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Probing the Quantized Berry Phases in 1H-NbSe$_2$ Using Scanning Tunneling Microscopy

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Monolayer NbSe2's Fermi-level band is an obstructed atomic phase whose Wannier center sits on an empty site, as established by a new STM deconvolution of orbital correlators.

desk verdict A genuinely new STM deconvolution method and a plausible OA identification, but Table S2 contains a three-tip spread that undercuts the word 'unambiguous.' read the letter →

arxiv 2501.09063 v1 pith:XM3XGYI2 submitted 2025-01-15 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el

Dumitru Călugăru , Yi Jiang , Haojie Guo , Sandra Sajan , Yongsong Wang , Haoyu Hu , Jiabin Yu , B. Andrei Bernevig
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Fernando de Juan Miguel M. Ugeda
This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.str-el
keywords obstructedatomicinsulatorWanniercenterinter-orbitalcorrelationfunctionsscanningtunnelingmicroscopymonolayer1H-NbSe2chargedensitydistributiontopologicalquantumchemistryflatband
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

Topologically trivial insulators split into unobstructed atomic phases, whose Wannier centers sit on occupied atoms, and obstructed atomic phases, whose centers sit at empty sites in the unit cell. This paper claims that the quasi-flat band crossing the Fermi level of monolayer 1H-NbSe2 is an obstructed atomic phase, and that this is the first unambiguous quantitative identification of such a phase. The evidence comes from converting constant-height STM current maps at two bias voltages into the band's charge density distribution, then deconvolving that density with Wannier orbital wave functions from ab initio calculations to extract the inter-orbital correlation matrix. The fitted correlators land at O1 ≈ 0.03 and O2 ≈ 0.33, which is the obstructed signature (Wannier center at the empty 1c site), not the unobstructed one (O1 ≈ 1/3, O2 ≈ 0). The identification matters because obstructed bands are predicted to be common and are tied to flat-band geometry, superconductivity, and electron-phonon effects.

What carries the argument

The load-bearing object is the orbital correlator matrix $\rho_{ij}(\Delta R)=\langle\Phi|\,c^{\dagger}_{R,i}c_{R+\Delta R,j}\,|\Phi\rangle$ for the fully filled quasi-flat band. The identity that converts the STM map into physics is $A(Q,z)=\sum_{\Delta R,i,j}B_{ij}(Q,z,\Delta R)\,\rho_{ij}(\Delta R)$: since the STM density is a spatially diagonal observable, the off-diagonal orbital correlations are recovered only through the finite spatial extent of the Wannier orbitals encoded in $B$. The obstructed character is read off two correlators: $O_1=\rho_{12}(0)$ and $O_2=\rho_{23}(a_1)$, which take values $(0,1/3)$ in the obstructed limit and $(1/3,0)$ in the fictitious unobstructed limit. A minimal three-orbital tight-binding model with real hoppings $t=-0.784$ eV and $E_0=1.733$ eV generates compact Wannier orbitals with the same correlator pattern, which is what makes the fitted numbers interpretable as an optimally compact obstructed atomic phase.

What would settle it

Re-fit the same STM-derived $A(Q,z)$ with Wannier functions obtained from a different exchange-correlation functional or with spin-orbit coupling included, and simultaneously check that the fitted tip height $z=3.25$ Å agrees with the physical setpoint for the stabilization parameters used in the measurement; if $O_1$ moves toward $1/3$, $O_2$ toward $0$, or the optimal $z$ becomes unphysical under a plausible alternative basis, the obstructed-atomic conclusion fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Fermi-level quasi-flat band of monolayer 1H-NbSe2 realizes an optimally compact obstructed atomic phase. Its Wannier center lies at the 1c position, an empty site in the unit cell, rather than at the Nb (1a) or Se (1b) positions. The central experimental relation writes the measured band charge density in Fourier space as $A(Q,z)=\sum_{\Delta R,i,j} B_{ij}(Q,z,\Delta R)\,\rho_{ij}(\Delta R)$, where $B$ is computed from ab initio Wannier wave functions and $\rho_{ij}(\Delta R)$ is the orbital-correlation matrix of the filled band. Fitting only the three dominant correlators gives $O_0=1/3$, $O_1=0.030$, and $O_2=0.332$ at the best-fit tip height $z=3.25$ Å. This is the obstructed-atomic limit ($O_1\to 0$, $O_2\to 1/3$), opposite to the unobstructed limit ($O_1\to 1/3$, $O_2\to 0$), and the band's wave function has more than 94% overlap with an optimally compact obstructed-atomic model on the triangular lattice.

Load-bearing premise

The result stands or falls on the assumption that the atomic-like orbital shapes used in the deconvolution, taken from density-functional calculations, are the true orbital content of the Fermi-level band, and that the darkest and brightest spots in the STM map correctly mark the Nb and Se sites; if either assumption fails, the extracted correlators are biased toward the theoretical prediction.

Editorial extensions

If this is right

  • The Fermi-level band of monolayer 1H-NbSe2 has its Wannier center at the empty 1c site, confirming the obstructed-atomic assignment for this material.
  • The same two-bias STM measurement, deconvolved with the ab initio spatial factor, gives a general route to identifying obstructed atomic bands in other monolayer transition-metal dichalcogenides.
  • The extracted correlators and the better-than-94% overlap with the optimally compact triangular-lattice model validate a minimal three-orbital description of the band, a controlled starting point for models of superconductivity and charge-density-wave order.
  • Because obstructed bands have an enlarged quantum geometric tensor, the measurement connects STM density maps to geometric lower bounds on superfluid weight and to electron-phonon coupling in flat bands.

Reading between the lines

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

  • Beyond the paper: repeating the same extraction below the 33 K charge-density-wave transition could show whether the 3×3 reconstruction moves or splits the Wannier center, turning the method into a direct probe of CDW order parameters.
  • Beyond the paper: the fitted correlation matrix is exactly the kind of input needed to evaluate geometric bounds on superfluid weight and electron-phonon coupling, so a single STM measurement could be converted into quantitative constraints on NbSe2's superconductivity.
  • Beyond the paper: re-fitting the same experimental density with Wannier functions from different density functionals, different cutoffs, or with spin-orbit coupling included would reveal how much of the obstructed-atomic conclusion is inherited from the ab initio input rather than from the measured maps.
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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

4 major / 5 minor

Summary. The manuscript claims to provide the first unambiguous quantitative experimental identification of an obstructed atomic (OA) phase, using monolayer 1H-NbSe2 as the test case. The authors measure constant-height STM tunneling current maps at two biases bracketing the Fermi-level quasi-flat band, combine them to obtain the band's charge density distribution (CDD), and Fourier-transform the CDD to obtain A(Q,z). They then fit A(Q,z) against a linear model A(Q,z) = sum_{ΔR,i,j} B_ij(Q,z,ΔR) ρ_ij(ΔR), where the spatial factor B_ij is computed from ab initio Wannier functions, to extract inter-orbital correlators ρ_ij(ΔR). With a three-correlator truncation (O0, O1, O2), the main-text fit gives O1≈0.030 and O2≈0.332, which the authors interpret as the OA limit (O1=0, O2=1/3) with the Wannier center at the empty 1c position. The paper also presents bias-dependent STS contrast as an independent check of the unit-cell site assignment, and argues that the method is general and transferable to other materials.

Significance. If the central claim holds, this is a significant conceptual and experimental advance: it provides a quantitative real-space probe of a band-topology classification that goes beyond Chern numbers, namely the obstructed versus unobstructed atomic distinction, and it demonstrates a general technique for extracting orbital-off-diagonal correlators from the spatially diagonal STM density. The theoretical derivation of the relation between the CDD and the correlator matrix is carried out carefully in the Supplementary Information, with explicit normalization constraints and symmetry analysis. The paper also deserves credit for reporting multiple experimental data sets with different tips and for including an independent bias-dependent STS validation of the site assignment. The main scientific risk is not the derivation but the robustness of the quantitative fitting pipeline and the consistency of the reported data with the headline claim of an 'unambiguous' OA identification.

major comments (4)
  1. [Supplementary Table S2; main text Fig. 3(i)-(j)] The statement that 'all measurements consistently yield O1≈0 and O2≈1/3' is not supported by the paper's own Table S2. Three Tip-2 data sets give fitted O1 = 0.124, 0.129, 0.132 and O2 = 0.309, 0.307, 0.306, while the main-text values O1≈0.030, O2≈0.332 come from Tip 3. O1 = 0.13 is about 40% of its unobstructed-atomic value 1/3, and in the parameterization O1=(1/3)cosθ, O2=(1/3)sinθ these Tip-2 values imply θ≈67° rather than the θ=90° OA limit. No error bars, goodness-of-fit values, or tip-selection/exclusion criteria are given, so a reader cannot determine whether Tip 2 is an outlier, a tip-dependent systematic effect, or evidence that the three-correlator model is misspecified. Since the central claim depends quantitatively on O1 being near zero, this internal inconsistency is load-bearing and must be resolved, either by including all data in a proper statistical treatment or by justifying the exclusion of Tip 2 with objective, pre-specified criteria.
  2. [Main text Fig. 3(i)-(j); SI IV.2, Eq. (S4.27)] The tip-sample distance z is a fitted parameter, selected as the minimum of the symmetrized error metric ϵsym, but no confidence intervals are reported for z, O1, or O2. Figure 3(i) shows that the extracted correlators vary substantially with the assumed z (the main text reports z=3.25 Å as the optimum, while Table S2 contains fitted z values of 3.25, 3.30, and 3.70 Å across data sets). For example, Tip 3 set 8, with fitted z=3.70 Å, gives O1=0.080 and O2=0.324, which is less cleanly in the OA limit. Because the error metric is symmetrized and the unsymmetrized error is reported as flat in z, the claim that the minimum at z=3.25 Å determines the correlators quantitatively needs an uncertainty propagation analysis. The authors should provide error bars and show that O1≈0, O2≈1/3 is stable within the experimental uncertainties in z and in the measured A(Q).
  3. [Main text Eq. (2); SI IV.1d] The deconvolution is not parameter-free and is partly circular: the spatial factor B_ij(Q,z,ΔR) is computed from the same ab initio DFT framework that predicts the OA phase, and the unit-cell origin in the experimental data is assigned using the DFT-computed brightness hierarchy (1b brightest, 1a darkest). If the Wannier functions are inaccurate, or if the s-like three-orbital basis does not capture the true orbital content of the band, the fitted correlators will be biased toward the DFT prediction. The bias-dependent STS comparison in Fig. 4 validates the site assignment but does not validate the quantitative O1 and O2 values. I recommend adding robustness tests, for example refitting with Wannier functions from different gauge choices or disentanglement windows, using an 11-band basis including Se p orbitals, and trying all three possible C3z origin assignments, to demonstrate that the O1≈0, O2≈1/3 conclusion is stable rather than imposed by the input basis.
  4. [SI IV.2, Eqs. (S4.24)-(S4.27); Table S1] The fit assumes that only O0, O1, and O2 are nonzero, an approximation justified by the DFT values in Table S1. Table S1 itself shows that O3-O7 are not strictly zero (e.g., O3=-0.075, O4=0.038), and if the true correlator matrix of the real material has non-negligible additional elements, the fitted O1 and O2 will absorb them. This model-misspecification possibility is consistent with the anomalous Tip-2 results. The paper should report per-data-set values of the error metrics ϵ and ϵsym, and perform a sensitivity analysis that includes O3-O7 in the fit, to show that the central OA identification is not an artifact of the three-correlator truncation.
minor comments (5)
  1. [Main text, Figs. 3(e)-(h)] The four Fourier-transform panels showing amplitude and phase are not labeled with the corresponding reciprocal-lattice vectors Q; adding such labels would make the figure much easier to interpret.
  2. [Main text, Eq. (2) and SI Eq. (S2.33)] The normalization conventions for A(Q,z) differ between the main text (1/Ω) and the SI (1/(NΩ0)); these should be stated consistently so that Eq. (2) can be reproduced directly from the SI.
  3. [Supplementary Table S2] The column header 'O_i (for 1≤i≤3)' is confusing because the table lists O0, O1, and O2; please relabel as O0, O1, O2.
  4. [Introduction and abstract] The abstract says 'first direct experimental evidence' while the introduction says 'first direct quantitative experimental evidence'; the wording should be unified, especially because 'quantitative' is central to the paper's claim.
  5. [Main text and Ref. [33]] The claimed 'more than 94% overlap' between the compact model and the ab initio wave function is attributed to '[33], to be published'. Since that reference is not yet available, please include the numerical details needed to verify this number in the supplement.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the correlator fit is overdetermined and can return either OA or UA values; the ab initio Wannier basis and origin assignment are theory-loaded but not definitional.

full rationale

The paper's central derivation is the inversion of Eq. (2), A(Q,z)=sum B_ij(Q,z,DeltaR) rho_ij(DeltaR), with B_ij computed from ab initio Wannier functions and rho_ij fitted to the STM-derived CDD. This is not circular by construction: the same Wannier basis supports both the OA limit (O1=0, O2=1/3) and the fictitious UA limit (O1=1/3, O2=0), as shown in the supplementary equations S3.14 and S3.21. The fit is also constrained by the normalization conditions O0=1/3 and O1^2+O2^2=1/9, leaving a one-parameter family; the experimental Fourier components select the value of theta, and a UA signal would favor O1=1/3. The main remaining concern is theory loading: the unit-cell origin is fixed using the DFT-computed brightness hierarchy, and the Wannier spatial factor comes from the same ab initio framework that predicted the OA phase. However, the manuscript provides an independent, falsifiable check in the -2 V conductance maps, which are not used in the correlator fit, and the fitted tip height is treated as a nuisance parameter. The citation of Ref. [33], an overlapping-author 'to be published' work, supplies the toy-model parameters and the 94% overlap claim, but these are interpretive rather than load-bearing for the experimental extraction. The Supplementary Table S2 discrepancy (Tip 2 gives O1=0.124-0.132 rather than O1 about 0.03) is a correctness and robustness concern about whether the identification is truly unambiguous, but it is not a circularity of the derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The analysis depends on DFT-derived Wannier functions and the Tersoff-Hamann approximation, plus a truncation to three correlators. These are reasonable but carry theory input into the experimental analysis. The free parameters are the tip height and the correlator amplitudes, which are fitted, not predicted.

free parameters (4)
  • Tip-sample distance z (assumed) = 3.25 Å (best fit); 3.25-3.70 Å across data sets
    Unknown in experiment; fitted by minimizing the fitting error ϵ_sym (Fig. 3j). The extracted correlators depend on z through B_ij(Q,z,ΔR).
  • Correlator O1 (or θ) = 0.030 (best fit); 0.014-0.132 across all sets
    Fitted from the STM CDD with constraint O1^2+O2^2=1/9; it is the order parameter distinguishing UA (O1=1/3) from OA (O1=0).
  • Correlator O2 (or θ) = 0.332 best fit; 0.306-0.333 across sets
    Fitted as the other constrained component; O2≈1/3 signals the OA phase.
  • Global scale γ = Not reported
    Arbitrary STM normalization; absorbed in the constrained fit (Eqs. S4.26-S4.27).
assumptions (5)
  • domain assumption Tersoff-Hamann approximation: the STM tip is a spherically symmetric metal with the same work function as the sample, and tunneling current is proportional to the integrated local spectral function.
    Invoked in Appendix I to express the measured current difference as the CDD of the quasi-flat band. This is a standard but non-trivial approximation for STM.
  • domain assumption The three mirror-even bands spanning the quasi-flat band are accurately represented by the three d-orbital Wannier functions (and their s-like combinations) obtained from DFT-PBE calculations.
    The spatial factor B_ij is computed from these Wannier functions. If the basis is incomplete or inaccurate, the extracted correlators are biased.
  • ad hoc to paper Only the correlators O0, O1, O2 are significant; all other correlators are set to zero.
    Justified by ab initio values in Table S1, but this truncation is necessary to make the fit well-posed and can suppress contributions from other orbital processes.
  • domain assumption The measured current difference I(V+)-I(V-) isolates the CDD of the quasi-flat band, and the proportionality constant (including tip height) is constant across the two successive scans.
    Required to sum A+ and A- (Eqs. S1.81-S1.84). Drift or bias-dependent tip changes would violate this.
  • domain assumption Spin-orbit coupling is negligible for the conclusions.
    Explicitly stated in Methods; accepted without detailed justification, though consistent with prior literature.

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Pith. "Pith review of Probing the Quantized Berry Phases in 1H-NbSe$_2$ Using Scanning Tunneling Microscopy." pith.science (2026). https://pith.science/paper/XM3XGYI2

@misc{pith2026250109063,
  author       = {Pith},
  title        = {Pith review of: Probing the Quantized Berry Phases in 1H-NbSe$_2$ Using Scanning Tunneling Microscopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XM3XGYI2}},
  note         = {Machine review of arXiv:2501.09063}
}
abstract

Topologically trivial insulators are classified into two primary categories: unobstructed and obstructed atomic insulators. While both types can be described by exponentially localized Wannier orbitals, a defining feature of obstructed atomic insulators is that the centers of charge of these orbitals are positioned at empty sites within the unit cell, rather than on atoms. Despite extensive theoretical predictions, the unambiguous and quantitative experimental identification of an obstructed atomic phase has remained elusive. In this work, we present the first direct experimental evidence of such a phase in 1H-NbSe$_2$. We develop a novel method to extract the inter-orbital correlation functions from the local spectral function probed by scanning tunneling microscopy (STM), leveraging the orbital wave functions obtained from ab initio calculations. Applying this technique to STM images, we determine the inter-orbital correlation functions for the atomic band of 1H-NbSe$_2$ that crosses the Fermi level. Our results show that this band realizes an optimally compact obstructed atomic phase, providing the first unambiguous experimental identification of such a phase. Our approach of deconvolving the STM signal using ab initio orbital wave functions is broadly applicable to other material platforms, offering a powerful tool for exploring other electronic phases.

Figures

Figures reproduced from arXiv: 2501.09063 by the authors.

Figure 1
Figure 1. FIG. 1. Crystal structure and electronic band structure of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Extracting the orbital correlators from STM. By probing the electron density in the middle of a bond, an STM tip can [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Bias-dependent contrast maps in NbSe [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 3 Pith papers

  1. [1]

    Notation We consider a two-dimensional5 crystalline material and letˆc† R,i denote the corresponding fermionic creation opera- tors6: ˆc† R,i creates an electron located in thei-th Wannier orbital of the unit cell atR. The Wannier orbital associated with ˆc† R,i is displaced byri from the origin of the unit cell (located atR), and its real-space wave func...

    work page

  2. [2]

    For convenience, we denote its wave function by ui (k) = uiα (k)

    The CDD of an isolated band Consider a single (isolated) band of the system, chosen without loss of generality to be the α-th band. For convenience, we denote its wave function by ui (k) = uiα (k) . (S2.20) Here, we suppress the band index for simplicity, as our discussion exclusively concerns this band. We aim to compute the CDD of a state where theα-th ...

    work page

  3. [3]

    Symmetry properties In this section, we consider the constraints imposed by the crystalline symmetries of the system on the CDD, correlation function, and spatial factor. We takeg = {R|p} to be a symmorphic (p = 0) or non-symmorphic (p ̸= 0) crystalline symmetry of the system, consisting of a rotationR followed by a translation by a non-lattice vectorp. A...

    work page

  4. [4]

    The general method can be outlined as follows: 32

    Fitting the correlation function We now explain how the correlation function can be extracted from experimental measurements ofA (r). The general method can be outlined as follows: 32

    work page

  5. [5]

    These symmetries constrainρij (∆R) according to Eq

    Begin by writing down the most general form of the real-space correlation function that obeys the symmetries of the system. These symmetries constrainρij (∆R) according to Eq. (S2.51). A cutoff distanceRmax is typically introduced, beyond which all the components of the correlation function are approximated as zero ρij (∆R) ≈ 0, for |∆R + rj − ri| > Rmax....

    work page

  6. [6]

    Compute the spatial factorBij (Q, z,∆R) using the method outlined in Appendix [II2a] with Wannier wave functions determined fromab initiosimulations

    work page

  7. [7]

    Due to finite experimental resolution, only a limited number of Fourier components can be resolved (i.e., A (Q, z) ≈ 0 for large |Q|)

    Fourier transform the experimentally measured CDD to obtainA (Q, z). Due to finite experimental resolution, only a limited number of Fourier components can be resolved (i.e., A (Q, z) ≈ 0 for large |Q|). Additionally, system symmetries reduce the number ofindependent real components ofA (Q, z). We denote this number by NA

    work page

  8. [8]

    (S2.33) forρij (∆R) in the least-squares sense

    Determine the correlation function by solving Eq. (S2.33) forρij (∆R) in the least-squares sense. We begin by writing Eq. (S2.33) as a matrix-vector product AQ = X I BQ,I ρI , (S2.57) where I = (i, j,∆R) is a composite index and ρI ≡ρij (∆R) , (S2.58) BQ,I ≡Bij (Q, z,∆R) , (S2.59) AQ ≡A (Q, z) . (S2.60) Since not all components ofAQ or ρI are independent,...

    work page

Show all 19 references
  1. [9]

    S8, with the corresponding lattice vectors given in Cartesian coordinates by a1 = a 1 2 , − √ 3 2 , 0 ! , a2 = a 1 2 , √ 3 2 , 0 ! , (S3.1) where a = 3.47Å is the lattice constant

    Crystal structure The crystal structure of NbSe2 is shown in Fig. S8, with the corresponding lattice vectors given in Cartesian coordinates by a1 = a 1 2 , − √ 3 2 , 0 ! , a2 = a 1 2 , √ 3 2 , 0 ! , (S3.1) where a = 3.47Å is the lattice constant. In the unit cell, the Nb atom ...

  2. [10]

    Electronic spectra of NbSe2 In this section, we analyze the ab initio electronic band structures and simulated CDD of NbSe2. The band structures presented in this work are computed using the Vienna Ab-initio Simulation Package (VASP) [50–54] with the generalized gradient appro...

  3. [11]

    This is expected since the Se atoms are closest to the STM tip, as shown in the crystal structure in Fig

    The Se site consistently exhibits the largest differential conductance at all energies. This is expected since the Se atoms are closest to the STM tip, as shown in the crystal structure in Fig. S8(a)

  4. [12]

    This is consistent with the BR analysis discussed earlier, which places the Wannier center of the quasi-flat band at the empty1c position

    Within the energy range of the quasi-flat band,−0.4 ≤ ω/eV ≤ 0.6, the 1c (empty) site shows significantly higher differential conductance than the Nb site. This is consistent with the BR analysis discussed earlier, which places the Wannier center of the quasi-flat band at the ...

  5. [13]

    This can be attributed to the bands in this energy range being induced from Wannier orbitals with Wannier centers located at both the Nb and the empty sites

    Near ω = −2 eV, the simulated differential conductance at the1a (Nb) and1c (empty) sites is comparable, with the intensity at the1b (Se) site being slightly larger. This can be attributed to the bands in this energy range being induced from Wannier orbitals with Wannier center...

  6. [14]

    This is due to contributions from the mirror-odd sector, whose Wannier centers are located at the1c site, as indicated in Fig

    For−1.5 ≤ ω/eV ≤ −1.0, the1c (empty) site surpasses the1a (Nb) site in intensity. This is due to contributions from the mirror-odd sector, whose Wannier centers are located at the1c site, as indicated in Fig. S9(a). We will use the hierarchy of intensities among the threeC3z-s...

  7. [15]

    ” in Eq. (S3.8) denote other terms with hopping amplitudes no larger than0.3 eV [33], which will be ignored for now. Because the different “local

    Wannier tight-binding models and the correlation function With the ab initio band structure of NbSe2 at hand, we construct tight-binding models for the system using Wannier90. The models presented here are similar to those reported in the literature for other TMD monolayers th...

  8. [16]

    In both cases, the strongly hybridized groups of orbitals are denoted by yellow circles. In the idealized OA and UA limits, the equal superposition of orbitals within the yellow circles corresponds to theˆγ† R,1 and ˆγ′† R,1 Wannier orbitals of the quasi-flat bands, as defined...

  9. [17]

    Extracting the CDD Fourier transformation A (Q, z) from experimental data As described in the main text, we measure the tunneling current of NbSe2 at constant height. According to the Tersoff-Hamann approximation discussed in Appendix [I2d], this tunneling current is proportio...

  10. [18]

    brightest

    Using the Fourier representations of the CDD and Blackman function from Eqs. (S2.31) and (S4.5), we rewrite A± meas (k) = Z d2r∥ X Q 1 (2π)2 Z d2k′ A± (Q, z) e iQ· T±r∥+r± ∥,0 h (k′L) e−ik′·r∥ e−ik·r∥ = X Q Z d2k′ A± (Q, z) eiQ·r± ∥,0 ˜h (k′L) δ T −1 ± Q − k′ − k = X Q A± (Q, ...

  11. [19]

    Here, zmeas denotes theunknown tip-sample distance, distinguishing it from theassumed tip-sample distance z

    Fitting the correlation function from experimental data Once the conventional unit cell has been identified in the experimental data, we proceed to extract the correlation function ρij (∆R)fromtheFouriertransformationofthequasi-flatbandCDD,givenby A (Q, zmeas) = A+ (Q, zmeas)+...

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