFT are plotted in black. (a) The band structures of the 6-band Se-onsite NNN Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Reference graph

Works this paper leans on

150 extracted references · 47 canonical work pages · cited by 1 Pith paper

  1. [1]

    effective

    New Perturbation Theory for the 6-Band Model and Effective 3-Band Model We first discuss the perturbation theory approach for the 6-band model. To do so, we separate the 3 Nb orbitals and the 3 Se orbitals explicitly, and rewrite 6-band model in the k-space Hamiltonian: H6band = X k (c† Nb,kc† Se,k) HNb(k) S(k) S†(k) HSe(k) ! cNb,k cSe,k ! (S3.1) where ou...

  2. [2]

    C 1, we used the new perturbation theory to derive an effective 3-band model that captures the dispersion of the top three bands

    Perturbative Analysis for the Obstructed Atomic Band In Appendix. C 1, we used the new perturbation theory to derive an effective 3-band model that captures the dispersion of the top three bands. In this section, we will provide a perturbative understanding of the obstructed 30 atomic band at the Fermi energy. Unless specified otherwise, we will still use...

  3. [3]

    482 cos kxa 2 cos √ 3kya 2 ! +25ei √ 3kya(cos(kxa) − 2) + 432i cos kxa 2 sin √ 3kya 2 ! − 50 cos(kxa) + 241 # ψNb,2(k) = 1 108 √ 6 ie− 1 2 (ikxa) sin kxa 2

    0 0 0 0 0 0 −5/4 0   , (S3.44) 34 Γ M K Γ -4 -2 0 2 4 Γ M K Γ -4 -2 0 2 4 Γ M K Γ -4 -2 0 2 4 FIG. S13. The band structures of DFT are plotted in black. The band structure given by Eq. (S3.38) and Eq. (S3.42) for H6,Se-onste,NNN in Eq. (S3.9) is plotted as orange dashed and solid lines, respectively, in (a). The band structure given by Eq. (S3.46) and ...

  4. [4]

    renormalized

    Lower 3-Band Model From the full DFT 6-band model, we can also numerically build the Wannier states for the lowest three bands from Wannier90, i.e., the lower 3 bands in Fig. 1(b). We add this model just for completeness, and the discussion of this part is analogous to that of Appendix. B 2. The trial states are chosen to be three Se p orbitals in Eq. (S2...

  5. [5]

    Other Materials The obstructed atomic band not only occurs in NbSe 2, but also exists in other 2D TMD materials. In Fig. S9, we plot the band structure for eight TMD materials. Among them, 1H-MoS 2, NbS 2, NbSe 2, TaS 2, TaSe 2, and WS2 have one isolated mz-even band near or at the Fermi energy is obstructed atomic—A 1@1c, and their Wannier functions can ...

  6. [6]

    Proposition: Suppose we have a 6 × 6 Hamiltonian H with H = P n=1,2,3 |uA,n⟩⟨uA,n|EA,n +P n=1,2,3 |uB,n⟩⟨uB,n|EB,n with EA,3 > EA,2 > EA,1 > EB,3 > EB,2 > EB,1

    A Proposition We now prove a spectral decomposition that we have used in this paper. Proposition: Suppose we have a 6 × 6 Hamiltonian H with H = P n=1,2,3 |uA,n⟩⟨uA,n|EA,n +P n=1,2,3 |uB,n⟩⟨uB,n|EB,n with EA,3 > EA,2 > EA,1 > EB,3 > EB,2 > EB,1. Suppose we have two rank-3 pro- jectors P1 = P a=1,2,3 |u1,a⟩⟨u1,a| and P2 = P a=1,2,3 |u2,a⟩⟨u2,a| such that (...

  7. [7]

    (S2.37) and Eq

    Conditions for a Compact W annier Basis We note that the basis of the 3-band model Eq. (S2.37) and Eq. (S2.67) is both very localized (nearest neighbor Wannier) as well as containing much fewer elements than the full symmetry allows. We now find the conditions of the 6-band model such that the 3-band models of Nb and Se have a orthonormal compact basis. (...

  8. [8]

    With errors of about only 0.2%, the values of the NN Nb-Se hoppings in Eq

    A Simple F orm of the 6-Band Model Hopping Matrices For an interesting fact, we present a simple form of the 6-Band Model hopping matrice. With errors of about only 0.2%, the values of the NN Nb-Se hoppings in Eq. (S2.16) are tτ Seτ Nb (τ Se) =   tN N,z,dz2 0 tN N,z,dx2 −y2 0 tN N,x,dxy 0 tN N,y,dz2 0 tN N,y,dx2 −y2   =   0.77754 0 −0.973852 0 −1...

Show all 150 references
  1. [9]

    An Alternative W ay to Understand the Approximated W annier F unctions of the 3-band Model If we rotate the Nb basis with R in Eq. (S2.35), i.e., c† R+τ Nb,1 c† R+τ Nb,2 c† R+τ Nb,3 = c† R+τ Nb,dz2 c† R+τ Nb,dxy c† R+τ Nb,dx2 −y2 R , (S3.80) then the NN hopping between Nb and ...

  2. [10]

    −0.62203 −0.62203   (S3.81) where we choose tSeNb = 1.244eV . (S3.82) With this observation, we can build a simple NN 6-band model, which only contains the onsite Nb energy, the onsite Se energy, and one NN hopping term: R†tτ Nbτ Nb (0)R = ENb13×3 tτ Seτ Se (0) = ESe13×3 tτ...

  3. [11]

    (S3.83), are cR+τ Se,z, cR−a1−a2+τ Se,y and cR−a2+τ Se,y

    To the first order in tSeNb/(ENb − ESe), we only include Se orbitals that directly hop to c† R+τ Nb,1, which, according to Eq. (S3.83), are cR+τ Se,z, cR−a1−a2+τ Se,y and cR−a2+τ Se,y. So to the first order of tSeNb/(ENb − ESe), we have a 4 × 4 model in the basis of c† R+τ Nb,...

  4. [12]

    Tang, J.-W

    E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum hall states, Phys. Rev. Lett. 106, 236802 (2011)

    2011

  5. [13]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011)

    2011

  6. [14]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010)

    2010

  7. [15]

    Provost and G

    J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76, 289 (1980)

    1980

  8. [16]

    Fubini, Sulle metriche definite da una forma hermitiana, Atti del Reale istituto Veneto di Scienze, Lettere ed Arti 63, 502 (1904)

    G. Fubini, Sulle metriche definite da una forma hermitiana, Atti del Reale istituto Veneto di Scienze, Lettere ed Arti 63, 502 (1904)

    1904

  9. [17]

    Study, K¨ urzeste Wege im komplexen Gebiet, Mathematische Annalen60, 321 (1905)

    E. Study, K¨ urzeste Wege im komplexen Gebiet, Mathematische Annalen60, 321 (1905)

    1905

  10. [18]

    Souza, T

    I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Physical Review B 62, 1666 (1999), cond-mat/9911007

    1999 arXiv

  11. [19]

    J. Yu, B. A. Bernevig, R. Queiroz, E. Rossi, P. T¨ orm¨ a, and B.-J. Yang, To appear

  12. [20]

    Neupert, L

    T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011)

    2011

  13. [21]

    D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum Hall effect in the absence of Landau levels, Nature Communications 2, 389 (2011), arXiv:1102.2658 [cond-mat.str-el]

    2011 arXiv

  14. [22]

    Regnault and B

    N. Regnault and B. A. Bernevig, Fractional chern insulator, Phys. Rev. X 1, 021014 (2011)

    2011

  15. [23]

    K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011)

    2011

  16. [25]

    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 Letters 127, 246403 (2021)

    2021

  17. [26]

    P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional chern insulators, Physical Review B 108, 205144 (2023)

    2023

  18. [27]

    Estienne, N

    B. Estienne, N. Regnault, and V. Cr´ epel, Ideal chern bands as landau levels in curved space, Physical Review Research 5, L032048 (2023)

    2023

  19. [28]

    Z. Liu, B. Mera, M. Fujimoto, T. Ozawa, and J. Wang, Theory of generalized landau levels and implication for non-abelian states (2024), 2405.14479 [cond-mat.mes-hall]

    2024 arXiv

  20. [29]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous hall states in twisted mote2, Nature 10.1038/s41586-023-06289-w (2023)

    2023 doi

  21. [30]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional chern insulator in moir´ e mote2, Nature 10.1038/s41586-023-06452-3 (2023)

    2023 doi

  22. [31]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, et al. , Observation of fractionally quantized anomalous hall effect, arXiv preprint arXiv:2308.02657 (2023)

    2023 arXiv

  23. [32]

    (See the expressions of tτ Nbτ Nbeff(0) and tτ Nbτ Nbeff(0) in Eq. (S3.23).) As we can see, Hef f,0(k) can be solved exactly, with the eigenstates created by (c† k,0, c† k,+, c† k,−) = (c† N b,k,dz2 , c† N b,k,dxy , c† N b,k,dx2 −y2 )RUw(k) , (S3.32) where Uw(k) =   e−ik·(a...

  24. [33]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Observation of integer and fractional quantum anomalous Hall states in twisted bilayer MoTe2, arXiv e-prints , arXiv:2308.06177 (2023)...

    2023 arXiv

  25. [34]

    Z. Ji, H. Park, M. E. Barber, C. Hu, K. Watanabe, T. Taniguchi, J.-H. Chu, X. Xu, and Z.-X. Shen, Local probe of bulk and edge states in a fractional chern insulator, Nature 635, 578 (2024)

    2024

  26. [35]

    Redekop, C

    E. Redekop, C. Zhang, H. Park, J. Cai, E. Anderson, O. Sheekey, T. Arp, G. Babikyan, S. Salters, K. Watanabe, T. Taniguchi, X. Xu, and A. F. Young, Direct magnetic imaging of fractional chern insulators in twisted mote 2 with a superconducting sensor (2024), arXiv:2405.10269 [...

    2024 arXiv

  27. [36]

    K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin hall effect in moir´ e mote2, Nature628, 522 (2024)

    2024

  28. [37]

    H. Park, J. Cai, E. Anderson, X.-W. Zhang, X. Liu, W. Holtzmann, W. Li, C. Wang, C. Hu, Y. Zhao, T. Taniguchi, K. Watanabe, J. Yang, D. Cobden, J.-H. Chu, N. Regnault, B. A. Bernevig, L. Fu, T. Cao, D. Xiao, and X. Xu, Ferromagnetism and topology of the higher flat band in a f...

    2024 arXiv

  29. [38]

    H. Park, W. Li, C. Hu, C. Beach, M. Gon¸ calves, J. F. Mendez-Valderrama, J. Herzog-Arbeitman, T. Taniguchi, K. Watan- abe, D. Cobden, L. Fu, B. A. Bernevig, N. Regnault, J.-H. Chu, D. Xiao, and X. Xu, Observation of high-temperature dissipationless fractional chern insulator ...

    2025 arXiv

  30. [39]

    F. Xu, Z. Sun, J. Li, C. Zheng, C. Xu, J. Gao, T. Jia, K. Watanabe, T. Taniguchi, B. Tong, L. Lu, J. Jia, Z. Shi, S. Jiang, Y. Zhang, Y. Zhang, S. Lei, X. Liu, and T. Li, Signatures of unconventional superconductivity near reentrant and fractional quantum anomalous hall insula...

    2025 arXiv

  31. [40]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous hall effect in multilayer graphene, Nature 626, 759 (2024)

    2024

  32. [41]

    J. Xie, Z. Huo, X. Lu, Z. Feng, Z. Zhang, W. Wang, Q. Yang, K. Watanabe, T. Taniguchi, K. Liu, Z. Song, X. C. Xie, J. Liu, and X. Lu, Tunable fractional chern insulators in rhombohedral graphene superlattices (2025), arXiv:2405.16944 [cond-mat.mes-hall]

    2025 arXiv

  33. [42]

    Y. Choi, Y. Choi, M. Valentini, C. L. Patterson, L. F. W. Holleis, O. I. Sheekey, H. Stoyanov, X. Cheng, T. Taniguchi, K. Watanabe, and A. F. Young, Electric field control of superconductivity and quantized anomalous hall effects in rhombohedral tetralayer graphene (2024), arX...

    2024 arXiv

  34. [43]

    Z. Lu, T. Han, Y. Yao, J. Yang, J. Seo, L. Shi, S. Ye, K. Watanabe, T. Taniguchi, and L. Ju, Extended quantum anomalous hall states in graphene/hbn moir´ e superlattices (2024), arXiv:2408.10203 [cond-mat.mes-hall]

    2024 arXiv

  35. [44]

    Xiao, G.-B

    D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled Spin and Valley Physics in Monolayers of ${\mathrm{MoS}} {2}$ and Other Group-VI Dichalcogenides, Physical Review Letters 108, 196802 (2012)

    2012

  36. [45]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDonald, Topological Insulators in Twisted Transition Metal Dichalco- genide Homobilayers, Physical Review Letters 122, 086402 (2019)

    2019

  37. [46]

    H. Pan, F. Wu, and S. Das Sarma, Band topology, Hubbard model, Heisenberg model, and Dzyaloshinskii-Moriya inter- action in twisted bilayer ${\mathrm{WSe}} {2}$, Physical Review Research 2, 033087 (2020)

    2020

  38. [47]

    H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Spontaneous fractional chern insulators in transition metal dichalcogenide moir´ e superlattices, Physical Review Research 3, L032070 (2021)

    2021

  39. [48]

    Zhang, T

    Y. Zhang, T. Liu, and L. Fu, Electronic structures, charge transfer, and charge order in twisted transition metal dichalco- genide bilayers, Physical Review B 103, 155142 (2021)

    2021

  40. [49]

    Devakul, V

    T. Devakul, V. Cr´ epel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature Commu- nications 12, 6730 (2021)

    2021

  41. [50]

    T. Wang, T. Devakul, M. P. Zaletel, and L. Fu, Topological magnets and magnons in twisted bilayer MoTe $ 2$ and WSe$ 2$ (2023)

    2023

  42. [51]

    A. P. Reddy, F. Alsallom, Y. Zhang, T. Devakul, and L. Fu, Fractional quantum anomalous Hall states in twisted bilayer ${\mathrm{MoTe}} {2}$ and ${\mathrm{WSe}} {2}$, Physical Review B 108, 085117 (2023)

    2023

  43. [52]

    J. Dong, J. Wang, P. J. Ledwith, A. Vishwanath, and D. E. Parker, Composite Fermi Liquid at Zero Magnetic Field in Twisted ${\mathrm{MoTe}} {2}$, Physical Review Letters 131, 136502 (2023)

    2023

  44. [53]

    W.-X. Qiu, B. Li, X.-J. Luo, and F. Wu, Interaction-Driven Topological Phase Diagram of Twisted Bilayer ${\mathrm{MoTe}} {2}$, Physical Review X 13, 041026 (2023)

    2023

  45. [54]

    T. Wang, M. Wang, W. Kim, S. G. Louie, L. Fu, and M. P. Zaletel, Topology, magnetism and charge order in twisted MoTe2 at higher integer hole fillings (2023)

    2023

  46. [55]

    A. P. Reddy and L. Fu, Toward a global phase diagram of the fractional quantum anomalous Hall effect, Physical Review B 108, 245159 (2023)

    2023

  47. [56]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Fractional Chern Insulator in Twisted Bilayer ${\mathrm{MoTe}} {2}$, Physical Review Letters 132, 036501 (2024)

    2024

  48. [57]

    J. Yu, J. Herzog-Arbeitman, M. Wang, O. Vafek, B. A. Bernevig, and N. Regnault, Fractional Chern insulators versus nonmagnetic states in twisted bilayer ${\mathrm{MoTe}} {2}$, Physical Review B 109, 045147 (2024)

    2024

  49. [58]

    C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, Maximally localized Wannier functions, interaction models, and fractional quan- tum anomalous Hall effect in twisted bilayer MoTe 2, Proceedings of the National Academy of Sciences 121, e2316749121 (2024)

    2024

  50. [59]

    Abouelkomsan, A

    A. Abouelkomsan, A. P. Reddy, L. Fu, and E. J. Bergholtz, Band mixing in the quantum anomalous Hall regime of twisted semiconductor bilayers, Physical Review B 109, L121107 (2024)

    2024

  51. [60]

    Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moir \’e frac- tional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer${\mathrm{MoTe}} {2}$, Physical Review B 109, 205121 (2024)

    2024

  52. [61]

    Zhang, C

    X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted MoTe2 and WSe2, Nature Communications 15, 4223 (2024)

    2024

  53. [62]

    A.-K. Wu, S. Sarkar, X. Wan, K. Sun, and S.-Z. Lin, Quantum-metric-induced quantum hall conductance inversion and reentrant transition in fractional chern insulators, Phys. Rev. Res. 6, L032063 (2024)

    2024

  54. [63]

    H. Li, Y. Su, Y. B. Kim, H.-Y. Kee, K. Sun, and S.-Z. Lin, Contrasting twisted bilayer graphene and transition metal dichalcogenides for fractional chern insulators: An emergent gauge picture, Phys. Rev. B 109, 245131 (2024)

    2024

  55. [64]

    Y. Park, Y. Kim, B. L. Chittari, and J. Jung, Topological flat bands in rhombohedral tetralayer and multilayer graphene on hexagonal boron nitride moir´ e superlattices, Physical Review B108, 155406 (2023). 45

    2023

  56. [65]

    Herzog-Arbeitman, Y

    J. Herzog-Arbeitman, Y. Wang, J. Liu, P. M. Tam, Z. Qi, Y. Jia, D. K. Efetov, O. Vafek, N. Regnault, H. Weng, Q. Wu, B. A. Bernevig, and J. Yu, Moir´ e fractional chern insulators. ii. first-principles calculations and continuum models of rhombohedral graphene superlattices, P...

    2024

  57. [66]

    Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov, N. Regnault, and B. A. Bernevig, Moir´ e, arXiv https://doi.org/10.48550/arXiv.2312.11617 (2023), 2312.11617

    2023 doi

  58. [67]

    J. Yu, J. Herzog-Arbeitman, Y. H. Kwan, N. Regnault, and B. A. Bernevig, Moir´ e, arXiv https://doi.org/10.48550/arXiv.2407.13770 (2024), 2407.13770

    2024 doi

  59. [68]

    Z. Guo, X. Lu, B. Xie, and J. Liu, Fractional chern insulator states in multilayer graphene moir´ e superlattices, Physical Review B 110, 075109 (2024)

    2024

  60. [69]

    B. Zhou, H. Yang, and Y.-H. Zhang, Fractional quantum anomalous hall effects in rhombohedral multilayer graphene in the moir´ e, arXiv https://doi.org/10.48550/arXiv.2311.04217 (2024), 2311.04217

    2024 doi arXiv

  61. [70]

    J. Dong, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohedral multilayer graphene i: Interaction-driven chern bands and fractional quantum hall states at zero magnetic field, arXiv https://doi.org/10.48550/arXiv.2311.0556...

    2024 doi arXiv

  62. [71]

    Soejima, J

    T. Soejima, J. Dong, T. Wang, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohe- dral multilayer graphene ii: General mechanism and a minimal model, arXiv https://doi.org/10.48550/arXiv.2403.05522 (2024), 2403.05522

    2024 doi

  63. [72]

    Huang, S

    K. Huang, S. D. Sarma, and X. Li, Impurity-induced thermal crossover in fractional chern insulators, arXiv https://doi.org/10.48550/arXiv.2409.04349 (2024), 2409.04349

    2024 doi arXiv

  64. [73]

    T. Tan, J. May-Mann, and T. Devakul, Wavefunction approach to the fractional anomalous hall crystal, arXiv https://doi.org/10.48550/arXiv.2409.06775 (2024), 2409.06775

    2024 doi arXiv

  65. [74]

    Z. Dong, A. S. Patri, and T. Senthil, Theory of quantum anomalous hall phases in pentalayer rhombohedral graphene moir´ e, arXiv https://doi.org/10.48550/arXiv.2311.03445 (2024), 2311.03445

    2024 doi arXiv

  66. [75]

    Huang, X

    K. Huang, X. Li, S. D. Sarma, and F. Zhang, Self-consistent theory of fractional quantum anomalous hall states in rhombohedral graphene, Physical Review B 110, 115146 (2024)

    2024

  67. [76]

    S. D. Sarma and M. Xie, Thermal crossover from a chern insulator to a fractional chern insulator in pentalayer graphene, Physical Review B 110, 155148 (2024)

    2024

  68. [77]

    Xie and S

    M. Xie and S. D. Sarma, Integer and fractional quantum anomalous hall effects in pentalayer graphene, Physical Review B 109, L241115 (2024)

    2024

  69. [78]

    Z. Dong, A. S. Patri, and T. Senthil, Stability of anomalous hall crystals in multilayer rhombohedral graphene, arXiv https://doi.org/10.48550/arXiv.2403.07873 (2024), 2403.07873

    2024 doi arXiv

  70. [79]

    K. Kudo, R. Nakai, and K. Nomura, Quantum anomalous, quantum spin, and quantum valley hall effects in pentalayer rhombohedral graphene, arXiv https://doi.org/10.48550/arXiv.2406.14354 (2024), 2406.14354

    2024 doi arXiv

  71. [80]

    Zhou and Y.-H

    B. Zhou and Y.-H. Zhang, New classes of quantum anomalous hall crystals in multilayer graphene, arXiv https://doi.org/10.48550/arXiv.2411.04174 (2024), 2411.04174

    2024 doi arXiv

  72. [81]

    Souza, T

    I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000)

    2000

  73. [82]

    Resta, Polarization Fluctuations in Insulators and Metals: New and Old Theories Merge, Physical Review Letters 96, 137601 (2006), cond-mat/0512247

    R. Resta, Polarization Fluctuations in Insulators and Metals: New and Old Theories Merge, Physical Review Letters 96, 137601 (2006), cond-mat/0512247

    2006 arXiv

  74. [83]

    R. M. Martin, Electronic Structure (2004)

    2004

  75. [84]

    Aebischer, D

    C. Aebischer, D. Baeriswyl, and R. M. Noack, Dielectric Catastrophe at the Mott Transition, Physical Review Letters 86, 468 (2001), cond-mat/0006354

    2001 arXiv

  76. [85]

    Verma and R

    N. Verma and R. Queiroz, Instantaneous Response and Quantum Geometry of Insulators, arXiv 10.48550/arxiv.2403.07052 (2024), 2403.07052

    2024 doi

  77. [86]

    Onishi and L

    Y. Onishi and L. Fu, Universal relation between energy gap and dielectric constant, arXiv 10.48550/arxiv.2401.04180 (2024), 2401.04180

    2024 doi arXiv

  78. [87]

    Souza, R

    I. Souza, R. M. Martin, and M. Stengel, Optical bounds on many-electron localization, arXiv 10.48550/arxiv.2407.17908 (2024), 2407.17908

    2024 doi arXiv

  79. [88]

    Peotta and P

    S. Peotta and P. T¨ orm¨ a, Superfluidity in topologically nontrivial flat bands, Nature Communications6, 8944 (2015)

    2015

  80. [89]

    Julku, S

    A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. T¨ orm¨ a, Geometric origin of superfluidity in the lieb-lattice flat band, Phys. Rev. Lett. 117, 045303 (2016)

    2016

  81. [90]

    Liang, T

    L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and P. T¨ orm¨ a, Band geometry, berry curvature, and superfluid weight, Phys. Rev. B 95, 024515 (2017)

    2017

  82. [91]

    X. Hu, T. Hyart, D. I. Pikulin, and E. Rossi, Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene, Phys. Rev. Lett. 123, 237002 (2019)

    2019

  83. [92]

    F. Xie, Z. Song, B. Lian, and B. A. Bernevig, Topology-bounded superfluid weight in twisted bilayer graphene, Phys. Rev. Lett. 124, 167002 (2020)

    2020

  84. [93]

    Julku, T

    A. Julku, T. J. Peltonen, L. Liang, T. T. Heikkil¨ a, and P. T¨ orm¨ a, Superfluid weight and berezinskii-kosterlitz-thouless transition temperature of twisted bilayer graphene, Phys. Rev. B 101, 060505 (2020)

    2020

  85. [94]

    Rossi, Quantum metric and correlated states in two-dimensional systems, Current Opinion in Solid State and Materials Science 25, 100952 (2021)

    E. Rossi, Quantum metric and correlated states in two-dimensional systems, Current Opinion in Solid State and Materials Science 25, 100952 (2021)

    2021

  86. [95]

    J. Yu, M. Xie, F. Wu, and S. Das Sarma, Euler-obstructed nematic nodal superconductivity in twisted bilayer graphene, Phys. Rev. B 107, L201106 (2023)

    2023

  87. [96]

    T¨ orm¨ a, Essay: Where Can Quantum Geometry Lead Us?, Physical Review Letters131, 240001 (2023)

    P. T¨ orm¨ a, Essay: Where Can Quantum Geometry Lead Us?, Physical Review Letters131, 240001 (2023). 46

    2023

  88. [97]

    H. Tian, X. Gao, Y. Zhang, S. Che, T. Xu, P. Cheung, K. Watanabe, T. Taniguchi, M. Randeria, F. Zhang, C. N. Lau, and M. W. Bockrath, Evidence for dirac flat band superconductivity enabled by quantum geometry, Nature 614, 440 (2023)

    2023

  89. [98]

    J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, and B. A. Bernevig, Non-trivial quantum geometry and the strength of electron–phonon coupling, Nature Physics 20, 1262 (2024)

    2024

  90. [99]

    Zhu and A

    P. Zhu and A. Alexandradinata, Anomalous shift and optical vorticity in the steady photovoltaic current, Phys. Rev. B 110, 115108 (2024)

    2024

  91. [100]

    P. M. Tam, J. Herzog-Arbeitman, and J. Yu, Quantum geometry and entanglement in two-dimensional insulators: A view from the corner charge fluctuation, arXiv preprint arXiv:2406.17023 (2024)

    2024 arXiv

  92. [101]

    Wu, K.-L

    X.-C. Wu, K.-L. Cai, M. Cheng, and P. Kumar, Corner charge fluctuations and many-body quantum geometry (2024), arXiv:2408.16057 [cond-mat.str-el]

    2024 arXiv

  93. [102]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. Vergniory, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017)

    2017

  94. [103]

    Elcoro, B

    L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nature communications 12, 5965 (2021)

    2021

  95. [104]

    H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nature communications 8, 50 (2017)

    2017

  96. [105]

    Watanabe, H

    H. Watanabe, H. C. Po, and A. Vishwanath, Structure and topology of band structures in the 1651 magnetic space groups, Science Advances 4, 10.1126/sciadv.aat8685 (2018)

    2018 doi

  97. [106]

    Kruthoff, J

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

    2017

  98. [107]

    Z. Song, Z. Fang, and C. Fang, ( d − 2)-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017)

    2017

  99. [108]

    Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020)

    2020

  100. [109]

    Y. Xu, L. Elcoro, G. Li, Z.-D. Song, N. Regnault, Q. Yang, Y. Sun, S. Parkin, C. Felser, and B. A. Bernevig, Three- dimensional real space invariants, obstructed atomic insulators and a new principle for active catalytic sites, arXiv preprint arXiv:2111.02433 (2021)

    2021 arXiv

  101. [111]

    J. A. Wilson, F. Di Salvo, and S. Mahajan, Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides, Advances in Physics 24, 117 (1975)

    1975

  102. [112]

    Rossnagel, On the origin of charge-density waves in select layered transition-metal dichalcogenides, Journal of Physics: Condensed Matter 23, 213001 (2011)

    K. Rossnagel, On the origin of charge-density waves in select layered transition-metal dichalcogenides, Journal of Physics: Condensed Matter 23, 213001 (2011)

    2011

  103. [113]

    X. Zhu, Y. Cao, J. Zhang, E. Plummer, and J. Guo, Classification of charge density waves based on their nature, Proceedings of the National Academy of Sciences 112, 2367 (2015)

    2015

  104. [114]

    Wilson, F

    J. Wilson, F. Di Salvo, and S. Mahajan, Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides, Adv. Phys. 24, 117 (1975)

    1975

  105. [115]

    Chatterjee, J

    U. Chatterjee, J. Zhao, M. Iavarone, R. Di Capua, J. P. Castellan, G. Karapetrov, C. D. Malliakas, M. G. Kanatzidis, H. Claus, J. P. C. Ruff, F. Weber, J. van Wezel, J. C. Campuzano, R. Osborn, M. Randeria, N. Trivedi, M. R. Norman, and S. Rosenkranz, Emergence of coherence in...

    2015

  106. [116]

    M. M. Ugeda, A. J. Bradley, Y. Zhang, S. Onishi, Y. Chen, W. Ruan, C. Ojeda-Aristizabal, H. Ryu, M. T. Edmonds, H.-Z. Tsai, A. Riss, S.-K. Mo, D. Lee, A. Zettl, Z. Hussain, Z.-X. Shen, and M. F. Crommie, Characterization of collective ground states in single-layer NbSe2, Nat. ...

    2016

  107. [117]

    C.-S. Lian, C. Si, and W. Duan, Unveiling Charge-Density Wave, Superconductivity, and Their Competitive Nature in Two-Dimensional NbSe2, Nano Lett. 18, 2924 (2018)

    2018

  108. [118]

    D. Lin, S. Li, J. Wen, H. Berger, L. Forr´ o, H. Zhou, S. Jia, T. Taniguchi, K. Watanabe, X. Xi, and M. S. Bahramy, Patterns and driving forces of dimensionality-dependent charge density waves in 2H-type transition metal dichalcogenides, Nat. Commun. 11, 2406 (2020)

    2020

  109. [119]

    Dreher, W

    P. Dreher, W. Wan, A. Chikina, M. Bianchi, H. Guo, R. Harsh, S. Ma˜ nas-Valero, E. Coronado, A. J. Mart ´ ınez-Galera, P. Hofmann, J. A. Miwa, and M. M. Ugeda, Proximity Effects on the Charge Density Wave Order and Superconductivity in Single-Layer NbSe2, ACS Nano 15, 19430 (2021)

    2021

  110. [120]

    Revolinsky, G

    E. Revolinsky, G. Spiering, and D. Beerntsen, Superconductivity in the niobium-selenium system, Journal of Physics and Chemistry of Solids 26, 1029 (1965)

    1965

  111. [121]

    F. Soto, H. Berger, L. Cabo, C. Carballeira, J. Mosqueira, D. Pavuna, P. Toimil, and F. Vidal, Electric and magnetic characterization of nbse2 single crystals: Anisotropic superconducting fluctuations above tc, Physica C: Superconductivity 460, 789 (2007)

    2007

  112. [122]

    Y. Cao, A. Mishchenko, G. L. Yu, E. Khestanova, A. P. Rooney, E. Prestat, A. V. Kretinin, P. Blake, M. B. Shalom, C. Woods, J. Chapman, G. Balakrishnan, I. V. Grigorieva, K. S. Novoselov, B. A. Piot, M. Potemski, K. Watanabe, T. Taniguchi, S. J. Haigh, A. K. Geim, and R. V. Go...

    2015

  113. [123]

    X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forr´ o, J. Shan, and K. F. Mak, Ising pairing in superconducting NbSe2 atomic layers, Nat. Phys. 12, 139 (2016)

    2016

  114. [124]

    K. Zhao, H. Lin, X. Xiao, W. Huang, W. Yao, M. Yan, Y. Xing, Q. Zhang, Z.-X. Li, S. Hoshino, J. Wang, S. Zhou, 47 L. Gu, M. S. Bahramy, H. Yao, N. Nagaosa, Q.-K. Xue, K. T. Law, X. Chen, and S.-H. Ji, Disorder-induced multifractal superconductivity in monolayer niobium dichalc...

    2019

  115. [125]

    W. Wan, P. Dreher, D. Mu˜ noz-Segovia, R. Harsh, H. Guo, A. J. Mart ´ ınez-Galera, F. Guinea, F. de Juan, and M. M. Ugeda, Observation of Superconducting Collective Modes from Competing Pairing Instabilities in Single-Layer NbSe2, Adv. Mater. 34, 2206078 (2022)

    2022

  116. [126]

    D. Lin, S. Li, J. Wen, H. Berger, L. Forr´ o, H. Zhou, S. Jia, T. Taniguchi, K. Watanabe, X. Xi, et al. , Patterns and driving forces of dimensionality-dependent charge density waves in 2 h-type transition metal dichalcogenides, Nature communications 11, 2406 (2020)

    2020

  117. [127]

    X. Xi, L. Zhao, Z. Wang, H. Berger, L. Forr´ o, J. Shan, and K. F. Mak, Strongly enhanced charge-density-wave order in monolayer nbse2, Nature nanotechnology 10, 765 (2015)

    2015

  118. [128]

    Calandra, I

    M. Calandra, I. I. Mazin, and F. Mauri, Effect of dimensionality on the charge-density wave in few-layer $2{H\text{- NbSe}} {2}$, Phys. Rev. B 80, 241108 (2009)

    2009

  119. [129]

    Leroux, I

    M. Leroux, I. Errea, M. Le Tacon, S.-M. Souliou, G. Garbarino, L. Cario, A. Bosak, F. Mauri, M. Calandra, and P. Rodi` ere, Strong anharmonicity induces quantum melting of charge density wave in $2H\ensuremath{-}{\mathrm{NbSe}} {2}$ under pressure, Phys. Rev. B 92, 140303 (2015)

    2015

  120. [130]

    Flicker and J

    F. Flicker and J. van Wezel, Charge order from orbital-dependent coupling evidenced by NbSe2, Nat. Commun. 6, 7034 (2015)

    2015

  121. [131]

    Flicker and J

    F. Flicker and J. van Wezel, Charge order in ${\mathbf{NbSe}} {2}$, Phys. Rev. B 94, 235135 (2016)

    2016

  122. [132]

    C.-S. Lian, C. Si, and W. Duan, Unveiling charge-density wave, superconductivity, and their competitive nature in two-dimensional nbse2, Nano letters 18, 2924 (2018)

    2018

  123. [133]

    Holbrook, J

    M. Holbrook, J. Ingham, D. Kaplan, L. Holtzman, B. Bierman, N. Olson, L. Nashabeh, S. Liu, X. Zhu, D. Rhodes, et al., Real-space imaging of the band topology of transition metal dichalcogenides, arXiv preprint arXiv:2412.02813 (2024)

    2024 arXiv

  124. [134]

    Z. Liu, P. Deng, Y. Xu, H. Yang, D. Pei, C. Chen, S. He, D. Liu, S.-K. Mo, T. Kim, C. Cacho, H. Yao, Z.-D. Song, X. Chen, Z. Wang, B. Yan, L. Yang, B. A. Bernevig, and Y. Chen, Massive 1d dirac line, solitons and reversible manipulation on the surface of a prototype obstructed...

    2024 arXiv

  125. [135]

    C˘ alug˘ aru, Y

    D. C˘ alug˘ aru, Y. Jiang, H. Guo, S. Sajan, Y. Wang, H. Hu, J. Yu, B. A. Bernevig, F. de Juan, and M. M. Ugeda, Probing the quantized berry phases in 1h-nbse 2 using scanning tunneling microscopy, arXiv preprint arXiv:2501.09063 (2025)

    2025 arXiv

  126. [136]

    G. Li, Y. Xu, Z. Song, Q. Yang, Y. Zhang, J. Liu, U. Gupta, V. S¨ u β, Y. Sun, P. Sessi, et al. , Obstructed surface states as the descriptor for predicting catalytic active sites in inorganic crystalline materials, Advanced Materials 34, 2201328 (2022)

    2022

  127. [137]

    Z. Yang, H. Sheng, Z. Guo, R. Zhang, Q. Wu, H. Weng, Z. Fang, and Z. Wang, Superconductivity in unconventional metals, arXiv preprint arXiv:2306.08347 (2023)

    2023

  128. [138]

    R. L. Barnett, A. Polkovnikov, E. Demler, W.-G. Yin, and W. Ku, Coexistence of gapless excitations and commensurate charge-density wave¡? format?¿ in the 2 h transition metal dichalcogenides, Physical review letters 96, 026406 (2006)

    2006

  129. [139]

    J. Zeng, H. Liu, H. Jiang, Q.-F. Sun, and X. Xie, Multiorbital model reveals a second-order topological insulator in 1 h transition metal dichalcogenides, Physical Review B 104, L161108 (2021)

    2021

  130. [140]

    Jung and Y.-H

    J. Jung and Y.-H. Kim, Hidden breathing kagome topology in hexagonal transition metal dichalcogenides, Physical Review B 105, 085138 (2022)

    2022

  131. [141]

    Qian, G.-B

    S. Qian, G.-B. Liu, C.-C. Liu, and Y. Yao, C n-symmetric higher-order topological crystalline insulators in atomically thin transition metal dichalcogenides, Physical Review B 105, 045417 (2022)

    2022

  132. [142]

    Sødequist, U

    J. Sødequist, U. Petralanda, and T. Olsen, Abundance of second order topology in c3 symmetric two-dimensional insu- lators, 2D Materials 10, 015009 (2022)

    2022

  133. [143]

    Schindler and B

    F. Schindler and B. A. Bernevig, Noncompact atomic insulators, Phys. Rev. B 104, L201114 (2021)

    2021

  134. [144]

    Jiang, Z

    K. Jiang, Z. Qi, H. Weng, and J. Hu, Mottness in obstructed atomic insulators without mott transition, Phys. Rev. B 108, 195102 (2023)

    2023

  135. [145]

    Herzog-Arbeitman, V

    J. Herzog-Arbeitman, V. Peri, F. Schindler, S. D. Huber, and B. A. Bernevig, Superfluid weight bounds from symmetry and quantum geometry in flat bands, Phys. Rev. Lett. 128, 087002 (2022)

    2022

  136. [146]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6, 15 (1996)

    1996

  137. [147]

    Kresse and J

    G. Kresse and J. Hafner, Ab initio molecular dynamics for open-shell transition metals, Phys. Rev. B 48, 13115 (1993)

    1993

  138. [148]

    Souza, N

    I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized wannier functions for entangled energy bands, Physical Review B 65, 035109 (2001)

    2001

  139. [149]

    Liu, W.-Y

    G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, Three-band tight-binding model for monolayers of group-vib transition metal dichalcogenides, Physical Review B—Condensed Matter and Materials Physics 88, 085433 (2013)

    2013

  140. [150]

    S. Fang, R. Kuate Defo, S. N. Shirodkar, S. Lieu, G. A. Tritsaris, and E. Kaxiras, Ab initio tight-binding hamiltonian for transition metal dichalcogenides, Physical Review B 92, 205108 (2015)

    2015

  141. [151]

    Cappelluti, R

    E. Cappelluti, R. Rold´ an, J. Silva-Guill´ en, P. Ordej´ on, and F. Guinea, Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer mos 2, Physical Review B—Condensed Matter and Materials Physics 88, 075409 (2013)

    2013

  142. [152]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

    1996

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