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REVIEW 4 major objections 4 minor 61 references

Topological Invariants in Nonlinear Thouless Pumping of Solitons

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes a single topological invariant, the ratio of a non-Abelian Chern number to the number of braiding nonlinear bands, that it claims governs soliton displacement in nonlinear Thouless pumps across weak and strong…

desk verdict Central non-Abelian Chern number is not gauge invariant for non-orthogonal nonlinear bands, so the paper's main formula D=C_NL/N is a conjecture, not a proven result. read the letter →

arxiv 2506.08502 v1 pith:EBKN7Q7B submitted 2025-06-10 physics.atom-ph cond-mat.quant-gasquant-ph

Fei-Fei Wu , Xian-Da Zuo , Qing-Qing Zhu , Tao Yuan , Yi-Yi Mao , Chao Zeng , Yi Jiang , Yu-Ao Chen
show 3 more authors
Jian-Wei Pan Wei Zheng Han-Ning Dai
This is my paper · ORCID
classification physics.atom-phcond-mat.quant-gasquant-ph
keywords nonlinearThoulesspumpingsolitontransportnon-AbelianChernnumberbandsfractionalKerrnonlinearityAubry-André-HarpermodelRice-Mele
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 claims that the quantized transport of solitons in a nonlinear Thouless pump is governed by one formula across all nonlinearity strengths: the displacement per cycle equals a non-Abelian Chern number divided by the number of braiding nonlinear bands, $D = C_{\mathrm{NL}}/N$. In the weak-nonlinearity limit, where the bands stay separated, this reduces to the Abelian Chern number of the occupied nonlinear band, recovering integer pumping. In the strong-nonlinearity regime, where nonlinear Kerr loops expand and the lowest bands intertwine into a degenerate braiding subspace, the same formula yields fractional displacements such as $-1/2$ and $-1/3$. The authors test this invariant numerically in several Aubry-André-Harper and Rice-Mele lattices and report agreement with dynamical simulations. A reader who wants a unified topological picture of nonlinear soliton transport would find that here a single integer-like quantity organizes both the weak and strong regimes.

What carries the argument

The non-Abelian Wilczek-Zee Berry connection on the degenerate nonlinear band subspace, with matrix elements $[A_k]_{ab} = i \langle \Phi_a | \partial_k \Phi_b \rangle$, and the associated non-Abelian Chern number $C_{\mathrm{NL}}$ obtained by integrating the trace of the Berry curvature over the $1+1$D Brillouin zone. This object replaces the Abelian Chern number of linear band theory: it is defined even when nonlinear eigenvectors are non-orthogonal, and the dimension $N$ of the degenerate subspace enters as the denominator in the pumping formula, encoding the braiding of bands.

What would settle it

Compute $C_{\mathrm{NL}}$ for the five-site AAH model at $g=1.6$ after applying a smooth U(2) gauge rotation to the two degenerate eigenvectors at every $(k,t)$ point; if the value changes under that rotation, the invariant is gauge-dependent and Eq. (1) fails. Alternatively, run the full nonlinear time evolution in a well-defined braiding phase and check whether the center-of-mass displacement equals $C_{\mathrm{NL}}/N$; a mismatch would falsify the formula.

Watch

Extended reading notes

Core claim

In the paper's own formulation, the discovery is a generalized TKNN expression for nonlinear soliton pumping: $D = C_{\mathrm{NL}}/N$, where $C_{\mathrm{NL}}$ is the non-Abelian Chern number of the lowest braiding nonlinear bands and $N$ is the number of those bands. The nonlinear bands are obtained by solving the instantaneous nonlinear Schrödinger equation via a modified Newton method; they are generically non-orthogonal and, at strong nonlinearity, form degenerate subspaces in which the standard Abelian Chern number is undefined. The paper defines a non-Abelian Berry connection on this subspace and computes $C_{\mathrm{NL}}$ as the integral of the trace of the non-Abelian Berry curvature over the two-dimensional Brillouin zone. It shows that in well-separated bands this reduces to the Abelian Chern number, while for $N$ braiding bands it gives a fractional pumped charge; numerical simulations of the three-, five-, and seven-site Aubry-André-Harper models and the Rice-Mele model match the formula in the well-defined phases.

Load-bearing premise

The formula stands on the claim that the non-Abelian Berry connection defined from the non-orthogonal nonlinear eigenvectors is a well-defined, gauge-invariant object with a quantized trace; if that trace is not a true topological integer, the numerator $C_{\mathrm{NL}}$ in Eq. (1) loses its topological meaning.

Editorial extensions

If this is right

  • In the weak nonlinear regime, the pumped displacement of a soliton is quantized to the integer Abelian Chern number of the occupied nonlinear band.
  • In the strong nonlinear regime, when $N$ lowest bands braid, the displacement is fractional, $C_{\mathrm{NL}}/N$; the paper reports $-1/2$ for two braiding bands in the five-site model and $-1/3$ for three braiding bands in the seven-site model.
  • There is an intermediate ill-defined region where loop bands touch non-filled bands, the Chern number is undefined, and soliton pumping is non-adiabatic and unquantized.
  • The invariant applies to both the nonlinear Aubry-André-Harper model and the nonlinear Rice-Mele model, yielding a phase diagram with well-defined topological regions separated by ill-defined ones.
  • The paper states that the correspondence between the topological invariant and soliton displacement can be checked in ultracold-atom and photonic-waveguide experiments by tuning the nonlinearity and measuring the displacement.

Reading between the lines

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

  • Beyond the paper, if $D = C_{\mathrm{NL}}/N$ survives closer scrutiny, it implies a nonlinearity-tuned sequence of rational pumping values set purely by the size of the braiding multiplet, which would allow engineering fractional transport without interparticle interactions.
  • The gauge-invariance of the non-orthogonal eigenvector connection is the point most likely to need tightening; a biorthogonal or metric-corrected connection could preserve quantization even if the raw Wilczek-Zee trace does not.
  • The ill-defined region, where loops touch non-filled bands, may correspond to a non-adiabatic Landau-Zener regime; measuring how sharply the integer-to-fractional transition occurs could probe the band-touching structure the paper identifies.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a unified topological invariant for nonlinear Thouless pumping of solitons in one-dimensional lattices with Kerr nonlinearity. The central claim, Eq. (1), is that the center-of-mass displacement D of a soliton after one pump cycle equals C_NL/N, where C_NL is a non-Abelian Chern number of the lowest braiding (degenerate) nonlinear bands and N is the number of such bands. In the weak-nonlinearity limit the formula is claimed to reduce to the Abelian Chern number and give integer pumping; in the strong-nonlinearity limit, for braiding bands, it is claimed to give fractional pumping such as D = -1/2, -1/3, or 0. The paper supports this with nonlinear band-structure calculations and dynamical simulations for three-site, five-site, and seven-site Aubry-André-Harper models and for a nonlinear Rice-Mele model.

Significance. If valid, the proposed formula would be a valuable bridge between nonlinear band topology and soliton transport, unifying integer, zero, and fractional pumping in a single expression. The numerical work covers several lattice sizes and model families, and the predictions are concrete and falsifiable. However, the central definition of C_NL in Eq. (3) is not gauge invariant for the non-orthogonal nonlinear eigenvectors explicitly used in the paper, so the claimed topological meaning of the numerator of Eq. (1) is not established. Moreover, Eq. (1) is asserted rather than derived, and the numerical evaluation of C_NL is not specified in a way that can be reproduced or checked. These are load-bearing issues, not presentation concerns.

major comments (4)
  1. [Section 'Chern number of nonlinear bands', Eq. (3)] The definition of the non-Abelian Berry connection uses non-orthogonal eigenvectors: [A_k]_ab = i <Φ_a|∂_k Φ_b>. Under a change of frame Φ_a' = Σ_b U_ba Φ_b, one obtains A_k' = U† A_k U + i U† S ∂_k U, where S_ab = <Φ_a|Φ_b>. Since the paper explicitly states that the nonlinear eigenvectors are not orthogonal, S is not the identity and the extra term is not a pure gauge term. Consequently Tr F and the integral in Eq. (3) are not invariant under smooth U(N) transformations, and the rotated vectors are not in general normalized. The paper's statement that 'they still preserve U(N) gauge due to the normalization of eigenvectors' is therefore incorrect. C_NL as defined is not a gauge-invariant topological invariant, and its quantization is not established. This invalidates the topological meaning of the numerator in Eq. (1).
  2. [Section 'Nonlinear AAH model', Eq. (1)] Equation (1) is asserted without derivation. The text states that the pumping charge is the average nonlinear Chern number over the dimension of the degenerate subspace, but no adiabatic or projection argument connects the center-of-mass displacement of a single soliton initial condition to Tr F / N. In the linear non-Abelian case, one would need to specify carefully how a single filled band or a single initial state in an N-fold degenerate subspace relates to the trace of the Berry curvature; here the nonlinear self-consistency makes that relation even less immediate. The numerical agreement in Figs. 2-4 cannot certify Eq. (1) because the quantity C_NL entering it is not well-defined.
  3. [Section 'Nonlinear AAH model', Figs. 3 and S2] The selection of the integer N and of which bands form the 'braiding' subspace is not governed by a clear, a priori criterion. For the five-site model, N=2 is used at g=1.6 and N=5 at g=3.0; for the seven-site model, N=3 at g=1.5 and N=5 at g=3.0. The paper does not provide an algorithmic rule for choosing N in terms of a spectral gap or a well-defined degeneracy structure, particularly because in the 'ill-defined' regions the lowest bands also touch loop bands. If N is chosen after inspecting the dynamical displacement D, the agreement between D and C_NL/N is partly circular.
  4. [Section 'Chern number of nonlinear bands' and Supplemental Material] The numerical computation of C_NL is not described in a reproducible way. The main text and the Supplement describe the modified Newton method for finding eigenvectors, but do not give the lattice discretization or the smooth-gauge procedure used to evaluate the integral in Eq. (3). Since Eq. (3) is gauge dependent for non-orthogonal frames, the reported values C_NL = -1, 0 cannot be independently checked. Reference [50] is a standard lattice-gauge formula for orthonormal tight-binding Bloch states; the paper does not explain how it is applied to non-orthogonal nonlinear eigenvectors.
minor comments (4)
  1. [Summary] In the concluding paragraph, 'fictionalization' should be 'fractionalization'.
  2. [Section 'Nonlinear Rice-Mele model'] The phrase 'a system with linear interaction' appears to be a typo; in context it should likely read 'a system with nonlinear interaction' or 'with a staggered potential'.
  3. [Eq. (2)] The sign convention in H = H_lin - diag(g|Ψ|^2) and the self-consistency condition H(Ψ)Ψ = EΨ are not stated explicitly; adding one sentence would improve clarity.
  4. [Fig. 2(b)] The panels of Fig. 2(b) for g=1.5, 2.3, and 3.0 are not labeled individually, making it hard to connect the dynamical evolution to the phase diagram in Fig. 2(c).

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the fractional pumped charges are sharp predictions from independently computed band-structure invariants checked against dynamics; one minor mutually-referential validation step and an unproven gauge-invariance premise are flagged.

  1. other [Nonlinear AAH model section, validation paragraph bridging Eq. (1) and Eq. (3).]
    "The pumping charge D is the average nonlinear Chern number over the dimension of degenerate subspace N, see Eq. (1), which is like the generalized Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [51], describing the Hall conductance for quantum Hall effect states. In Fig. 2(c), we find the nonlinear Chern number agrees well with the pumping charges, indicating the definition of the nonlinear topological invariant, Eq. (3), is valid."

    Mutually-referential validation: the paper concludes that the definition of C_NL in Eq. (3) is 'valid' because C_NL agrees with the pumping charges D, but the only relation connecting D to C_NL is the asserted Eq. (1), D = C_NL/N, justified solely by analogy ('like the generalized TKNN formula'). The invariant is therefore declared valid on the strength of the very law it is invoked to support. This is a soft circularity, not a construction: C_NL comes from the band structure via the Fukui-Hatsugai-Suzuki formula, N comes from the band-isolation criterion, and D comes from independent real-space dynamical simulations, and the fractional matches (-1/2, -1/3) are sharp rather than forced. The loop weakens the epistemic status of both Eq. (3) and Eq.

full rationale

The paper's central relation D = C_NL/N is not fitted to data by construction: C_NL is computed from nonlinear band structures with the lattice Berry-curvature (Fukui-Hatsugai-Suzuki) formula, the denominator N is the number of mutually braiding bands in the occupied subspace selected by the isolation criterion ('the lowest bands remain isolated from the other higher bands throughout the 2D BZ'), and D is obtained from independent real-space dynamical simulations. The non-trivial fractional predictions (C_NL = -1, N = 2 giving D = -1/2 in the five-site AAH model; C_NL = -1, N = 3 giving D = -1/3 in the seven-site model) are sharp quantitative agreements, so the prediction ring is genuine. No load-bearing self-citation exists: the only author citation (ref [38]) is contextual among earlier nonlinear pumping works, and the non-Abelian connection and lattice Chern formula are cited to independent literature (Wilczek-Zee; Fukui-Hatsugai-Suzuki; TKNN). No uniqueness theorem is imported and no ansatz is smuggled via the authors' prior work. Two caveats are flagged explicitly. First, the validation of Eq. (3) quotes agreement with D through the same relation (Eq. 1) that the invariant is used to support, a partial mutual-consistency loop; this is reported as a step but is mitigated by the independence of the two computations. Second, the gauge-invariance premise for non-orthogonal eigenvectors is asserted, not proved: 'Despite the non-orthogonality, they still preserve U(N) gauge due to the normalization of eigenvectors'; under a U(N) rotation the connection acquires an overlap term U-dagger S dU with S unequal to the identity, so the trace of the curvature is not invariant and C_NL is not shown to be quantized or frame-independent. This gap, if it stands, undermines the topological meaning of the numerator in Eq. (1); it is a mathematical-validity issue rather than an input-output equality, so under the hard rules it is weighed but not scored as circularity. The Supplemental Material completeness caveat (about 100 random guesses per {k,t} with post-examination) is likewise a numerical-completeness limitation, not circularity. Overall: no significant circularity, score 2.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a small number of domain assumptions about adiabaticity and Bloch structure, plus one ad hoc assumption about the validity of the non-Abelian Berry connection for non-orthogonal nonlinear eigenvectors. The main free choice is the integer N, which is determined by inspection of the band structure rather than derived. No new physical entities are introduced.

free parameters (1)
  • N (number of braiding bands) = 2 (5-site), 3 (7-site)
    N in Eq. (1) is read off the band structure by identifying which nonlinear bands are degenerate over the full BZ. The paper gives no first-principles rule for choosing N, and in the medium-nonlinearity 'ill-defined' regions N is not defined. The good agreement of D = C/N depends on this choice.
assumptions (4)
  • domain assumption Adiabatic following: under slow periodic driving, the soliton wavefunction remains the instantaneous nonlinear ground state.
    Invoked in 'Under the adiabatic approximation, Ψ(t,x) is the unitary gauge transformation of instantaneous nonlinear eigenvectors'.
  • ad hoc to paper The nonlinear eigenvectors are normalized and span a U(N) gauge space despite being non-orthogonal; the non-Abelian Berry connection built from them is gauge invariant with quantized trace.
    The paper asserts this in the paragraph after Eq. (3) but does not prove it. Standard non-Abelian Chern numbers require an orthonormal frame; the non-orthogonality of nonlinear eigenvectors is not addressed.
  • domain assumption Bloch theorem applies to the time-dependent nonlinear eigenstates, so the wavefunction can be written as Φ(k) e^{ikx}.
    Used to define momentum-space Hamiltonian (4) and the 2D BZ integration in Eq. (3).
  • domain assumption Newton's down-hill method finds all relevant nonlinear eigenstates.
    The nonlinear band structure, and hence C_NL, depends on the completeness of the numerically found eigenvectors.

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Pith. "Pith review of Topological Invariants in Nonlinear Thouless Pumping of Solitons." pith.science (2026). https://pith.science/paper/EBKN7Q7B

@misc{pith2026250608502,
  author       = {Pith},
  title        = {Pith review of: Topological Invariants in Nonlinear Thouless Pumping of Solitons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EBKN7Q7B}},
  note         = {Machine review of arXiv:2506.08502}
}
read the original abstract

Recent explorations of quantized solitons transport in optical waveguides have thrust nonlinear topological pumping into the spotlight. In this work, we introduce a unified topological invariant applicable across both weakly and strongly nonlinear regimes. In the weak nonlinearity regime, where the nonlinear bands are wellseparated, the invariant reduces to the Abelian Chern number of the occupied nonlinear band. Consequently, the pumped charge is quantized to an integer value. As the nonlinearity increases, the nonlinear bands start to intertwine, leading to a situation where the invariant is expressed as the non-Abelian Chern number divided by the number of interacting bands. This could result in a fractional quantization of the pumped charge. Our unified topological invariant approach not only advances the understanding of the soliton dynamics, but also provides implications for the future design of nonlinear topological systems.

Figures

Figures reproduced from arXiv: 2506.08502 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the nonlinear Chern number and the corre [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a), The diagram of the linear AAH model with three sites per unit cell. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a), Schematic of the unit cell with five sites. (b), The [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

61 extracted references · 53 canonical work pages

  1. [50]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn.74, 1674-1677 (2005)

    work page 2005

  2. [1]

    D. J. Thouless, Phys. Rev. B27, 6083 (1983)

    1983

  3. [2]

    Niu and D

    Q. Niu and D. J. Thouless, J. Phys. A: Math. Gen.17, 2453 (1984)

    work page 1984

  4. [3]

    Niu, Phys

    Q. Niu, Phys. Rev. Lett.64, 1812 (1990)

    1990

  5. [4]

    R.Citroand M.Aidelsburger,Nat.Rev.Phys.5,87-101(2023)

    work page 2023

  6. [5]

    Lohse, C

    M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, Nat. Phys.12, 350-354 (2016)

    work page 2016

  7. [6]

    Nakajima, T

    S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L.Wang, M.Troyer,and Y.Takahashi, Nat.Phys.12, 296-300 (2016)

    work page 2016

  8. [7]

    H.-I.Lu, M.Schemmer, L.M.Aycock, D.Genkina, S.Sugawa, and I. B. Spielman, Phys. Rev. Lett.116, 200402 (2016)

    work page 2016

Show all 61 references
  1. [8]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil- berberg, Phys. Rev. Lett.109, 106402 (2012)

    2012

  2. [9]

    Zilberberg, S

    O. Zilberberg, S. Huang, J. Guglielmon, M.-H Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, Nature553, 59-62 (2018)

    2018

  3. [10]

    Prodan,and C

    W.-T Cheng, E. Prodan,and C. Prodan,Phys.Rev.Lett.125, 224301 (2020)

    2020

  4. [11]

    Fedorova,H.-X Qiu,S

    Z. Fedorova,H.-X Qiu,S. Linden,andJ. Kroha,Nat.Commun. 11, 3758 (2020)

    2020

  5. [12]

    C.-L. Deng, Y. Liu, Y.-R. Zhang, X.-G. Li, T. Liu, C.-T. Chen, T. Liu, C.-W. Lu, Y.-Y. Wang, T.-M. Li, C.-P. Fang, S.-Y. Zhou, J.-C. Song, Y.-S. Xu, Y. He, Z.-H. Liu, K.-X. Huang, Z.-C. Xiang, J.-C. Wang, D.- N. Zheng, G.-M. Xue, K. Xu, H. F. Yu, and H. Fan, Phys. Rev. Lett.13...

    2024

  6. [13]

    Q.-S. Wu, A. A. Soluyanov, and T. Bzdušek, Science365, 1273-1277 (2019)

    2019

  7. [14]

    You, S.-J

    O.-B. You, S.-J. Liang, B.-Y. Xie, W.-L. Gao, W.-M. Ye, J. Zhu, and S. Zhang, Phys. Rev. Lett.128, 244302 (2022)

    2022

  8. [15]

    Brosco, L

    V. Brosco, L. Pilozzi, R. Fazio, and C. Conti, Phys. Rev. A 103, 063518 (2021)

    2021

  9. [16]

    Tangpanitanon, V.M

    J. Tangpanitanon, V.M. Bastidas, S. Al-Assam, P. Roushan, D. Jaksch, and D. G. Angelakis, Phys. Rev. Lett.117, 213603 (2016)

    2016

  10. [17]

    Lin, Y.-G

    L. Lin, Y.-G. Ke, and C.-H. Lee, Phys. Rev. A.101, 023620 (2020)

    2020

  11. [18]

    Ke, and C.-H

    Y.-G. Ke, and C.-H. Lee, Nat. Phys.19, 1387-1388 (2023)

    2023

  12. [19]

    Walter, Z.-J

    A.S. Walter, Z.-J. Zhu, M. Gächter, J. Minguzzi, S. Roschin- ski, K. Sandholzer, K. Viebahn, and T. Esslinger, Nat. Phys. 19, 1471-1475 (2023)

    2023

  13. [20]

    Ke, J.-X

    Y.-G. Ke, J.-X. Huang, W.-J. Liu, Yuri Kivshar, and C.-H. Lee, Phys. Rev. Lett.131, 103604 (2023)

    2023

  14. [21]

    Viebahn, A.-S

    K. Viebahn, A.-S. Walter, E. Bertok, Z.-J. Zhu, M. Gächter, A. A. Aligia, F. Heidrich-Meisner, and T. Esslinger, Phys. Rev. X14, 021049 (2024)

    2024

  15. [22]

    W. Liu, S. Hu, L. Zhang, Y. Ke, and C. Lee, Phys. Rev. Res.5, 013020 (2023)

    2023

  16. [23]

    Smirnova, D

    D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Appl. Phys. Rev.7, 021306 (2020)

    2020

  17. [24]

    Zhou,D.Z

    D. Zhou,D.Z. Rocklin,M. Leamy,andY. Yao, Nat.Commun. 13, 3379 (2022)

    2022

  18. [25]

    Mukherjee and M

    S. Mukherjee and M. C. Rechtsman, Science368, 856-859 (2020)

    2020

  19. [26]

    Mukherjeeand M.C

    S. Mukherjeeand M.C. Rechtsman, Phys.Rev.X11,041057 (2021)

    2021

  20. [27]

    Lumer, Y

    Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Phys. Rev. Lett.111, 243905 (2013)

    2013

  21. [28]

    Leykam and Y

    D. Leykam and Y. D. Chong, Phys. Rev. Lett.117, 143901 (2016)

    2016

  22. [29]

    J. L. Marzuola, M. Rechtsman, B. Osting, and M. Bandres, arXiv:1904.10312, (2019)

    2019 arXiv

  23. [30]

    Chaunsali and G

    R. Chaunsali and G. Theocharis, Phys. Rev. B100, 014302 (2019)

    2019

  24. [31]

    S. Wong, T. A. Loring, and A. Cerjan, Phys. Rev. B108, 195142 (2023)

    2023

  25. [32]

    Jürgensen, S

    M. Jürgensen, S. Mukherjee, and M. C. Rechtsman, Nature 596, 63-67 (2021)

    2021

  26. [33]

    Jürgensen, S

    M. Jürgensen, S. Mukherjee, C. Jörg, and M. C. Rechtsman, Nat. Phys.19, 420-426 (2023)

    2023

  27. [34]

    Q.-D. Fu, P. Wang, Y. V. Kartashov, V. V. Konotop, and F.-W. Ye, Phys. Rev. Lett.128, 154101 (2022)

    2022

  28. [35]

    Jürgensen and M

    M. Jürgensen and M. C. Rechtsman, Phys. Rev. Lett. 128, 113901 (2022)

    2022

  29. [36]

    Tuloup, R

    T. Tuloup, R. W. Bomantara, and J. Gong, New J. Phys.25, S1 083048 (2023)

    2023

  30. [37]

    Mostaan, F

    N. Mostaan, F. Grusdt, and N. Goldman, Nat. Commun. 13, 5997 (2022)

    2022

  31. [38]

    Yuan, H.-N

    T. Yuan, H.-N. Dai, and Y.-A. Chen, Acta Phys. Sin. 72(16): 160302, (2023)

    2023

  32. [39]

    Q.-D. Fu, P. Wang, Y. V. Kartashov, V. V. Konotop, and F.-W. Ye, Phys. Rev. Lett.129, 183901 (2022)

    2022

  33. [40]

    Xiao, B.-N

    X.-R.You, L.-Y. Xiao, B.-N. Huang, Y.-G. Ke,andC.-H. Lee, Phys. Rev. A.111, 033306 (2025)

    2025

  34. [41]

    Zhang, Z

    Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A80, 063815 (2009)

    2009

  35. [42]

    X. Cao, C. Jia, Y. Hu, and Z. Liang, Phys. Rev. A110, 013305 (2024)

    2024

  36. [43]

    Ravets, N

    S. Ravets, N. Pernet, N. Mostaan, N. Goldman,and J. Bloch, Phys. Rev. Lett.134, 093801 (2025)

    2025

  37. [44]

    Y.-L. Tao, Y. Zhang, and Y. Xu, arXiv:2502.06131 (2025)

    2025 arXiv

  38. [45]

    X. Cao, C. Jia, H. Lyu, Y. Hu, and Z. Liang, Phys. Rev. A 111, 023329 (2025)

    2025

  39. [46]

    K. Sone, M. Ezawa, Y. Ashida, N. Yoshioka,andT. Sagawa, Nat. Phys.20, 1164–1170 (2024)

    2024

  40. [47]

    Wu and Q

    B. Wu and Q. Niu, Phys. Rev. A61, 023402 (2000)

    2000

  41. [48]

    See Supplemental Material for the modified Newton method, the degenerate bands of the three-site AAH model with strong nonlinearity, the Thouless pumping of solitons in the seven-site nonlinear AAH model and the corresponding nonlinear Chern number, the energy structures of th...

  42. [49]

    Wilczek and A

    F. Wilczek and A. Zee, Phys. Rev. Lett.52, 2111 (1984)

    1984

  43. [51]

    Thouless, M

    D.J. Thouless, M. Kohmoto, M.P. Nightingale,and M. den Nijs, Phys. Rev. Lett.49, 405 (1982)

    1982

  44. [52]

    K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.45, 494 (1980)

    1980

  45. [53]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett.48, 1559 (1982)

    1982

  46. [54]

    L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, Science370, 701-704 (2020)

    2020

  47. [55]

    Bai, J.-Z

    K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Phys. Rev. Lett.133, 116602 (2024)

    2024

  48. [56]

    M. S. Kirsch, Y.-Q. Zhang, M. Kremer, L. J. Maczewsky, S. K. Ivanov, Y. V. Kartashov, L. Torner, D. Bauer, A. Sza- meit, and M. Heinrich, Nat. Phys.17, 995-1000 (2021)

    2021

  49. [57]

    Z.-J. Zhu, Y. Kiefer, S. Jele, M. Gächter, G. Bisson, K. Viebahn, and T. Esslinger, arXiv:2409.02984 (2024)

    2024

  50. [58]

    Topological Invariants in Nonlinear Thouless Pumping of Solitons

    X.-L. Zhang, F. Yu, Z.-G. Chen, Z.-N. Tian, Q.-D. Chen, H.-B. Sun, and G.-C Ma, Nat. Photon.16, 390-395 (2022). S1 Supplementary material for “Topological Invariants in Nonlinear Thouless Pumping of Solitons” Fei-Fei Wuet. al. (September 11, 2025) MODIFIED NEWTON METHOD Duetot...

    2022

  51. [59]

    Start with a random vector,Φguess, as the initial guess for the eigenvector

  52. [60]

    Check whether the results converge once the number of iterations hits the maximum limit and only the converged results are retained

    UseΦ guess in the Newton’s down-hill method. Check whether the results converge once the number of iterations hits the maximum limit and only the converged results are retained

  53. [61]

    The number of variables is linearly dependent on the size of the unit cell, leading to the search of all eigenvectors difficult and large computation for the lager model

    Repeat steps 1 and 2 multiple times to obtain a set of eigenvectors and eigenvalues for the instant Hamiltonian. The number of variables is linearly dependent on the size of the unit cell, leading to the search of all eigenvectors difficult and large computation for the lager ...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.