Pith. sign in

REVIEW 3 major objections 6 minor 45 references

Phase transition of Kitaev spin liquid described by quantum geometric tensor

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The quantum geometric tensor marks Kitaev spin-liquid phase transitions.

desk verdict A correct, careful re-derivation with a useful susceptibility connection, but the 'robustness' claim is tested only within the vortex-free effective model, not the actual Kitaev spin liquid. read the letter →

arxiv 2412.20889 v1 pith:L6T2GSQS submitted 2024-12-30 cond-mat.str-el

Meng-Meng Lu , Zheng-Chuan Wang This is my paper · ORCID
classification cond-mat.str-el
keywords KitaevspinliquidBerrycurvatureFubini-StudymetricquantumgeometrictensortopologicalphasetransitionUhlmannBuresmagneticsusceptibility
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

This paper argues that the quantum geometric tensor—a mathematical object that measures how a quantum state changes when parameters are varied—carries a measurable signature of the topological phase transition in the Kitaev spin liquid. Working in the effective magnetic field space of the fermionic representation, the authors show that the xy-component of the total Berry curvature jumps exactly along the phase transition lines of the Kitaev phase diagram, and that this jump is insensitive to the strength of the local three-spin perturbation. They further relate this curvature to the derivative of the effective magnetic susceptibility at zero frequency, giving a physical observable that could be probed, and show that the zz-component of the Fubini-Study metric peaks at the triple point of the phase diagram. The same geometric quantities are generalized to finite temperature through the Uhlmann curvature and the Bures metric, which display extrema when tuning across phase boundaries.

What carries the argument

The central object is the quantum geometric tensor $Q_{\alpha\beta}=\langle\partial_\alpha\psi|(1-P)|\partial_\beta\psi\rangle = g_{\alpha\beta} - \frac{i}{2}F_{\alpha\beta}$, whose real part is the Fubini-Study metric and whose imaginary part is the Berry curvature, evaluated in the space of the effective magnetic field $\mathbf{B}_k=(\alpha_k,-\beta_k,\xi_k)$ of the fermionic two-band Hamiltonian. The calculation chain runs: Jordan-Wigner transformation plus the perturbative three-spin term $H'=-K\sum_p\sum_{\ell=1}^4 P_p^{(\ell)}$ reduces the Kitaev model in a field to $H_k=\boldsymbol{\sigma}\cdot\mathbf{B}_k$; the spectral Berry curvature becomes $F_{k,\alpha\beta}=\epsilon_{\alpha\beta\gamma}B_{k,\gamma}/(2\varepsilon_k^3)$; and the total curvature is the Brillouin-zone integral. The paper's key identity connects this geometric quantity to a physical response: $F_{\alpha\beta} = -i\,d\chi_{\alpha\beta}(\omega)/d\omega\big|_{\omega=0}$, the zero-frequency derivative of the effective spin susceptibility. The same tensor, extended to finite temperature via the Uhlmann curvature and the Bures metric, produces the extrema and peaks used as phase-transition markers.

What would settle it

The central claim would be settled by a full non-perturbative calculation—exact diagonalization or tensor network—of the Kitaev model in a magnetic field at parameters such as $J_z=0$ and $K=1$: if the xy-component of the Berry curvature of the actual ground state does not jump exactly along the phase boundaries obtained from the complete Hamiltonian, the geometric marker is an artifact of the perturbative and vortex-free truncations. Alternatively, the predicted zero-frequency susceptibility derivative can be checked in candidate Kitaev materials, where the transition should appear as a singular feature in the low-frequency magnetic response.

Watch

Extended reading notes

Core claim

The paper claims that the quantum geometric tensor of the Kitaev spin liquid in a magnetic field is a reliable marker of its topological phase transitions. Working with the Jordan-Wigner transformed fermionic Hamiltonian $H_k=\boldsymbol{\sigma}\cdot\mathbf{B}_k$, with effective field $\mathbf{B}_k=(\alpha_k,-\beta_k,\xi_k)$ and $\alpha_k=4K(\sin k_x-\sin k_y)$, the authors compute the total Berry curvature $F_{xy}$ as an integral over the Brillouin zone and find that it jumps exactly along the three $A_i$–$B$ phase boundaries of the Kitaev phase diagram when $J_x+J_y+J_z=1$. They show that this jump is unchanged when the three-spin coupling $K$ is varied, and interpret that as robustness against local perturbation. They also derive the relation $F_{\alpha\beta}=-i\,d\chi_{\alpha\beta}(\omega)/d\omega\big|_{\omega=0}$, linking the curvature to the derivative of the effective magnetic susceptibility. In the same geometric framework, the zz-component of the Fubini-Study metric peaks at the crossing point of the three phases, and the finite-temperature Uhlmann curvature shows an extremum as $J_x$ is tuned from the $A_z$ phase into the $B$ phase.

Load-bearing premise

The results stand on the assumption that the perturbative three-spin Hamiltonian together with the Jordan-Wigner transformation restricted to the vortex-free sector faithfully represents the Kitaev spin liquid in a magnetic field across the whole scanned parameter range, with $K$ taken as large as 1 and $J_z$ as low as 0.

Editorial extensions

If this is right

  • The jump of $F_{xy}$ along each $A_i$–$B$ line supplies a geometric order parameter for the Kitaev topological transition, computable directly from the fermionic band structure.
  • Because $F_{\alpha\beta}=-i\,d\chi_{\alpha\beta}(\omega)/d\omega\big|_{\omega=0}$, the transition should appear as a singular feature in the low-frequency magnetic susceptibility of the quasifermion system, making the geometric marker experimentally accessible in principle.
  • The insensitivity of the curvature jump to the coupling $K$ means the geometric signature survives local perturbations, consistent with the robustness expected of topological order.
  • The zz-component of the Fubini-Study metric, peaking at the triple point of the phase diagram, provides a second independent geometric signature that could locate multicritical points.
  • The finite-temperature Bures metric contains an added Fisher–Rao term that vanishes at zero temperature, so the geometric description continuously interpolates between the mixed-state and pure-state regimes.

Reading between the lines

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

  • The identification of $F_{\alpha\beta}$ with the zero-frequency susceptibility derivative suggests a general recipe: any two-band model that can be cast as a spin in an effective magnetic field will show phase boundaries as jumps of the geometric tensor components, so the Kitaev case may be one instance of a broader principle.
  • Only the three-spin coupling $K$ was varied in the robustness test; adding other symmetry-allowed perturbations (Heisenberg or $\Gamma$ terms) would test whether the geometric marker survives realistic deviations from the pure Kitaev model.
  • The predicted susceptibility singularity could be looked for in candidate Kitaev materials such as $\alpha$-RuCl$_3$ under an applied in-plane magnetic field, where the field drives the system across the phase boundary; a nontrivial extension because the effective model assumes a clean vortex-free sector.
  • The peak of $g_{zz}$ at the triple point may be a general indicator of multicriticality in exactly solvable spin liquids, worth testing in other models with three-phase coexistence.
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

3 major / 6 minor

Summary. The paper studies the Kitaev honeycomb model in an external magnetic field and proposes a geometric signature of its topological phase transitions. Using the Jordan-Wigner transformation and a perturbative three-spin term, the authors obtain a two-band free-fermion Hamiltonian of the form H_k = σ·B_k, with the effective magnetic field B_k serving as the parameter space. They compute the Berry curvature, relate it to the derivative of an effective spin susceptibility, compute the mean Uhlmann curvature at finite temperature, and derive the quantum geometric tensor and the Fubini-Study metric. The central numerical claim is that the xy-component of the total Berry curvature jumps along the three phase-transition lines of the Kitaev phase diagram and that the jump location is unchanged as the three-spin coupling K is varied; the zz-component of the Fubini-Study metric is reported to peak at the three-phase crossing point. Analytic relations are also derived connecting the Uhlmann curvature to the spectral function and the susceptibility.

Significance. If the central claim is correct and can be shown to hold in the physical spin model, the paper would provide a concrete geometric marker for the topological phase transition in a Kitaev spin liquid, going beyond the pure-state Berry curvature by relating it to a measurable susceptibility and by extending the analysis to mixed states. The explicit link between the Berry curvature and dχ/dω at ω=0 (Eq. (20)) is a useful identity within the two-band model, though it is a mathematical relation rather than a new physical effect. The paper does not supply code or machine-checked derivations, and the parameter scan is not accompanied by convergence tests; nevertheless, the mapping from the phase diagram to a geometric quantity is conceptually appealing and the numerical correspondence in Figs. 3-5 is plausible if the effective model is trusted.

major comments (3)
  1. [Sec. II A / II C (Eq. (4), Fig. 5)] The entire numerical analysis is carried out in the vortex-free sector of the effective three-spin Hamiltonian H' = -K Σ_p Σ_ℓ P_p^(ℓ), which is only the third-order perturbative replacement of the Zeeman term (Eq. (3)). The scan 0 ≤ K ≤ 1 with J_z as low as 0 is far outside the regime where this perturbative mapping is controlled, and the paper does not justify that higher-order corrections and vortex excitations are negligible or that the physical ground state stays in the vortex-free sector for these parameters. The central robustness claim is therefore demonstrated only for the auxiliary free-fermion model; to establish the claim for the Kitaev spin liquid, the authors should either restrict the parameter range to the perturbative regime or compare the gap-closing lines and the xy-component of the Berry curvature with exact diagonalization or tensor-network results of the original spin Hamiltonian.
  2. [Sec. II C / Conclusion] The statement that the Berry-curvature behavior 'will not be influenced by local perturbation' overstates what is shown. Varying K changes the coefficient of the three-spin term that is already present in the effective Hamiltonian, and the observed K-independence of the jumping lines is a property of this particular fermionic two-band model. It does not imply stability against generic local perturbations such as Heisenberg or Γ exchange, and the conclusion should be rephrased accordingly; a concrete test would be to add such a perturbation and check whether the jumping pattern survives.
  3. [Sec. IV A / Appendix C (Eqs. (49), (85))] The explicit components of the quantum geometric tensor in Eq. (85) are garbled: the spherical-coordinate and Cartesian forms are mixed with misplaced denominators (e.g., in the lines for Q_{k,xx}, Q_{k,xy}, and Q_{k,zz}), making the formulas unverifiable. In addition, Eq. (49) defines φ_k = arctan(-ξ_k/α_k), which contradicts Eq. (8) where B_y/B_x = -β_k/α_k; this error propagates into the eigenstates (51)-(52) and the QGT components. Since the Fubini-Study metric results in Fig. 7 rest on this calculation, these expressions must be corrected and the numerical integration made reproducible.
minor comments (6)
  1. [Throughout] There are numerous typos and awkward sentences (e.g., 'Andnerson', 'vertox-free', 'mangetic field'), and the English needs careful editing.
  2. [Sec. IV B (Eq. (56))] The notation {,} is introduced as a 'fermionic commutator' but the defining equation ∂α ρ = (1/2){ρ, L_α} requires the anticommutator; please correct the terminology.
  3. [Sec. IV B (Eq. (67))] The reference 'Eq.(??)' should be a specific equation (likely Eq. (63) or (65)), and the reduction from Eq. (66) to Eq. (67) is not shown; please add the missing step.
  4. [Sec. III (Eq. (41))] The expression U_{k,αβ} = -i tanh²(ε_k/T) tanh(ε_k/2T) dχ_{k,αβ}/dω absorbs the imaginary unit in an unusual way; the sign and branch conventions should be stated explicitly.
  5. [Fig. 6a] The peak at T=0, K=0 is attributed to time-reversal symmetry; a sentence explaining why the Uhlmann curvature vanishes at K=0 would be helpful.
  6. [References] Ref. 36 appears unrelated to the discussed metric literature; please check all references for relevance and completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Berry-curvature and metric signatures are computed, not fitted, from the same two-band Hamiltonian that defines the Kitaev phase diagram.

full rationale

The central chain from the effective three-spin Hamiltonian (Eq. 4) through the Jordan-Wigner form H_k = sigma·B_k (Eq. 7) to the spectral Berry curvature (Eq. 24), the total F_xy (Eq. 26), and the comparison with the Kitaev phase diagram is fully self-contained. Equation (20), F_{k,alpha beta} = -i d chi_{k,alpha beta}(omega)/d omega at omega = 0, is derived in the text from the Lehmann representation (Eqs. 15-19) and is an identity for the two-band model, not a fitted relation. The jump lines of F_xy occur where |B_k| = 0; the gap-closing wavevectors are the high-symmetry points (0,pi), (pi,0), (pi,pi), where sin k_x = sin k_y = 0, so alpha_k = 4K(sin k_x - sin k_y) = 0 and the critical lines J_x = 0.5, J_y = 0.5, J_z = 0.5 are independent of K by direct inspection of Eq. (28), not by a self-citation or adjustable parameter. The only self-citation (ref. 41, Z.C. Wang on a 'sub-geometric phase') appears in a passing remark and is not load-bearing. The unresolved 'Eq.(??)' in Sec. IV B is a typographical omission, not a circular step. The physical caveat that the vortex-free effective Hamiltonian may not represent the full Kitaev spin model for large K is a modeling-validity concern and is outside the definition of circularity. No quantity is fitted to the target result, so the central claim is a self-consistent computation rather than a circular derivation.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard algebraic mappings and linear response theory, with one domain assumption (vortex-free sector and perturbative validity) and one modeling choice (effective-field parameter space). No entities are invented. The free parameter K is a hand-chosen coupling strength, not fitted to data.

free parameters (1)
  • three-spin coupling K = 0.5 default; varied as 0.3, 0.4 in Fig. 5
    K is the strength of the perturbative three-spin term. The paper treats it as an independent parameter up to 1 and varies it to claim robustness, although the perturbative derivation assumes small K.
assumptions (6)
  • standard math Jordan-Wigner transformation maps the spin Kitaev model to a quadratic fermionic Hamiltonian (Eqs. (6)-(9)).
    Well-established exact mapping for the honeycomb model.
  • domain assumption The external magnetic field is treated perturbatively, yielding the three-spin term H' = -K Σ_p Σ_{ℓ=1}^4 P_p^{(ℓ)} (Eqs. (4)-(5)).
    Valid only for small magnetic field; the paper does not specify the field amplitude and uses K up to 1.
  • domain assumption The ground state lies in the vortex-free sector, so the total ground state is a product over k of single-particle states (Appendix B).
    True for the pure Kitaev model; the paper assumes it remains valid for the perturbed model in the scanned regime.
  • standard math Linear response theory (Lehmann representation) applies to the k-space spin-density operator, giving the susceptibility relation Eqs. (18)-(20).
    Standard Kubo-formula machinery; the derivation is presented in Appendix A.
  • standard math The Uhlmann curvature formula Eq. (34) and the Bures metric formula Eq. (60) are taken from refs. 40, 44, and 46.
    Borrowed from cited reviews and prior work on mixed-state geometry.
  • ad hoc to paper The parameter space for the quantum geometric tensor is chosen as the components of the effective magnetic field B_k (Eq. (8)).
    This is a modeling choice; the resulting curvature is not the response to the physical magnetic field.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Phase transition of Kitaev spin liquid described by quantum geometric tensor." pith.science (2026). https://pith.science/paper/L6T2GSQS

@misc{pith2026241220889,
  author       = {Pith},
  title        = {Pith review of: Phase transition of Kitaev spin liquid described by quantum geometric tensor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6T2GSQS}},
  note         = {Machine review of arXiv:2412.20889}
}
abstract

Weinvestigate the topological phase transition of Kitaev spin liquid in an external magnetic field by calculating the Berry curvature and the Fubini-Study metric. Employing Jordan-Wigner transformation and effective perturbative theory to transform the Hamiltonian into fermionic quadratic form, the Berry curvature is calculated by choosing the effective magnetic field as the parameter, and we find that the xy-component of the Berry curvature has the same behavior around the critical lines with the phase diagram and the behavior of Berry curvature around the critical line will not be influenced by local perturbation, i.e. it has the robustness against the local perturbation. Especially, we relate the Berry curvature with the derivative of effective magnetic susceptibility which can be regarded as the signature of topological phase transition besides, we related the second nonlinear susceptibility with the non-Abelian Berry connection. Then we analytically calculate the generalized Berry curvature in the mixed state called mean Uhlmann curvature which can be related with the spectral function, the curves that mean Uhlmanncurvaturechangingtemperaturewithdifferentcouplingconstant reveal that it will have an extrema when adjust $J_x$ from $A$ phase to $B$ phase. At last we analytically calculate the quantum geometric tensor in the effective magnetic field space whose imaginary part is the Berry curvature and real part is the Fubini-Study metric, and we find that the zz-component of Fubini-Study metric and the phase diagram are highly correlated which will peak at the cross point of three phases, then the Fubini-Study metric is extended to the finite temperature with arising an additional term called Fisher-Rao metric caused by mixed state.

Figures

Figures reproduced from arXiv: 2412.20889 by the authors.

Figure 1
Figure 1. FIG. 1: Honeycomb model. In Fig.(b), we can see that the red, the green and black line are the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The phase transition lines of Kitaev model [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The xy-component of BC with constrain [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The top view of Fig [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: BC with different [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: xy-component of mean Ulhmann curvature [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Different components of Fubini-Study metric with [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 19 canonical work pages

  1. [1]

    P. W. Anderson, 1973, Materials Research Bulletin, 8(2), 153-160. DOI: 10.1016/0025-5408(73)90167-0 https://doi.org/10.1016/0025-5408(73)90167-0

    doi:10.1016/0025-5408(73)90167-0 1973

  2. [2]

    DOI: 10.1103/RevModPhys.89.041003 https://doi.org/10.1103/RevModPhys.89.041003

    Yi Zhou et al., 2017, Reviews of Modern Physics, 89, 041003. DOI: 10.1103/RevModPhys.89.041003 https://doi.org/10.1103/RevModPhys.89.041003

    doi:10.1103/revmodphys.89.041003 2017

  3. [3]

    Balents, 2010, Nature (London), 464, 199

    L. Balents, 2010, Nature (London), 464, 199. DOI: 10.1038/nature08917 https://doi.org/10.1038/nature08917

    doi:10.1038/nature08917 2010

  4. [4]

    Kitaev, 2006, Annals of Physics, 321, 2

    A. Kitaev, 2006, Annals of Physics, 321, 2. DOI: 10.1016/j.aop.2005.10.005 https://doi.org/10.1016/j.aop.2005.10.005

    doi:10.1016/j.aop.2005.10.005 2006

  5. [5]

    X. G. Wen, 1989, Phys. Rev. B, 39, 7223. DOI: 10.1103/PhysRevB.39.7223 https://doi.org/10.1103/PhysRevB.39.7223

    work page doi:10.1103/physrevb.39.7223 1989

  6. [6]

    X. G. Wen, 1991, Phys. Rev. B, 44, 2664. DOI: 10.1103/PhysRevB.44.2664 https://doi.org/10.1103/PhysRevB.44.2664

    doi:10.1103/physrevb.44.2664 1991

  7. [7]

    X. G. Wen, 2002, Phys. Rev. B, 65, 165113. DOI: 10.1103/PhysRevB.65.165113 https://doi.org/10.1103/PhysRevB.65.165113

    doi:10.1103/physrevb.65.165113 2002

  8. [8]

    P. W. Anderson, 1987, Science, 235, 1196. DOI: 10.1126/science.235.4793.1196 https://doi.org/10.1126/science.235.4793.1196

    arXiv 1987

Show all 45 references
  1. [9]

    R. B. Laughlin, 1983, Phys. Rev. Lett., 50, 1395. DOI: 10.1103/PhysRevLett.50.1395 https://doi.org/10.1103/PhysRevLett.50.1395

    1983 doi

  2. [10]

    X. G. Wen, F. Wilczek, and A. Zee, 1989, Phys. Rev. B, 39, 11413. DOI: 10.1103/PhysRevB.39.11413 https://doi.org/10.1103/PhysRevB.39.11413

    1989 doi

  3. [11]

    Kalmeyer and R

    V. Kalmeyer and R. B. Laughlin, 1987, Phys. Rev. Lett., 59, 2095. DOI: 10.1103/PhysRevLett.59.2095 https://doi.org/10.1103/PhysRevLett.59.2095

    1987 doi

  4. [12]

    Kitaev, 2003, Annals of Physics, 303, 2

    A. Kitaev, 2003, Annals of Physics, 303, 2. DOI: 10.1016/S0003-4916(02)00018-0 https://doi.org/10.1016/S0003-4916(02)00018-0

    2003 doi

  5. [13]

    Vatshal Srivastav, Utso Bhattacharya, and Amit Dutta, Phys. Rev. B, 100, 144203 (2019). DOI: 10.1103/PhysRevB.100.144203 https://doi.org/10.1103/PhysRevB.100.144203

    2019 doi

  6. [14]

    Rong-Yang Sun, Tomonori Shirakawa, and Seiji Yunoki, Phys. Rev. B, 107, L041109 (2023). DOI: 10.1103/PhysRevB.107.L041109 https://doi.org/10.1103/PhysRevB.107.L041109

    2023 doi

  7. [15]

    Julien Vidal, Sébastien Dusuel, and Kai Phillip Schmidt, Phys. Rev. B, 79, 033109 (2009). DOI: 10.1103/PhysRevB.79.033109 https://doi.org/10.1103/PhysRevB.79.033109

    2009 doi

  8. [16]

    Claudio Castelnovo and Claudio Chamon, Phys. Rev. B, 77, 054433 (2008). DOI: 10.1103/PhysRevB.77.054433 https://doi.org/10.1103/PhysRevB.77.054433

    2008 doi

  9. [17]

    K. J. Satzinger,et al., Science, 374, 1237 (2021). DOI: 10.1126/science.abf1910 https://www.science.org/doi/10.1126/science.abi8378

    2021 doi

  10. [18]

    Semeghini et al., Science, 374, 1242 (2021)

    G. Semeghini et al., Science, 374, 1242 (2021). DOI: 10.1126/science.abm1063 https://www.science.org/doi/10.1126/science.abi8794

    2021 doi

  11. [19]

    Kazem Negahdari and Abdollah Langari, Phys

    M. Kazem Negahdari and Abdollah Langari, Phys. Rev. B, 107, 134404 (2023). DOI: 10.1103/PhysRevB.107.134404 https://doi.org/10.1103/PhysRevB.107.134404

    2023 doi

  12. [20]

    Lahtinen et al., 2008, Ann

    J. Lahtinen et al., 2008, Ann. Phys., 323, 2286. DOI: 10.1016/j.aop.2007.10.001 https://doi.org/10.1016/j.aop.2007.10.001

    2008 doi

  13. [21]

    Gegenwart, Phys

    Yogesh Singh and P. Gegenwart, Phys. Rev. B, 82, 064412 (2010). DOI: 10.1103/PhysRevB.82.064412 https://doi.org/10.1103/PhysRevB.82.064412

    2010 doi

  14. [22]

    Manni, J

    Yogesh Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett., 108, 127203 (2021). DOI: 10.1103/PhysRevLett.108.127203 https://doi.org/10.1103/PhysRevLett.108.127203

    2021 doi

  15. [23]

    K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. Vijay Shankar, Y. F. Hu, K. S. Burch, Hae-Young Kee, and Young-June Kim,Phys. Rev. B, 90, 041112(R) (2014). DOI: 10.1103/PhysRevB.90.041112 https://doi.org/10.1103/PhysRevB.90.041112

    2014 doi

  16. [24]

    Francesco Bascone et al., 2019, J. Stat. Mech., 094002. DOI: 10.1088/1742-5468/ab3f0d https://doi.org/10.1088/1742-5468/ab3f0d

    2019 doi

  17. [25]

    Francesco Bascone et al., 2019, Phys. Rev. B 99, 205155. DOI: https://doi.org/10.1103/PhysRevB.99.205155 https://doi.org/10.1103/PhysRevB.99.205155

    2019 doi

  18. [26]

    Han-Dong Chen and Zohar Nussinov, 2008, J. Phys. A: Math. Theor., 41, 075001. DOI: 10.1088/1751-8113/41/7/075001 https://doi.org/10.1088/1751-8113/41/7/075001

    2008 doi

  19. [27]

    Chen and J

    H.-D. Chen and J. Hu, 2007, Phys. Rev. B, 76, 193101. DOI: 10.1103/PhysRevB.76.193101 https://doi.org/10.1103/PhysRevB.76.193101

    2007 doi

  20. [28]

    Feng, G.-M

    X.-Y. Feng, G.-M. Zhang, and T. Xiang, 2007, Phys. Rev. Lett., 98, 087204. DOI: 10.1103/PhysRevLett.98.087204 https://doi.org/10.1103/PhysRevLett.98.087204

    2007 doi

  21. [29]

    Simon Michel and Michael Potthoff, Phys. Rev. B, 106, 235423 (2022). DOI: 10.1103/PhysRevB.106.235423 https://doi.org/10.1103/PhysRevB.106.235423

    2022 doi

  22. [30]

    M. V. Berry, 1984, Proc. R. Soc. London, Ser. A, 392, 45. DOI: 10.1098/rspa.1984.0023 https://doi.org/10.1098/rspa.1984.0023

    1984

  23. [31]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, 2010, Rev. Mod. Phys., 82, 1959. DOI: 10.1103/RevModPhys.82.1959 https://doi.org/10.1103/RevModPhys.82.1959

    2010 doi

  24. [32]

    DOI: 10.1038/s41598-018-27362-9 https://doi.org/10.1038/s41598-018-27362-9

    Angelo Carollo, Bernardo Spagnolo and Davide Valenti, Scientific Reports, 8, 9852 (2018). DOI: 10.1038/s41598-018-27362-9 https://doi.org/10.1038/s41598-018-27362-9

    2018 doi

  25. [33]

    DOI: 10.1038/s41598-019-45546-9 https://doi.org/10.1038/s41598-019-45546-9

    Luca Leonforte, Davide Valenti, Bernardo Spagnolo and Angelo Carollo, Scientific Reports, 9, 9106 (2019). DOI: 10.1038/s41598-019-45546-9 https://doi.org/10.1038/s41598-019-45546-9

    2019 doi

  26. [34]

    J. P. Provost and G. Vallee, 1980, Comm. Math. Phys., 76, 289. DOI: 10.1007/BF02193559 https://doi.org/10.1007/BF02193559

    1980 doi

  27. [35]

    Wilczek and A

    F. Wilczek and A. Zee, 1984, Phys. Rev. Lett., 52, 2111. DOI: 10.1103/PhysRevLett.52.2111 https://doi.org/10.1103/PhysRevLett.52.2111

    1984 doi

  28. [36]

    Ran, arXiv:1012.1337 (2010)

    C. Ran, arXiv:1012.1337 (2010). arXiv:1012.1337 https://arxiv.org/abs/1012.1337

    2010 arXiv

  29. [37]

    Kolodrubetz, P

    M. Kolodrubetz, P. Mehta, and A. Polkovnikov, 2017, Physics Reports, 697, 1–87. DOI: 10.1016/j.physrep.2017.08.003 https://doi.org/10.1016/j.physrep.2017.08.003

    2017 doi

  30. [38]

    M. V. Berry and A. Shapere, 2010, Phys. Rev. B, 82, 245113. DOI: 10.1103/PhysRevB.82.245113 https://doi.org/10.1103/PhysRevB.82.245113

    2010 doi

  31. [40]

    Amico and R

    L. Amico and R. Fazio, 2020, Physics Reports, 838, 1–72. DOI: 10.1016/j.physrep.2019.12.001 https://doi.org/10.1016/j.physrep.2019.12.001

    2020 doi

  32. [41]

    Z. C. Wang, 2019, Scientific Reports, 9, 13258. DOI: 10.1038/s41598-019-49617-3 https://doi.org/10.1038/s41598-019-49617-3

    2019 doi

  33. [42]

    A. S. Sørensen and E. Demler, 2013, Phys. Rev. B, 87, 245103. DOI: 10.1103/PhysRevB.87.245103 https://doi.org/10.1103/PhysRevB.87.245103

    2013 doi

  34. [43]

    R. C. Ashoori et al., 2016, Nature Physics, 12, 778-782. DOI: 10.1038/nphys3758 https://doi.org/10.1038/nphys3758

    2016 doi

  35. [44]

    Xu-Yang Hou, Zheng Zhou, Xin Wang, Hao Guo, and Chih-Chun Chien, 2024, Phys. Rev. B 110, 035144 DOI: 10.1103/PhysRevB.110.035144 https://journals.aps.org/prb/abstract/10.1103/PhysRevB.110.035144

    2024 doi

  36. [45]

    DOI:10.1038/s41467-024-51146-7 https://doi.org/10.1038/s41467-024-51146-7

    Han Li, Enze Lv, Ning Xi, Yuan Gao, Yang Qi, Wei Li, Gang Su, 2024, Nat Commun 15, 7011. DOI:10.1038/s41467-024-51146-7 https://doi.org/10.1038/s41467-024-51146-7

    2024 doi

  37. [46]

    Donald Bures, 1969, Transactions of the American Mathematical Society, 135, 199-212 https://api.semanticscholar.org/CorpusID:53529518 DOI:10.1090/S0002-9947-1969-0236719-2

    1969 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.