Pith. sign in

REVIEW 5 major objections 4 minor 60 references

Quantum geometry, localization, and topological bounds of spin fluctuations

T0 review · 5 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read Crystal dislocations strongly enhance the real-space quantum metric of magnons, establishing a tunable link between lattice topology, Hilbert-space geometry, and spin-wave localization.

desk verdict Interesting setup, but the main quantitative claim lacks a clean-lattice baseline and the hexatic phase labeling is wrong. read the letter →

arxiv 2512.17454 v1 pith:RJ45QP62 submitted 2025-12-19 cond-mat.mes-hall cond-mat.mtrl-sci

Carlos Saji , Roberto E. Troncoso This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords quantummetricdislocationsmagnonsgeometrictensorBottindextopologicaldefectshexaticphaselocalization
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 sets out to show that crystalline dislocations—missing half-planes of atoms in a hexagonal magnet—act as tunable sources of quantum geometry for magnons, the spin-wave excitations of the ordered magnet. Its central claim is that dislocations strongly enhance the quantum metric, the real part of the quantum geometric tensor that measures how much nearby quantum states differ, and that this enhancement appears in both periodic arrays and randomly disordered arrays of dislocations. The enhancement tracks the localization of magnonic states, accompanies the breakdown of topological order measured by the Bott index, and leads to disorder-induced topological transitions and a hexatic phase at high defect densities. If correct, the quantum metric becomes a bridge between crystalline topology and magnonic behavior, and defects become a knob for controlling geometric and topological properties of spin fluctuations.

What carries the argument

The carrying object is the real-space quantum metric density, g(r) = -(1/2π)([X_μ,P][X_μ,P])_{r,r}, which replaces momentum derivatives with commutators with the position operator and lets quantum geometry be defined without translational symmetry. For bosonic magnons, the projector P is built from the Bogoliubov transformation, and topology is assessed via the Bott index B, a real-space invariant that replaces the Chern number in disordered systems. These tools connect the geometric tensor to localization and to the transition out of topological order.

What would settle it

Compute G and g(r) for a pristine hexagonal lattice using exactly the same real-space commutator formula and area normalization, then compare with dislocation arrays at the same parameters; if the integrated metric does not grow relative to the pristine value as dislocation density increases, the central enhancement claim fails. A complementary check is to run a disordered configuration without topological defects, such as random vacancies, and see whether G stays flat.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the local Hilbert-space geometry of a magnon system is reshaped by topological lattice defects: dislocations generate strain fields that break translational symmetry, and the resulting real-space quantum metric density grows sharply near defect cores, signaling increased overlap of Bloch states and reduced localization length. The integrated metric G rises with dislocation density and peaks at the transition where the Chern or Bott topological invariant jumps to zero, so the geometric enhancement is proposed as the microscopic mechanism linking defect structure, state localization, and topological phase transitions. This is established for a he

Load-bearing premise

The claim that dislocations 'enhance' the quantum metric rests on comparing the metric in defect-laden systems against an implicit baseline; the paper does not benchmark the same real-space formula against a pristine lattice, so part of the apparent growth could stem from the normalization or finite-size effects rather than a genuine geometric response.

Editorial extensions

If this is right

  • Dislocations act as tunable geometric degrees of freedom: increasing their density or rearranging them raises the quantum metric and can drive magnon localization at defect cores.
  • Both ordered and disordered dislocation arrays show the enhancement, so geometry responds to defect topology rather than to translational order by itself.
  • Quantum metric enhancement coincides with the loss of topological protection: the Bott index jumps to zero just where G is largest, identifying geometry as a marker or precursor of the topological transition.
  • In disordered arrays, dislocation-driven geometry expands the accessible topological phase space and enables disorder-induced topological phases before the hexatic phase sets in.
  • The same geometric quantity obeys the topological bound C ≤ V_g ≤ G, so defect-induced metric growth carries implications for quantities bounded by geometry, such as superfluid weight and structure factors.

Reading between the lines

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

  • If the enhancement is physical rather than a normalization artifact, it suggests a general principle: any source of local strain that breaks translation symmetry could be used to engineer quantum geometry—disclinations, grain boundaries, or moiré strain should show analogous metric growth.
  • The real-space quantum metric could serve as a local order parameter for topological transitions in aperiodic systems, complementing global invariants like the Bott index; one could test whether g(r) localizes specifically at defect cores when the transition occurs.
  • A concrete experimental extension is to measure nonlinear thermal Hall or magnon transport across a sample with tuned dislocation density, looking for a peak at the same concentration where the metric enhancement is predicted.
  • Because the metric bounds the superfluid weight in flat bands, dislocation-engineered flat magnon bands might offer a bosonic testbed for geometric pairing phenomena analogous to those studied in twisted graphene.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The manuscript studies how lattice dislocations affect the real-space quantum geometric tensor of magnons in a two-dimensional hexagonal ferromagnet. Using a Bogoliubov–de Gennes Hamiltonian with exchange, Dzyaloshinskii–Moriya, pseudo-dipolar, and anisotropy terms, the authors compute the quantum metric, quantum volume, Chern number, and Bott index for ordered and disordered dislocation arrays. The central claim is that dislocations strongly enhance the quantum metric, that this enhancement tracks magnon localization, and that in disordered arrays it is tied to the breakdown of topological order. The paper also introduces a real-space quantum metric for bosonic systems and classifies high-density disordered states as a hexatic phase.

Significance. If the central claim is established, the paper would identify a new, experimentally relevant tuning knob — crystalline dislocations — for the quantum geometry of magnonic systems, connecting lattice topology, Hilbert-space geometry, and localization. The manuscript usefully imports the real-space quantum metric approach into the bosonic/BdG setting and combines it with the Bott index for disordered systems. However, the headline 'enhancement' claim is currently not benchmarked against the pristine lattice, and the real-space metric is not validated against the momentum-space definition. The disordered-array part also contains internal inconsistencies in the hexatic classification and in the abstract's claim of disorder-induced topological phases. These issues are fixable but require substantial additional analysis.

major comments (5)
  1. [Fig. 2(b) and Fig. 3(e)] The central claim that dislocations 'strongly enhance' the quantum metric is a comparative statement, yet no pristine-lattice baseline is shown under the same normalization and band-window conditions. Fig. 1(c) shows G for the clean lattice as a function of ΔK, but Fig. 2(b) plots the dislocation array separately, and Fig. 3(e) shows G versus ρ with no defect-free reference at all. The apparent growth of G with dislocation density could be a normalization or finite-size effect of Eq. (4), or simply reflect the increasing number of localized defect-core regions contributing to the sum. To support the headline, the authors should overlay pristine and dislocated results on the same axes, state the normalization explicitly, and demonstrate convergence with system size.
  2. [Eq. (4) and 'Magnonic quantum geometry'] The real-space quantum metric in Eq. (4) is introduced by analogy with electronic systems, but no check is reported that in the translationally invariant limit it reduces to the momentum-space integrated metric G = (2π)^{-1}∫BZ Tr[g(k)] d²k for the same band. Because the bosonic projector involves the paraunitary T, the commutator expression is not trivially inherited from the electronic formula. Without this validation, the numerical values of G in Figs. 2 and 3 cannot be interpreted as the quantum metric of the magnon bands. The authors should either derive Eq. (4) from the BdG eigenstates or demonstrate numerically its equivalence to Eq. (3) in the clean limit.
  3. [Abstract and Conclusions] The abstract states that disorder enables 'transitions to disorder-induced topological phases,' but the results in Fig. 3(f) and the Conclusions explicitly describe the opposite: the Bott index jumps to B = 0, i.e., the system becomes topologically trivial at high dislocation density. There is no evidence in the manuscript for a disorder-induced topological phase; the observed behavior is disorder-induced trivialization. The abstract and the corresponding introductory sentence should be revised to match the actual findings, or new data must be provided if a disorder-induced topological phase is intended.
  4. [Hexatic phase, Fig. 3(c)] The identification of a hexatic phase appears to rest on inconsistent or erroneous data. The text states that 'G6(r) ∼ r^{−η}, with η = 2.5 and η = 3.1 for ρ = 0.25 and ρ = 3.1,' but ρ = 3.1 is an impossible concentration and ρ = 0.25 was earlier described as quasi-long-range order. More importantly, in the standard KTHNY theory the hexatic phase has an algebraic decay exponent η_6 ≤ 1/4; values of 2.5 and 3.1 would correspond to exponentially decaying correlations, i.e., an isotropic phase, not a hexatic. The authors must correct the typographical errors, report the fitted exponents for the actual concentrations, and justify the phase classification. This matters because the conclusion that high-density states are 'hexatic' is used to interpret the topological trivialization.
  5. [Fig. 3(e,f) and Methods] The disordered-array results appear to be computed for a single realization per dislocation concentration. For random configurations, the quantum metric and Bott index can fluctuate strongly from sample to sample, especially near a topological transition. The sharp enhancement of G and the discontinuous jump of B in Fig. 3 could be realization-specific rather than generic. The authors should perform an ensemble average over many disorder realizations, report error bars, and specify the system size and the number of realizations used. If averages have been performed, the procedure must be described.
minor comments (4)
  1. [Fig. 3 caption and text] The caption says 'Lx = Ly = XX', which is an unresolved placeholder. The actual system dimensions should be given.
  2. [Hexatic phase paragraph] The sentence 'with η = 2.5 and η = 3.1 for ρ = 0.25 and ρ = 3.1' contains an obvious typo: the second concentration should likely be ρ = 0.6 from Fig. 3(c), not ρ = 3.1. Please correct.
  3. [Spin fluctuations and Magnonic quantum geometry] Minor language issues: 'the nth Bloch band un k, where n = 1,...,N, form a set' should be 'forms a set'; 'positive definiteness' should be 'positive-definiteness'; 'the local quantum metric g(r), Eq. (4), computed for ρ = 0.25 and ρ = 0.6, is displayed (in yellow) at panels (a) and (b)' is awkwardly phrased.
  4. [References] The real-space quantum metric formula in Eq. (4) is attributed to Refs. [48,51-54], but the precise reference for the bosonic/BdG generalization should be specified; the current text says only 'analogous to its definition for periodic systems.'

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central quantities are computed from the Hamiltonian without fitting; self-citations are background model/method references, not load-bearing reductions of the central claim.

full rationale

None of the paper's load-bearing steps reduce to its own inputs. The bulk QGT (Eq. 3), Chern number, integrated quantum metric, and quantum volume are computed directly from the BdG Hamiltonian by paraunitary diagonalization; the model parameters (J, K, F, ΔK, D) are hand-set, not fitted to G, Vg, C, or the Bott index. The real-space metric g(r) (Eq. 4) is taken from external literature on real-space quantum geometry (refs. 48, 51–54), not from the authors' prior work, and although it is interpreted as a localization functional (ref. 29), the paper computes it from the eigenstates of the specific ordered and disordered Hamiltonians rather than assuming the enhancement. The ordered-array part uses the standard momentum-space QGT, so no definitional equivalence between 'dislocation' and 'large G' is hidden in the formula. The self-citations (refs. 34, 35, 55, all involving Troncoso; ref. 35 also Saji) supply the starting spin model, the Bosonic Bott-index method, and earlier identification of dislocation-bound magnon modes; these are independently published inputs, not the current conclusion. The central claim would be better supported by a defect-free baseline under the same Eq. (4) normalization—Figs. 2(b) and 3(e) have no clean-lattice reference—and the manuscript contains reproducibility problems such as 'Lx = Ly = XX' and 'η = 2.5 and η = 3.1 for ρ = 0.25 and ρ = 3.1'. These are evidentiary/audit weaknesses, not circularity: no prediction is fitted, no uniqueness theorem is imported from the authors, and no output quantity is rewritten as an input. Score 1 reflects only the minor self-citations and the missing control comparison, not circular reduction.

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

The paper introduces no new entities or forces. It relies on standard magnon spin-wave theory, a hand-set set of model couplings, and a real-space quantum metric definition borrowed from electronic literature. The most fragile input is the interpretation of the disordered high-density state as hexatic, which is asserted from power-law exponents that do not match the standard hexatic range.

free parameters (5)
  • Anisotropy strength K = 20J (J=1)
    Model parameter chosen by hand to open the topological magnon gap; central results depend on this regime.
  • Pseudo-dipolar coupling F = 7J in Figs. 1-2; 3J and 5J in Fig. 3
    Chosen to realize topological bands and to vary in the disorder scan; not derived from experiment.
  • Dzyaloshinskii-Moriya coupling D = varied (e.g., D=0 in Fig. 1a; nonzero in Fig. 1d)
    Tuned to control the Chern number; free hand-set parameter.
  • Sublattice splitting ΔK = varied; ΔK=5J in Fig. 2
    Tuned to close/open the topological gap and to shift the Chern number; free parameter.
  • Dislocation concentration ρ = 0.1, 0.25, 0.6
    Set to generate Lloyd-algorithm disorder configurations; the claimed transition to trivial topology occurs at high ρ.
assumptions (5)
  • domain assumption Holstein–Primakoff bosonization and truncation at quadratic order (1/S expansion) around a collinear ferromagnetic state.
    Underpins Eq. (1); magnon interactions and quantum corrections are neglected. Validity requires large spin and low temperature, not stated quantitatively.
  • standard math Colpa paraunitary diagonalization preserves bosonic commutation relations (T†ζT=ζ).
    Standard linear algebra for quadratic boson Hamiltonians [39]; used to define the magnon bands used in Eq. (3).
  • domain assumption The momentum-space QGT formula Eq. (3) and real-space metric Eq. (4) are valid for the bosonic BdG problem.
    Eq. (4) is adopted from the electronic real-space metric literature [48,53,54] and called a bosonic analogue; the derivation for the paraunitary case is not shown in the Letter.
  • domain assumption Bott index B computed from the projector P = TΠ(N)ΣzT†Σz is a reliable topological invariant for disordered bosonic magnon bands.
    Method from the authors' prior work [34]; no independent verification is given here.
  • ad hoc to paper Lloyd's-algorithm configurations at ρ=0.25–0.6 represent the hexatic phase.
    The paper infers hexatic order from G6(r) ~ r^{-η} with η=2.5/3.1, values inconsistent with standard KTHNY hexatic exponents (η6 ≤ 1/4). This phase classification is load-bearing for the disorder-topology narrative.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum geometry, localization, and topological bounds of spin fluctuations." pith.science (2026). https://pith.science/paper/RJ45QP62

@misc{pith2026251217454,
  author       = {Pith},
  title        = {Pith review of: Quantum geometry, localization, and topological bounds of spin fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJ45QP62}},
  note         = {Machine review of arXiv:2512.17454}
}
read the original abstract

We study how topological crystalline defects--dislocations--reshape the real-space quantum geometric tensor and act as tunable sources of quantum geometry. We show that dislocations strongly enhance the quantum metric, establishing a direct link between lattice topology and the Hilbert-space geometry of states. We characterize the quantum geometry of topological magnons in ordered arrays of dislocations, demonstrating that defect-induced geometric enhancement controls their localization and topological protection. In disordered arrays, dislocation-driven geometry expands the accessible topological phase space and enables transitions to disorder-induced topological phases. Our results identify the quantum metric as a tunable bridge between crystalline topology, magnonic excitations, and emergent topological matter in aperiodic solid-state and synthetic systems.

Figures

Figures reproduced from arXiv: 2512.17454 by the authors.

Figure 1
Figure 1. (a) Magnonic band spectrum with D = ∆K = 0 with the topological gap. (b) QM density in momentum space Tr[gµν(k)] and Berry curvature Ω(k) for the case (a). Quantum metric G, quantum volume Vg and Chern number C as a function of ∆K (c) and DMI (d) for the lower band. In all plots, the parameters are set to J = 1, K = 20J and F = 7J. Goldstone modes—and at the Dirac points, where the gap is of topological origin. The … view at source ↗
Figure 2
Figure 2. Bulk magnon structure of the magnetic triangular arrays of dislocations, along first BZ loop Γ- [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Geometric and topological properties of magnonic states in the presence of randomly distributed dislocations on a [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 1 canonical work pages

  1. [1]

    J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathe- matical Physics76, 289–301 (1980)

    1980

  2. [2]

    Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121–137 (2011)

    R. Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121–137 (2011)

    2011

  3. [3]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Rev. Mod. Phys.82, 1539 (2010)

    2010

  4. [4]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    1959

  5. [5]

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

    2010

  6. [6]

    Qi and S.-C

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

    2011

  7. [7]

    T¨ orm¨ a, Essay: Where can quantum geometry lead us?, Phys

    P. T¨ orm¨ a, Essay: Where can quantum geometry lead us?, Phys. Rev. Lett.131, 240001 (2023)

    2023

  8. [8]

    Liu, X.-B

    T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, Quantum geometry in condensed matter, National Science Review 12, 10.1093/nsr/nwae334 (2024)

    work page doi:10.1093/nsr/nwae334 2024

Show all 60 references
  1. [9]

    J. Yu, B. A. Bernevig, R. Queiroz, E. Rossi, P. T¨ orm¨ a, and B.-J. Yang, Quantum geometry in quantum materials, npj Quantum Materials10, 10.1038/s41535-025-00801-3 (2025)

    2025 doi

  2. [10]

    Ahn, G.-Y

    J. Ahn, G.-Y. Guo, and N. Nagaosa, Low-frequency di- vergence and quantum geometry of the bulk photovoltaic effect in topological semimetals, Phys. Rev. X10, 041041 (2020)

    2020

  3. [11]

    Watanabe and Y

    H. Watanabe and Y. Yanase, Chiral photocurrent in parity-violating magnet and enhanced response in topo- logical antiferromagnet, Phys. Rev. X11, 011001 (2021)

    2021

  4. [12]

    Gao, Y.-F

    A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen, M. Huang, D. B´ erub´ e, H. Li, C. Tzschaschel, T. Dinh, Z. Sun, S.-C. Ho, S.-W. Lien, B. Singh, K. Watanabe, T. Taniguchi, D. C. Bell, H. Lin, T.-R. Chang, C. R. Du, A. ...

    2023

  5. [13]

    N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao, Quantum-metric-induced nonlinear transport in a topo- logical antiferromagnet, Nature621, 487–492 (2023)

    2023

  6. [14]

    J. Wang, H. Zeng, W. Duan, and H. Huang, Intrin- sic nonlinear hall detection of the n´ eel vector for two- dimensional antiferromagnetic spintronics, Phys. Rev. Lett.131, 056401 (2023)

    2023

  7. [15]

    K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agar- wal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B108, L201405 (2023)

    2023

  8. [16]

    Kaplan, T

    D. Kaplan, T. Holder, and B. Yan, Unification of nonlinear anomalous hall effect and nonreciprocal magnetoresistance in metals by the quantum geometry, Phys. Rev. Lett.132, 026301 (2024)

    2024

  9. [17]

    Jiang, T

    Y. Jiang, T. Holder, and B. Yan, Revealing quantum geometry in nonlinear quantum materials, Reports on Progress in Physics88, 076502 (2025)

    2025

  10. [18]

    Ulrich, J

    Y. Ulrich, J. Mitscherling, L. Classen, and A. P. Schnyder, Quantum geometric origin of the intrinsic nonlinear hall effect (2025), arXiv:2506.17386 [cond-mat.mes-hall]

    2025 arXiv

  11. [19]

    Roy, Band geometry of fractional topological insulators, Phys

    R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B90, 165139 (2014)

    2014

  12. [20]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact landau level description of geometry and interaction in a flatband, Phys. Rev. Lett.127, 246403 (2021)

    2021

  13. [21]

    Estienne, N

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

    2023

  14. [22]

    T¨ orm¨ a, L

    P. T¨ orm¨ a, L. Liang, and S. Peotta, Quantum metric and effective mass of a two-body bound state in a flat band, Phys. Rev. B98, 220511 (2018)

    2018

  15. [23]

    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

  16. [24]

    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 bi- layer graphene, Phys. Rev. B101, 060505 (2020)

    2020

  17. [25]

    Abouelkomsan, K

    A. Abouelkomsan, K. Yang, and E. J. Bergholtz, Quantum metric induced phases in moir´ e materials, Phys. Rev. Res. 5, L012015 (2023)

    2023

  18. [26]

    M. R. Slot, Y. Maximenko, P. M. Haney, S. Kim, D. T. Walkup, E. Strelcov, S. T. Le, E. M. Shih, D. Yildiz, S. R. Blankenship, K. Watanabe, T. Taniguchi, Y. Barlas, N. B. Zhitenev, F. Ghahari, and J. A. Stroscio, A quantum ruler for orbital magnetism in moir´ e quantum matter, ...

    2023

  19. [27]

    Zhang, X.-Z

    Z. Zhang, X.-Z. Li, and W.-Y. He, Orbital magnetization as the origin of the nonlinear hall effect and its implica- tions (2025), arXiv:2503.07960 [cond-mat.mes-hall]

    2025

  20. [28]

    D. Q. To, D. Q. Ho, J. M. O. Zide, L. Gundlach, M. B. Jungfleisch, G. W. Bryant, A. Janotti, and M. F. Doty, Quantum geometric origins of the orbital degrees of free- dom of hybrid bosonic quasiparticles in magnetic systems (2025), arXiv:2509.25635 [cond-mat.mes-hall]

    2025

  21. [29]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized wannier functions for composite energy bands, Phys. Rev. B56, 12847 (1997)

    1997

  22. [30]

    P. A. McClarty, Topological magnons: A review, Annual Review of Condensed Matter Physics13, 171–190 (2022)

    2022

  23. [31]

    F. Zhuo, J. Kang, A. Manchon, and Z. Cheng, Topologi- cal phases in magnonics, Advanced Physics Research4, 10.1002/apxr.202300054 (2023)

    2023 doi

  24. [32]

    Onoseet al., Observation of the magnon thermal hall effect, Science329, 297 (2010)

    Y. Onoseet al., Observation of the magnon thermal hall effect, Science329, 297 (2010)

    2010

  25. [33]

    Sch¨ utte and M

    C. Sch¨ utte and M. Garst, Magnon–skyrmion scattering in chiral magnets, Physical Review B90, 094423 (2014)

    2014

  26. [34]

    X. S. Wang, A. Brataas, and R. E. Troncoso, Bosonic bott index and disorder-induced topological transitions of magnons, Physical Review Letters125, 217202 (2020)

    2020

  27. [35]

    C. Saji, N. Vidal-Silva, A. S. Nunez, and R. E. Troncoso, Topological magnonic dislocation modes, Physical Review B111, 165146 (2025)

    2025

  28. [36]

    Y. Ran, Y. Zhang, and A. Vishwanath, One-dimensional topologically protected modes in topological insulators with lattice dislocations, Nature Physics5, 298–303 (2009)

    2009

  29. [37]

    X. S. Wang and X. R. Wang, Anomalous magnon nernst ef- fect of topological magnonic materials, Journal of Physics D: Applied Physics51, 194001 (2018)

    2018

  30. [38]

    Holstein and H

    T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940)

    1940

  31. [39]

    Colpa, Diagonalization of the quadratic boson hamilto- nian, Phys

    J. Colpa, Diagonalization of the quadratic boson hamilto- nian, Phys. A: Stat. Mech. Appl.93, 327 (1978)

    1978

  32. [40]

    Peotta and P

    S. Peotta and P. T¨ orm¨ a, Superfluidity in topologi- cally nontrivial flat bands, Nature Communications6, 10.1038/ncomms9944 (2015)

    2015 doi

  33. [41]

    Onishi and L

    Y. Onishi and L. Fu, Topological bound on the structure factor, Phys. Rev. Lett.133, 206602 (2024)

    2024

  34. [42]

    Onishi and L

    Y. Onishi and L. Fu, Fundamental bound on topological gap, Phys. Rev. X14, 011052 (2024)

    2024

  35. [43]

    Lloyd, Least squares quantization in pcm, IEEE Trans- actions on Information Theory28, 129–137 (1982)

    S. Lloyd, Least squares quantization in pcm, IEEE Trans- actions on Information Theory28, 129–137 (1982)

    1982

  36. [44]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics6, 1181–1203 (1973)

    1973

  37. [45]

    A. P. Young, Melting and the vector coulomb gas in two dimensions, Phys. Rev. B19, 1855 (1979)

    1979

  38. [46]

    B. I. Halperin and D. R. Nelson, Theory of two- dimensional melting, Phys. Rev. Lett.41, 121 (1978)

    1978

  39. [47]

    J. M. Kosterlitz, Nobel lecture: Topological defects and phase transitions, Rev. Mod. Phys.89, 040501 (2017)

    2017

  40. [48]

    L. A. Oliveira and W. Chen, Real space quantum metric of solids, Phys. Rev. B111, 094202 (2025)

    2025

  41. [49]

    J. Wang, Y. Chen, and H. Huang, Quantum metric en- hancement and hierarchical scaling in one-dimensional quasiperiodic systems (2025), arXiv:2507.04213 [cond- mat.mes-hall]

    2025 arXiv

  42. [50]

    J. Sun, H. Guo, and B.-J. Yang, Geometric superfluid weight in quasicrystals (2025), arXiv:2507.20540 [cond- mat.supr-con]

    2025

  43. [51]

    Marsal, P

    Q. Marsal, P. Holmvall, and A. M. Black-Schaffer, Quan- tum metric and localization in a quasicrystal (2025)

    2025

  44. [52]

    J. M. Romeral, A. W. Cummings, and S. Roche, Scaling of the integrated quantum metric in disordered topological phases, Phys. Rev. B111, 134201 (2025)

    2025

  45. [53]

    Marrazzo and R

    A. Marrazzo and R. Resta, Local theory of the insulating state, Phys. Rev. Lett.122, 166602 (2019)

    2019

  46. [54]

    M. S. M. de Sousa, A. L. Cruz, and W. Chen, Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker, Phys. Rev. B107, 205133 (2023)

    2023

  47. [55]

    H. D. Rosales and R. E. Troncoso, Robustness of topo- logical magnons in disordered arrays of skyrmions, Phys. Rev. B110, 134435 (2024)

    2024

  48. [56]

    T. A. Loring and M. B. Hastings, Disordered topological insulators via c*-algebras, EPL (Europhysics Letters)92, 67004 (2010)

    2010

  49. [57]

    M. Kang, S. Kim, Y. Qian, P. M. Neves, L. Ye, J. Jung, D. Puntel, F. Mazzola, S. Fang, C. Jozwiak, A. Bostwick, E. Rotenberg, J. Fuji, I. Vobornik, J.-H. Park, J. G. Checkelsky, B.-J. Yang, and R. Comin, Measurements of the quantum geometric tensor in solids, Nature Physics 21...

    2024

  50. [58]

    Varshney, R

    H. Varshney, R. Mukherjee, A. Kundu, and A. Agarwal, Intrinsic nonlinear thermal hall transport of magnons: A quantum kinetic theory approach, Phys. Rev. B108, 165412 (2023)

    2023

  51. [59]

    Vi˜ nas Bostr¨ om, T

    E. Vi˜ nas Bostr¨ om, T. S. Parvini, J. W. McIver, A. Rubio, S. V. Kusminskiy, and M. A. Sentef, Direct optical probe of magnon topology in two-dimensional quantum magnets, Phys. Rev. Lett.130, 026701 (2023)

    2023

  52. [60]

    Koller, V

    E. Koller, V. Leeb, N. B. Perkins, and J. Knolle, Ra- man circular dichroism and quantum geometry of chiral quantum spin liquids (2025)

    2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.