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REVIEW 3 major objections 4 minor 42 references

Real space quantum metric of solids

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The overlap of local states at slightly different positions defines a real-space quantum metric; in homogeneous systems its density average is the electron momentum variance, measurable by ARPES.

desk verdict A clean real-space metric construction with a genuine ARPES identity, married to under-tested lattice-curvature numerics that need a scheme-independence check. read the letter →

arxiv 2411.10909 v2 pith:EANIV3EX submitted 2024-11-16 cond-mat.mes-hall cond-mat.dis-nn

Lucas A. Oliveira , Wei Chen This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.dis-nn
keywords real-spacequantummetriclocalstateformalismARPESgeometrydisorder-inducedcurvatureRicciscalarWannierfunctionsCherninsulator
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 proposes that a quantum metric can be defined in the real space of any solid, not only in momentum space. The metric comes from overlapping two local states, each formed by projecting a position state onto the occupied eigenstates or by applying a local annihilation operator to the ground state. In homogeneous continuous systems, the density-averaged metric equals the electron momentum variance $\langle k_\mu k_\nu\rangle/\hbar^2$, so it is directly measurable by angle-resolved photoemission. On lattices, the metric is constructed with central differences, and a single impurity deforms it, producing a spatially oscillating Ricci scalar whose sign at the impurity can be tuned by chemical potential and impurity strength. The claim therefore places a measurable, disorder-tunable geometric structure on the real-space manifold of ordinary solids.

What carries the argument

The central object is the normalized local state $|\psi_\sigma(x)\rangle = P|x,\sigma\rangle/\sqrt{n_\sigma(x)}$, with the equivalent many-body form $|\Phi_\sigma(x)\rangle = c_{x\sigma}|G\rangle/\sqrt{n_\sigma(x)}$; the metric is extracted from the second-order small-distance expansion of their overlap. For crystalline systems, the construction uses a central-difference Taylor expansion of the local states at neighboring lattice sites, yielding a finite-difference formula for $g_{\mu\nu}(x,\sigma)$ that reduces to the derivative expression in the continuum limit. The same central differences then define Christoffel symbols, Riemann tensor, Ricci scalar, and the volume form, and a density-weighted average $g_{\mu\nu}(x) = \sum_\sigma P_\sigma(x)g_{\mu\nu}(x,\sigma)$ connects the metric to the ARPES spectral function via the momentum variance $\langle k_\mu k_\nu\rangle/\hbar^2$.

What would settle it

For the continuous-system claim, measure the ARPES spectral function of a homogeneous two-dimensional electron gas and compute $\mathrm{Tr}\,g = \langle k_x^2 + k_y^2\rangle/\hbar^2$ with Eq. (36); the predicted value from Eq. (39) for $D=2$ is $k_F^2/(2\hbar^2)$, and disagreement beyond the known Fermi momentum would falsify the metric--momentum-variance identification. For the lattice-curvature claim, recompute the Ricci scalar profile of the disordered 2D metal with a fourth-order finite-difference scheme instead of the Appendix B central differences; if the sign of $R(x_{\mathrm{imp}})$ or the oscillation pattern changes, the computed curvature is an artifact of the difference scheme rather than a robust geometric property.

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Extended reading notes

Core claim

The paper's central claim is that the overlap of local states at slightly different positions defines a genuine real-space quantum metric, $|\langle\psi_\sigma(x)|\psi_\sigma(x+\delta x)\rangle| = 1 - \frac{1}{2} g_{\mu\nu}(x,\sigma)\,\delta x^\mu \delta x^\nu$, where $|\psi_\sigma(x)\rangle = P|x,\sigma\rangle/\sqrt{n_\sigma(x)}$ and $P$ is the projector onto filled eigenstates. In homogeneous continuous systems, the density-averaged metric reduces to $g_{\mu\nu} = \langle k_\mu k_\nu\rangle/\hbar^2$, making it an ARPES-accessible property. This metric exists even when the momentum-space quantum metric is trivial, such as in a free electron gas. In disordered lattices, the same construction yields a deformed manifold with nonzero Ricci scalar around impurities, while the Euler characteristic remains zero for a single impurity; the many-body local state $|\Phi_\sigma(x)\rangle = c_{x\sigma}|G\rangle/\sqrt{n_\sigma(x)}$ produces the same metric through the density-normalized single-particle correlation function.

Load-bearing premise

The lattice construction assumes that local states at neighboring sites can be expanded to second order in the lattice constant using central differences, and that the resulting discrete metric is smooth enough for curvature quantities to carry geometric meaning; near a strong impurity this expansion has no controlled error estimate.

Editorial extensions

If this is right

  • In any homogeneous metal or insulator, the trace of the density-averaged metric, $\mathrm{Tr}\,g = \langle k^2\rangle/\hbar^2$, is a coordinate-invariant characteristic of the occupied Fermi sea that can be extracted from a full-Brillouin-zone ARPES measurement.
  • Systems with no momentum-space quantum geometry, such as a free electron gas or particles in a box, still carry a real-space metric, so the real-space construction is the more universal notion.
  • In 2D metals, the impurity-induced metric deformation is long-ranged and oscillates with distance, while in the Chern insulator it decays quickly; the spatial profile of the metric distinguishes metallic from insulating response to disorder.
  • The sign of the Ricci scalar at an impurity site can be flipped by changing the chemical potential, the mass term, or the impurity potential, demonstrating a concrete parameter-controlled route to local curvature engineering without changing the Euler characteristic.
  • For insulators, the Wannier representation recasts the metric as a sum of products of neighboring Wannier functions, linking the real-space quantum metric to established localization measures.

Reading between the lines

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

  • Inference: If the ARPES identification holds, existing high-resolution photoemission data on simple metals could be reprocessed with Eq. (36) to produce real-space quantum metric maps without new experiments, giving a testable prediction against band-structure calculations.
  • Inference: The lattice metric is built from a second-order central-difference scheme, and only its continuum limit is guaranteed to be coordinate-invariant; a natural stress test is to compare the Ricci scalar profile computed with fourth-order finite differences to see whether the sign and oscillation pattern survive.
  • Inference: Because the many-body form of the metric is an expansion of the density-normalized correlation function, cold-atom quantum gas microscopes and momentum-resolved electron scattering, which measure such correlations directly, could provide experimental access to the real-space metric in lattice models.
  • Inference: The single-impurity results keep the Euler characteristic at zero; an extrapolation beyond the paper is that impurity clusters, lines, or disordered regions with net local curvature could produce a nonzero Euler characteristic, potentially realizing a real-space analogue of topology change.
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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. The manuscript proposes a local-state formalism in which the projector onto occupied single-particle states is applied to a position ket, |ψ_σ(x)⟩ = P|x,σ⟩/√n_σ(x), and defines a real-space quantum metric g_μν(x,σ) through the overlap of these states at nearby positions (Eq. 8). The same metric is shown to arise from the normalized action of the annihilation operator on the many-body ground state (Sec. II D). In continuous homogeneous systems the density-averaged metric equals the momentum variance ⟨k_μ k_ν⟩/ℏ² and is expressed through the ARPES spectral function (Eq. 36); the free-electron-gas result is g_μμ = k_F²/[(D+2)ℏ²] (Eq. 39). For lattice systems, the authors introduce a central-difference version of the metric (Eq. B7) and compute g_μν, volume form, Ricci scalar, and Euler characteristic around a single impurity for a 2D metal and a Chern insulator, concluding that disorder can locally curve the real-space manifold.

Significance. The continuous part of the paper is elegant and essentially derivative-free: the ARPES identity (Eq. 36) follows directly from the definition of the metric and the spectral function, with no fitted parameters, and it gives a concrete experimental protocol for measuring a real-space quantum metric. If the lattice construction could be made scheme-independent, the extension to disordered systems would open a genuinely new way to discuss local quantum geometry in solids. At present, however, the load-bearing claims about disorder-induced curvature rest on an uncontrolled discretization choice, and the box example contains an internal scaling contradiction; these issues must be resolved before the paper's broader conclusions can be accepted.

major comments (3)
  1. [§III.A, Eqs. (23)-(25) and Fig. 3] The claim that g_xx scales as n_max^6 contradicts the displayed formula (25). For large n_max, the leading behavior of Eq. (25) is g_xx ≈ (π/L)^2 (1/3) n_max^2 plus O(1) terms, in agreement with the homogeneous D=1 result g = k_F^2/(3ℏ^2) with k_F = π n_max/L. Hence g_xx/n_max^6 cannot saturate at a nonzero constant; it should tend to zero. The prefactor of the 'normalized' vector \hat u in Eq. (23), 2^{D/2}/(√V n_max), is also inconsistent with the normalization of Eq. (21), where the denominator is √(Σ sin²). Please correct the scaling statement, the prefactor, and the associated figure.
  2. [Appendix B, Eqs. (B3)-(B7), and Section IV] The lattice metric is defined by a central-difference stencil rather than by a limiting procedure. For a strong impurity (e.g., U_imp = 10 in Fig. 4), the local state can vary on the scale of the lattice constant, so the second-order expansion in Eq. (B3) has no controlled remainder. Moreover, forward and backward overlaps are discarded because of an imposed inversion symmetry (Eq. (B2)), but in a disordered environment |⟨ψ(x)|ψ(x+a_μ)⟩| and |⟨ψ(x)|ψ(x-a_μ)⟩| are not equal; the symmetry is an assumption, not a consequence. A different second-order stencil, such as a higher-order central difference or a combination of next-nearest-neighbor differences, will in general yield a different g_μν, different Christoffel symbols, and a different Ricci scalar R(x_imp) in Figs. 4-5, potentially with a different sign. Please provide a stencil-independence or convergence test, or clearly state that the computed geometry is defined relative to the chosen finite-difference derivative and is not claimed to be coordinate-invariant.
  3. [Section IV, Eq. (10)] The numerical result χ = 0 'up to numerical precision' is not a meaningful check of the construction. For any smooth metric on a torus, the Gauss-Bonnet theorem forces χ = 0, so the vanishing integral is automatic regardless of the specific discretized metric. The authors should either remove the topological conclusion or explain what property of the discretization the check is designed to test.
minor comments (4)
  1. [§III.D, Eq. (37)] The D=2 line appears to be an incorrect Fourier transform of the Fermi-sea projector; the correct result is (a²k_F/(2πℏ r)) J_1(k_F r/ℏ), not a²(1 - cos(k_F r/ℏ))/(2π r²). The two expressions agree only at leading order in r.
  2. [§II.A] The sentence 'it does not evolve with time' is not correct for a non-eigenstate; |ψ_σ(x)⟩ evolves under e^{-iHt}. The metric remains a meaningful instantaneous quantity, but the time-evolution claim should be removed.
  3. [Section IV, Figs. 4 and 5] The manuscript does not state the system size, boundary conditions, or color scales; please provide these details so the finite-size and convergence aspects of the calculations can be judged.
  4. [Eq. (23)] As noted in the first major comment, \hat u is not normalized as written; even if the scaling issue is resolved, the notation should be corrected to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the real-space metric is defined from local-state overlaps and the ARPES identity is derived, not fitted.

full rationale

The paper's central construction is self-contained. The real-space quantum metric g_mu_nu(x,sigma) is defined in Eq. (8) directly from the overlap of local states |psi_sigma(x)> = P|x,sigma>/sqrt(n_sigma(x)), and the many-body version in Eq. (18) is shown to give the same overlap. The homogeneous-system result in Eq. (30) follows by substituting the Bloch representation into this definition, and the ARPES identity in Eq. (36) is obtained by expressing the density-averaged metric of Eq. (12) in terms of the spectral function A(k,omega) introduced in Eqs. (32)-(35). No parameter is fitted to the quantity being predicted, and no step assumes the momentum-variance result in order to derive it. The lattice implementation in Appendix B, Eqs. (B3)-(B7), uses a central-difference discretization to define the metric on a lattice; this is a methodological choice that may affect the numerical values of curvature but does not reduce the derivation to its own output. Self-citations appear (e.g., Refs. 7, 14, 21, 23, 40) but they are contextual references for related markers and topological properties, not load-bearing justifications of the paper's main claims. The paper even flags its own limitation that atomic-scale variation of the metric may not be experimentally detectable, which further indicates that the claims are not being asserted by construction. Accordingly, no circular step meeting the required evidence threshold is found.

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

No data were fitted: all quantities derive from stated model Hamiltonians. The central definition rests on standard quantum-geometric expansion, noninteracting Fermi-sea ground states, and a specific discrete central-difference prescription whose error is uncontrolled. No new physical entities (particles, forces, dimensions) are introduced.

assumptions (4)
  • standard math Overlap of nearby normalized states expands as 1 - (1/2)g_μνδx^μδx^ν (Provost-Vallee).
    Used in Eq. (8) and throughout; standard background result (Ref. 1).
  • domain assumption Ground state is a noninteracting filled Fermi sea of single-particle eigenstates.
    The projector P in Eq. (3) and the many-body state in Eq. (14) require noninteracting electrons; interactions are deferred to future work (Section V).
  • ad hoc to paper Local states on neighboring lattice sites can be expanded to second order in the lattice constant by central differences.
    Appendix B, Eqs. (B3)-(B7); this is the paper's specific discretization choice and has no controlled error estimate for strongly localized or disordered states.
  • domain assumption For homogeneous continuous solids, Bloch states and a fictitious lattice constant a provide a valid regularization.
    Section III B, Eqs. (26)-(30); standard band-structure regularization, with a cancelling in the final metric.

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Pith. "Pith review of Real space quantum metric of solids." pith.science (2026). https://pith.science/paper/EANIV3EX

@misc{pith2026241110909,
  author       = {Pith},
  title        = {Pith review of: Real space quantum metric of solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EANIV3EX}},
  note         = {Machine review of arXiv:2411.10909}
}
read the original abstract

By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.

Figures

Figures reproduced from arXiv: 2411.10909 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics for two ways to construct quantum geom [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematics of the real space quantum metric [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The spatial profile of quantum metric [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The spatial pattern of the (density-averaged) diagonal [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The Ricci scalar on the impurity site [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

Works this paper leans on

42 extracted references · 8 canonical work pages

  1. [1]

    author author J. P. \ Provost \ and\ author G. Vallee ,\ title title Riemannian structure on manifolds of quantum states , \ https://projecteuclid.org:443/euclid.cmp/1103908308 journal journal Comm. Math. Phys. \ volume 76 ,\ pages 289--301 ( year 1980 ) NoStop

    arXiv 1980

  2. [2]

    author author Y. Q. \ Ma , author S. J. \ Gu , author S. Chen , author H. Fan , \ and\ author W. M. \ Liu ,\ title title The euler number of bloch states manifold and the quantum phases in gapped fermionic systems , \ 10.1209/0295-5075/103/10008 journal journal EPL \ volume 103 ,\ pages 10008 ( year 2013 ) NoStop

    doi:10.1209/0295-5075/103/10008

  3. [3]

    author author Y. Q. \ Ma ,\ title title Quantum distance and the euler number index of the bloch band in a one-dimensional spin model , \ 10.1103/PhysRevE.90.042133 journal journal Phys. Rev. E \ volume 90 ,\ pages 042133 ( year 2014 ) NoStop

    doi:10.1103/physreve.90.042133

  4. [4]

    Yang , author Y

    author author L. Yang , author Y. Q. \ Ma , \ and\ author X. G. \ Li ,\ title title Geometric tensor and the topological characterization of the bloch band in a two-band lattice model , \ https://doi.org/10.1016/j.physb.2014.09.022 journal journal Physica B: Condensed Matter \ volume 456 ,\ pages 359--364 ( year 2015 ) NoStop

    doi:10.1016/j.physb.2014.09.022 2014

  5. [5]

    Pi\'echon , author A

    author author F. Pi\'echon , author A. Raoux , author J. N. \ Fuchs , \ and\ author G. Montambaux ,\ title title Geometric orbital susceptibility: Quantum metric without berry curvature , \ 10.1103/PhysRevB.94.134423 journal journal Phys. Rev. B \ volume 94 ,\ pages 134423 ( year 2016 ) NoStop

    doi:10.1103/physrevb.94.134423

  6. [6]

    Panahiyan , author W

    author author S. Panahiyan , author W. Chen , \ and\ author S. Fritzsche ,\ title title Fidelity susceptibility near topological phase transitions in quantum walks , \ 10.1103/PhysRevB.102.134111 journal journal Phys. Rev. B \ volume 102 ,\ pages 134111 ( year 2020 ) NoStop

    doi:10.1103/physrevb.102.134111

  7. [7]

    von Gersdorff \ and\ author W

    author author G. von Gersdorff \ and\ author W. Chen ,\ title title Measurement of topological order based on metric-curvature correspondence , \ 10.1103/PhysRevB.104.195133 journal journal Phys. Rev. B \ volume 104 ,\ pages 195133 ( year 2021 ) NoStop

    doi:10.1103/physrevb.104.195133

  8. [8]

    Mera , author A

    author author B. Mera , author A. Zhang , \ and\ author N. Goldman ,\ title title Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers , \ 10.21468/SciPostPhys.12.1.018 journal journal SciPost Phys. \ volume 12 ,\ pages 018 ( year 2022 ) NoStop

    doi:10.21468/scipostphys.12.1.018

Show all 42 references
  1. [9]

    author author M. S. M. \ de Sousa \ and\ author W. Chen ,\ title title Opacity of graphene independent of light frequency and polarization due to the topological charge of the dirac points , \ 10.1103/PhysRevB.108.165201 journal journal Phys. Rev. B \ volume 108 ,\ pages 16520...

    doi

  2. [10]

    Tan , author D

    author author X. Tan , author D. W. \ Zhang , author Z. Yang , author J. Chu , author Y. Q. \ Zhu , author D. Li , author X. Yang , author S. Song , author Z. Han , author Z. Li , author Y. Dong , author H. F. \ Yu , author H. Yan , author S. L. \ Zhu , \ and\ author Y. Yu ,\ ...

    doi

  3. [11]

    Ozawa \ and\ author N

    author author T. Ozawa \ and\ author N. Goldman ,\ title title Extracting the quantum metric tensor through periodic driving , \ 10.1103/PhysRevB.97.201117 journal journal Phys. Rev. B \ volume 97 ,\ pages 201117 ( year 2018 ) NoStop

    2018 doi

  4. [12]

    Ghosh , author Y

    author author B. Ghosh , author Y. Onishi , author S. Y. \ Xu , author H. Lin , author L. Fu , \ and\ author A. Bansil ,\ title title Probing quantum geometry through optical conductivity and magnetic circular dichroism , \ https://arxiv.org/abs/2401.09689 \ ( year 2024 ) ,\ h...

    2024 arXiv

  5. [13]

    Komissarov , author T

    author author I. Komissarov , author T. Holder , \ and\ author R. Queiroz ,\ title title The quantum geometric origin of capacitance in insulators , \ 10.1038/s41467-024-48808-x journal journal Nature Communications \ volume 15 ,\ pages 4621 ( year 2024 ) NoStop

    doi

  6. [14]

    author author W. Chen ,\ https://arxiv.org/abs/2409.04893 title Dielectric and optical markers originated from quantum geometry , \ ( year 2024 a ),\ http://arxiv.org/abs/2409.04893 arXiv:2409.04893 [cond-mat.mes-hall] NoStop

    2024 arXiv

  7. [15]

    Marzari \ and\ author D

    author author N. Marzari \ and\ author D. Vanderbilt ,\ title title Maximally localized generalized wannier functions for composite energy bands , \ 10.1103/PhysRevB.56.12847 journal journal Phys. Rev. B \ volume 56 ,\ pages 12847--12865 ( year 1997 ) NoStop

    doi

  8. [16]

    Marzari , author A

    author author N. Marzari , author A. A. \ Mostofi , author J. R. \ Yates , author I. Souza , \ and\ author D. Vanderbilt ,\ title title Maximally localized wannier functions: Theory and applications , \ 10.1103/RevModPhys.84.1419 journal journal Rev. Mod. Phys. \ volume 84 ,\ ...

    doi

  9. [17]

    Souza \ and\ author D

    author author I. Souza \ and\ author D. Vanderbilt ,\ title title Dichroic f -sum rule and the orbital magnetization of crystals , \ 10.1103/PhysRevB.77.054438 journal journal Phys. Rev. B \ volume 77 ,\ pages 054438 ( year 2008 ) NoStop

    doi

  10. [18]

    Matsuura \ and\ author S

    author author S. Matsuura \ and\ author S. Ryu ,\ title title Momentum space metric, nonlocal operator, and topological insulators , \ 10.1103/PhysRevB.82.245113 journal journal Phys. Rev. B \ volume 82 ,\ pages 245113 ( year 2010 ) NoStop

    doi

  11. [19]

    Kolodrubetz , author V

    author author M. Kolodrubetz , author V. Gritsev , \ and\ author A. Polkovnikov ,\ title title Classifying and measuring geometry of a quantum ground state manifold , \ 10.1103/PhysRevB.88.064304 journal journal Phys. Rev. B \ volume 88 ,\ pages 064304 ( year 2013 ) NoStop

    doi

  12. [20]

    Kolodrubetz , author D

    author author M. Kolodrubetz , author D. Sels , author P. Mehta , \ and\ author A. Polkovnikov ,\ title title Geometry and non-adiabatic response in quantum and classical systems , \ https://doi.org/10.1016/j.physrep.2017.07.001 journal journal Phys. Rep. \ volume 697 ,\ pages...

    2017 doi

  13. [21]

    Chen ,\ title title Quantum geometrical properties of topological materials , \ 10.1088/1361-648X/ad8619 journal journal J

    author author W. Chen ,\ title title Quantum geometrical properties of topological materials , \ 10.1088/1361-648X/ad8619 journal journal J. Phys.: Condens. Matter \ volume 37 ,\ pages 025605 ( year 2024 b ) NoStop

    2024 doi

  14. [22]

    Marrazzo \ and\ author R

    author author A. Marrazzo \ and\ author R. Resta ,\ title title Local theory of the insulating state , \ 10.1103/PhysRevLett.122.166602 journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 166602 ( year 2019 ) NoStop

    2019 doi

  15. [23]

    author author M. S. M. \ de Sousa , author A. L. \ Cruz , \ and\ author W. Chen ,\ title title Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker , \ 10.1103/PhysRevB.107.205133 journal journal Phys. Rev. B \ volume 107 ,\ pages 205133 ( ...

    doi

  16. [24]

    Kohn ,\ title title Density functional and density matrix method scaling linearly with the number of atoms , \ 10.1103/PhysRevLett.76.3168 journal journal Phys

    author author W. Kohn ,\ title title Density functional and density matrix method scaling linearly with the number of atoms , \ 10.1103/PhysRevLett.76.3168 journal journal Phys. Rev. Lett. \ volume 76 ,\ pages 3168--3171 ( year 1996 ) NoStop

    doi

  17. [25]

    author author R. Resta ,\ title title Kohn's theory of the insulating state: A quantum-chemistry viewpoint , \ 10.1063/1.2176604 journal journal The Journal of Chemical Physics \ volume 124 ,\ pages 104104 ( year 2006 ) NoStop

    2006 doi

  18. [26]

    Thonhauser \ and\ author D

    author author T. Thonhauser \ and\ author D. Vanderbilt ,\ title title Insulator/chern-insulator transition in the haldane model , \ 10.1103/PhysRevB.74.235111 journal journal Phys. Rev. B \ volume 74 ,\ pages 235111 ( year 2006 ) NoStop

    doi

  19. [27]

    Resta ,\ title title The insulating state of matter: a geometrical theory , \ 10.1140/epjb/e2010-10874-4 journal journal EPJ B \ volume 79 ,\ pages 121--137 ( year 2011 ) NoStop

    author author R. Resta ,\ title title The insulating state of matter: a geometrical theory , \ 10.1140/epjb/e2010-10874-4 journal journal EPJ B \ volume 79 ,\ pages 121--137 ( year 2011 ) NoStop

    2011 doi

  20. [28]

    Bianco \ and\ author R

    author author R. Bianco \ and\ author R. Resta ,\ title title Mapping topological order in coordinate space , \ 10.1103/PhysRevB.84.241106 journal journal Phys. Rev. B \ volume 84 ,\ pages 241106 ( year 2011 ) NoStop

    2011 doi

  21. [29]

    author author T. B. \ Smith , author L. Pullasseri , \ and\ author A. Srivastava ,\ title title Momentum-space gravity from the quantum geometry and entropy of bloch electrons , \ 10.1103/PhysRevResearch.4.013217 journal journal Phys. Rev. Res. \ volume 4 ,\ pages 013217 ( yea...

    doi

  22. [30]

    author author S. M. \ Carroll ,\ @noop title Spacetime and Geometry: An Introduction to General Relativity \ ( publisher Cambridge University Press, Cambridge ,\ year 2019 ) NoStop

    2019

  23. [31]

    Het\'enyi \ and\ author P

    author author B. Het\'enyi \ and\ author P. L\'evay ,\ title title Fluctuations, uncertainty relations, and the geometry of quantum state manifolds , \ 10.1103/PhysRevA.108.032218 journal journal Phys. Rev. A \ volume 108 ,\ pages 032218 ( year 2023 ) NoStop

    doi

  24. [32]

    author author G. D. \ Mahan ,\ @noop title Many-Particle Physics \ ( publisher Springer ,\ year 2000 ) NoStop

    2000

  25. [33]

    author author D. Vanderbilt ,\ @noop title Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators \ ( publisher Cambridge University Press, Cambridge ,\ year 2018 ) NoStop

    2018

  26. [34]

    Chen ,\ title title Absence of equilibrium edge currents in theoretical models of topological insulators , \ 10.1103/PhysRevB.101.195120 journal journal Phys

    author author W. Chen ,\ title title Absence of equilibrium edge currents in theoretical models of topological insulators , \ 10.1103/PhysRevB.101.195120 journal journal Phys. Rev. B \ volume 101 ,\ pages 195120 ( year 2020 ) NoStop

    2020 doi

  27. [35]

    Molignini , author B

    author author P. Molignini , author B. Lapierre , author R. Chitra , \ and\ author W. Chen ,\ title title Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker , \ 10.21468/SciPostPhysCore.6.3.059 journal journal SciPost Phys. Core \ volum...

    doi

  28. [36]

    Prodan , author T

    author author E. Prodan , author T. L. \ Hughes , \ and\ author B. A. \ Bernevig ,\ title title Entanglement spectrum of a disordered topological chern insulator , \ 10.1103/PhysRevLett.105.115501 journal journal Phys. Rev. Lett. \ volume 105 ,\ pages 115501 ( year 2010 ) NoStop

    doi

  29. [37]

    Costa , author G

    author author M. Costa , author G. R. \ Schleder , author M. Buongiorno Nardelli , author C. Lewenkopf , \ and\ author A. Fazzio ,\ title title Toward realistic amorphous topological insulators , \ 10.1021/acs.nanolett.9b03881 journal journal Nano Lett. \ volume 19 ,\ pages 89...

    doi

  30. [38]

    Ul c c akar , author J

    author author L. Ul c c akar , author J. Mravlje , \ and\ author T. Rejec ,\ title title Kibble-zurek behavior in disordered chern insulators , \ 10.1103/PhysRevLett.125.216601 journal journal Phys. Rev. Lett. \ volume 125 ,\ pages 216601 ( year 2020 ) NoStop

    doi

  31. [39]

    d'Ornellas , author R

    author author P. d'Ornellas , author R. Barnett , \ and\ author D. K. K. \ Lee ,\ title title Quantized bulk conductivity as a local chern marker , \ 10.1103/PhysRevB.106.155124 journal journal Phys. Rev. B \ volume 106 ,\ pages 155124 ( year 2022 ) NoStop

    doi

  32. [40]

    author author L. A. \ Oliveira \ and\ author W. Chen ,\ title title Robustness of topological order against disorder , \ 10.1103/PhysRevB.109.094202 journal journal Phys. Rev. B \ volume 109 ,\ pages 094202 ( year 2024 ) NoStop

    2024 doi

  33. [41]

    author author M. V. \ Berry ,\ title title Quantal phase factors accompanying adiabatic changes , \ 10.1098/rspa.1984.0023 journal journal Proc. R. Soc. A \ volume 392 ,\ pages 45--57 ( year 1984 ) NoStop

    1984

  34. [42]

    Xiao , author M

    author author D. Xiao , author M. C. \ Chang , \ and\ author Q. Niu ,\ title title Berry phase effects on electronic properties , \ 10.1103/RevModPhys.82.1959 journal journal Rev. Mod. Phys. \ volume 82 ,\ pages 1959--2007 ( year 2010 ) NoStop

    1959 doi

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