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Nonlinearity-induced Fractional Thouless Pumping of Solitons

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Pith's one-line read A soliton in an off-diagonal Aubry-André-Harper model can be pumped across one unit cell in one, two, three, or four periods even though all linear bands have Chern number zero, because the soliton itself creates the effective topological…

desk verdict A credible numerical discovery that fractional soliton pumping can occur even when all linear bands have zero Chern number, undermined by missing finite-size and precision analysis. read the letter →

arxiv 2502.06131 v1 pith:T7E2BN74 submitted 2025-02-10 nlin.PS cond-mat.mes-hallphysics.optics

Yu-Liang Tao , Yongping Zhang , Yong Xu This is my paper · ORCID
classification nlin.PScond-mat.mes-hallphysics.optics
keywords fractionalThoulesspumpingsolitontransportnonlinearity-inducedtopologyoff-diagonalAubry-André-HarpermodeldiscretenonlinearSchrödingerequationChernnumberphotonicwaveguidearraysparity-timesymmetry
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 shows that fractional Thouless pumping can occur in a nonlinear chain whose linear bands are all topologically trivial, overturning the usual requirement that the relevant Bloch bands carry a nonzero Chern number. In an off-diagonal Aubry-André-Harper model with periodically modulated nearest-neighbor hoppings, a soliton is transported across one unit cell over one, two, three, or four pump periods, giving average displacements of $1$, $1/2$, $1/3$, or $1/4$ unit cell per cycle. The mechanism is that the soliton's own density changes the effective on-site potentials, turning the modified linear Hamiltonian into a topologically nontrivial one with Chern number 1. If the claim holds, it expands quantized and fractional transport to systems without linear topological bands and gives a concrete route to observation in photonic waveguide arrays.

What carries the argument

The central object is the supercell Hamiltonian $H_{\rm sc}(\theta)$: the linear off-diagonal AAH chain plus the soliton-induced on-site potentials $\tilde m_x(\theta)=g_x|\psi_x(\theta)|^2$, taken periodic over $L$ sites. Because $H_{\rm sc}$ is translationally invariant under a shift of $L$ sites, it has a band structure indexed by a supercell momentum, and the relevant band's Chern number is computed over the two-dimensional parameter space of supercell momentum and the rescaled pump phase $\tilde\theta=\theta'/l$. In each reported integer and fractional case this Chern number is 1, and the corresponding Wannier function approximates the soliton's trajectory; the parity-time symmetry of the model then fixes the center of mass at half the pump period to be exactly $n/2$ unit cells.

What would settle it

Compute the soliton center-of-mass displacement $\delta x_c$ after one full pump period for the $(p,q)=(5,2)$ case on supercells with $L=15$ sites, then on larger supercells such as $L=30$ and $L=45$ and with periodic versus open boundary conditions; if the per-cycle displacement moves away from $1/2$ as $L$ increases, the claimed fractional quantization is a finite-size artifact.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that nonlinearity alone can induce both integer and fractional Thouless pumping of solitons in a system whose linear Hamiltonian has zero Chern number in every band. Numerically, a soliton bifurcating from the lowest linear band moves by exactly one unit cell after one pump period for $(p,q)=(3,1)$, by one unit cell after two periods for $(p,q)=(5,2)$ (an average of $1/2$ cell per cycle), by one unit cell after three periods for $(p,q)=(7,3)$ ($1/3$ per cycle), and by one unit cell after four periods for $(p,q)=(9,4)$ ($1/4$ per cycle). The paper attributes these displacements to soliton-induced on-site potentials $\tilde m_x(\theta)=g_x|\psi_x(\theta)|^2$ that, when added to the linear Hamiltonian in a supercell, produce a band with Chern number 1. Thus the soliton effectively creates its own topological pump even though the bare linear bands are trivial. The same mechanism is reported for a diagonal AAH model in the Supplemental Material.

Load-bearing premise

The reported displacements are computed on finite supercells and are assumed to remain exactly quantized as the system size grows, which the paper does not directly test.

Editorial extensions

If this is right

  • Quantized and fractional Thouless pumping no longer requires a nonzero Chern number in the bare linear Hamiltonian; the relevant topology can be generated by the soliton itself.
  • Because only nearest-neighbor hoppings are modulated and the nonlinear coefficients stay constant, the predicted $1/2$, $1/3$, and $1/4$ pumps are compatible with existing photonic waveguide arrays.
  • The parity-time symmetry argument pins the center-of-mass displacement at $\theta = m\Theta/2$ to exactly $n/2$ unit cells, giving a sharp experimental signature at half the pump period.
  • The same mechanism appears in both off-diagonal and diagonal AAH models, suggesting the phenomenon is generic across AAH-type nonlinear lattices rather than specific to the off-diagonal modulation.

Reading between the lines

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

  • A natural next step, not taken in the paper, is to map how the fractional value $n/m$ changes as the norm $N$ or the nonlinear coefficients $g_x$ are tuned; the paper reports the phenomenon at intermediate nonlinearity but does not identify thresholds or transitions between $1/2$, $1/3$, and $1/4$.
  • If the supercell Chern-number explanation holds in the thermodynamic limit, the same reasoning could apply to other nonlinear wave equations whose solitons reshape the local potential, making nonlinearity-induced fractional pumping a broader design principle for transport in trivial systems.
  • An experimentalist could test the claim directly by measuring output centroid positions after one, two, three, and four periods in a waveguide array; a clean one-unit-cell displacement with no linear-band Chern number would confirm the mechanism.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies a one-dimensional nonlinear off-diagonal Aubry-André-Harper (AAH) model with periodically modulated nearest-neighbor hoppings and static, site-dependent on-site potentials and nonlinearities. Using instantaneous soliton solutions obtained by Newton's method, the authors report that a soliton can be pumped across one unit cell over one, two, three, or four pump periods, corresponding to average displacements of 1, 1/2, 1/3, and 1/4 unit cells per period, despite all bands of the linear Hamiltonian having zero Chern number. They attribute the effect to soliton-induced on-site potentials that effectively make a modified supercell Hamiltonian topologically nontrivial, and they support the half-period case with a parity-time symmetry argument in the Supplemental Material. The paper also reports stability checks, adiabatic time-evolution agreement, and an analogous effect in a diagonal AAH model.

Significance. If the central claim holds, the paper significantly extends the known correspondence between soliton pumping and linear-band topology: it shows that trivial linear bands can nonetheless support quantized, fractional soliton transport purely through nonlinearity-induced potentials. The proposal is experimentally plausible because it requires only modulated nearest-neighbor hoppings and static nonlinearities, and the manuscript provides four parameter families, stability checks, and a symmetry-based argument for the 1/2 case. The main weakness is that the claimed exact quantization rests on numerical displacement measurements whose system size, boundary conditions, and numerical tolerance are never reported, and the topological explanation is a self-consistent construction rather than an independent predictor. The result is likely correct, but the evidence as presented is incomplete.

major comments (4)
  1. [§4, Figs. 3–4] The central quantitative claim is that the soliton displacement is exactly 1, 1/2, 1/3, or 1/4 unit cells per period, but the manuscript reports no numerical values, tolerances, lattice sizes, boundary conditions, or convergence checks. The density plots appear to span only a few dozen sites (e.g., the p=5 case in Fig. 3(c) shows roughly 15 sites), so open-boundary pinning or finite-size corrections could change the displacement. Because the novelty is quantized transport in a trivial linear band, an approximate or size-dependent displacement would invalidate the central claim. Please provide a table of δxc(θ) at the final time for all four parameter sets, state the total number of sites and boundary conditions, and include a convergence study as a function of system size and of the numerical tolerance used in Newton's method and in the adiabatic evolution.
  2. [§2, 'supercell' paragraph after Fig. 2] The Chern number of the modified Hamiltonian H_sc is computed from a Hamiltonian that includes the soliton's own induced potential \tilde{m}_x(θ)=g_x|ψ_x(θ)|^2, which is itself extracted from the observed soliton trajectory. This makes the Chern number a self-consistent reinterpretation of the motion rather than an independent predictor of the displacement. The manuscript should specify how the Chern number is calculated (band index, discretization in k_sc and θ, gauge conventions), verify that H_sc has a spectral gap along the entire pump cycle, and test the predictive content of the topological description—for instance, by checking that perturbing the parameters away from the reported values changes the displacement in the way the Chern number would predict.
  3. [SM S-1, Eqs. (S7)–(S9)] The parity-time symmetry argument as written does not prove the statement, made in §4, that the center-of-mass displacement 'amounts to exactly 1/2 unit cell per period.' Equation (S9) only gives the displacement at the midpoint θ = mΘ/2; the per-period average requires an additional monotonicity or branch-continuation argument. Moreover, Eq. (S7) assumes that the parity-reflected branch at θ' = lΘ−θ is the same branch followed by the adiabatic evolution from θ=0, which is not established. The conditions under which Eq. (S2) holds for the specific off-diagonal hoppings and g_x profiles used in the p=5,7,9 cases should be stated explicitly, including the allowed reflection centers x_r for each (p,q).
  4. [§2, Fig. 2(c) and Fig. 3(b)] The paper repeatedly asserts that all bands of the linear Hamiltonian have zero Chern number, but no Chern numbers are shown or tabulated. The band-structure plots alone do not demonstrate triviality, especially because the off-diagonal model can host nontrivial bands for other parameter choices. Please provide the computed Chern numbers for the linear bands for each parameter set, or cite the specific known result and state the parameter conditions under which all bands are trivial.
minor comments (5)
  1. [Abstract and §1] The phrase 'nonlinearity is not required to change during a pump period' is a strength and should be kept prominent, but the abstract's 'one, two, three or four pump periods' should clarify whether the integer case is the p=3 case and the fractional cases are the p=5,7,9 cases.
  2. [Eq. (2) and Fig. 2(b)] The horizontal axis in Fig. 2(b) is labeled 0 to 1, but the text uses θ with period Θ; please define the plotted variable (θ/Θ) in the caption.
  3. [Eq. (1)] The definition of J_x should be written more explicitly, since J_x = J_{[(x−1) mod p]+1} is used but the notation 'Jx = J[(x−1) mod p]+1' is easy to misread as an index shift.
  4. [§2, Fig. 2(e)] In the caption of Fig. 2(e), 'versus θ over four periods' is confusing because the horizontal axis may be θ/Θ; please clarify the range and the period of \tilde{m}_x.
  5. [SM S-2] The diagonal AAH results in Fig. S1 include a case with displacement −2 unit cells per period; this case is not discussed in the main text and deserves a sentence explaining the sign and magnitude.

Circularity Check

2 steps flagged · score 6.0 of 10

The Chern-number explanation is post hoc: the modified Hamiltonian is built from the soliton trajectory it is invoked to explain.

  1. self definitional [Main text, 'Nonlinearity-induced integer Thouless pumping of solitons', H_sc definition paragraph after Fig. 2(d).]
    "To incorporate the effects of a soliton, we define a modified linear Hamiltonian Hsc(θ) in a supercell consisting of L sites as Hlin(θ) with an additional on-site potential m̃x(θ)=gx|ψx(θ)|2 [26]."

    The 'modified linear Hamiltonian' is defined by adding the soliton's own density as the potential. The band structure and therefore the Chern number of the 'relevant band' therefore depend on the very trajectory the paper wants to explain. A potential that translates with the soliton will produce a band whose Wannier center translates with it, so the later finding that the Chern number is 'consistent with the displacement' is a restatement of the input: the self-induced potential already encodes the observed motion. The Chern number is not derived from the original linear Hamiltonian (which has C=0) or from any independent principle; it is computed from a Hamiltonian that contains the displacement as an input.

  2. fitted input called prediction [Main text, 'Nonlinearity-induced fractional Thouless pumping of solitons', after Fig. 3(d).]
    "We find that the Chern number of the band associated with the soliton in the modified linear Hamiltonian is 1, indicating that the Wannier function of this band (an approximation to the soliton solution) is pumped by 15 sites, i.e., three original unit cells after 6Θ. This is consistent with the displacement of 1/2 per period."

    The parenthetical 'an approximation to the soliton solution' makes the reduction explicit: the band used for the topological invariant is constructed from the same soliton solution whose displacement is the claimed result. The 15-site/6Θ motion is already inserted into the periodic potential m̃ through the observed trajectory, so reading off Chern number 1 and then saying it 'indicates' the 15-site pumping reverses the logical order. The fractional displacement is not predicted by the topology; the topology is forced by the displacement already present in the soliton solution.

full rationale

The direct numerical observation is not circular: the instantaneous solitons are obtained by solving the discrete nonlinear Schrödinger equation with fixed gx and time-modulated hoppings, and the center-of-mass displacement is computed from the resulting wavefunctions. That computation is self-contained and would stand even without the Hsc Chern-number argument. The circularity is confined to the explanatory layer. The paper attributes integer and fractional pumping to 'changes in on-site potentials induced by a soliton solution' and then verifies this by constructing Hsc from the very same soliton solution. Since m̃ = g|ψ|², any translation of the soliton is literally inserted into the Hamiltonian, making the Chern number 1 a property of a Hamiltonian that already contains the observed trajectory. The Supplemental Material PT-symmetry proof is not circular: it derives the half-period position from the m-period displacement and the symmetry, but the m-period displacement itself remains an input. Self-citation [39] is used only to note a model difference and is not load-bearing. The missing system-size and tolerance reporting is a finite-size/correctness risk, not a circularity, and does not affect this score.

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

The central claim depends on a large set of hand-chosen numerical parameters (N, Ja, mx, gx, p, q); no derivation specifies the regime where pumping occurs. The paper's explanation additionally imposes a periodic extension of the soliton-induced potential to define a topological invariant. There are no new physical entities, but the effective Hamiltonian H_sc is a state-dependent construct.

free parameters (5)
  • Nonlinearity norm N = 2.3, 2.2, 2.1, 2.0 depending on case
    Hand-chosen to make solitons stable and pumping occur; no systematic rule is given.
  • Modulation amplitude Ja = 0.05 (0.1 for p=9)
    Set to fixed values; pumping depends qualitatively on this amplitude.
  • On-site potentials mx = Variable profiles listed per case
    Chosen by hand; no derivation of the range that yields pumping.
  • Nonlinear coefficients gx = Variable profiles listed per case
    Site-dependent values tuned to produce the soliton branch; central to the effective potential.
  • Index pair (p,q) = (3,1), (5,2), (7,3), (9,4)
    Model parameters that define the unit cell and the fractional quantization that is observed.
assumptions (4)
  • domain assumption Adiabatic following: a soliton initialized on an instantaneous branch remains on that branch as θ is varied slowly.
    Used to equate the time evolution to the sequence of instantaneous stationary solutions; cited to refs [46-50].
  • standard math The displacement of a band or soliton branch equals the Chern number of the corresponding band of the (modified) Hamiltonian over the (θ,k) torus.
    Adopted from linear Thouless pumping theory and applied to the supercell Hamiltonian H_sc.
  • ad hoc to paper The soliton-induced potential \tilde{m}_x(θ) can be periodically extended with period L such that H_sc is translationally invariant.
    The extension \tilde{m}_{x+L}=\tilde{m}_x and \tilde{m}_{x+pj}=\tilde{m}_x(θ-jΘ) is imposed to define a Chern number; this periodicity is not derived from the nonlinear dynamics.
  • domain assumption Parity-time symmetry of both the linear Hamiltonian and the nonlinear coefficients g_x, as required in Supplement S-1.
    The proof of the exact half-period displacement requires H_lin(lΘ-θ) to be the reflected H_lin(θ) and g_x=g_{2x_r-x}; the chosen parameter sets are asserted to satisfy this.

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Pith. "Pith review of Nonlinearity-induced Fractional Thouless Pumping of Solitons." pith.science (2026). https://pith.science/paper/T7E2BN74

@misc{pith2026250206131,
  author       = {Pith},
  title        = {Pith review of: Nonlinearity-induced Fractional Thouless Pumping of Solitons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7E2BN74}},
  note         = {Machine review of arXiv:2502.06131}
}
abstract

Recent studies have shown that a soliton can be {\it fractionally} transported by slowly varying a system parameter over one period in a nonlinear system. This phenomenon is attributed to the nontrivial topology of the corresponding energy bands of a linear Hamiltonian. Here we find the occurrence of fractional Thouless pumping of solitons in a nonlinear off-diagonal Aubry-Andr\'{e}-Harper model. Surprisingly, this happens despite the fact that all the energy bands of the linear Hamiltonian are topologically trivial, indicating that nonlinearity can induce fractional Thouless pumping of solitons. Specifically, our results show that a soliton can be pumped across one unit cell over one, two, three or four pump periods, implying an average displacement of $1$, $1/2$, $1/3$ or $1/4$ unit cells per cycle, respectively. We attribute these behaviors to changes in on-site potentials induced by a soliton solution, leading to the nontrivial topology for the modified linear Hamiltonian. Given that our model relies solely on varying nearest-neighbor hoppings, it is readily implementable on existing state-of-the-art photonic platforms.

Figures

Figures reproduced from arXiv: 2502.06131 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic illustration of nonlinearity-induced [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Schematics of the off-diagonal AAH model, a 1D [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Nonlinearity-induced fractional Thouless pumping [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Nonlinearity-induced fractional Thouless pumping [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 2 Pith papers

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  1. Topological Invariants in Nonlinear Thouless Pumping of Solitons

    physics.atom-ph 2025-06 reject novelty 6.0 of 10

    The displacement of a soliton after one nonlinear pumping cycle is claimed to equal the non-Abelian Chern number of the braiding nonlinear bands divided by the number of braiding bands.

  2. Strongly tilted field induced fractional quantized-drift in non-interacting system

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    Strong tilting of a time-modulated superlattice activates Landau-Zener tunneling that equally populates several bands, making the per-cycle drift equal to the average Chern number, yielding fractional drifts such as 1...

Reference graph

Works this paper leans on

50 extracted references · 34 canonical work pages · cited by 2 Pith papers

  1. [38]

    Jürgensen, S

    M. Jürgensen, S. Mukherjee, C. Jörg, and M. C. Rechts- man, Quantized fractional thouless pumping of solitons, Nat. Phys. 19, 420 (2023). 6

    work page 2023

  2. [1]

    D. N. Christodoulides and R. I. Joseph, Discrete self- focusing in nonlinear arrays of coupled waveguides, Opt. Lett. 13, 794 (1988)

    work page 1988

  3. [2]

    H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Discrete spatial optical soli- tons in waveguide arrays, Phys. Rev. Lett. 81, 3383 (1998)

    work page 1998

  4. [3]

    J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 422, 147 (2003)

    work page 2003

  5. [4]

    D.N.Christodoulides, F.Lederer,andY.Silberberg,Dis- cretizing light behaviour in linear and nonlinear waveg- uide lattices, Nature424, 817 (2003)

    work page 2003

  6. [5]

    Y. S. Kivshar and G. P. Agrawal,Optical Solitons: From Fibers to Photonic Crystals (Elsevier Science, 2003)

    work page 2003

  7. [6]

    Lederer, G

    F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. As- santo, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep.463, 1 (2008)

    work page 2008

  8. [7]

    P. G. Kevrekidis, The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computa- tions and physical perspectives,Vol.232(Springer,Berlin, Heidelberg, 2009)

    work page 2009

Show all 50 references
  1. [8]

    V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Mod. Phys. Lett. B18, 627 (2004)

    2004

  2. [9]

    Morsch and M

    O. Morsch and M. Oberthaler, Dynamics of Bose- Einstein condensates in optical lattices, Rev. Mod. Phys. 78, 179 (2006)

    2006

  3. [10]

    Dauxois and M

    T. Dauxois and M. Peyrard,Physics of Solitons (Cam- bridge University Press, 2006)

    2006

  4. [11]

    P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero- González, Emergent nonlinear phenomena in Bose- Einstein condensates: theory and experiment (Springer, Berlin, Heidelberg, 2008)

    2008

  5. [12]

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fesh- bach resonances in ultracold gases, Rev. Mod. Phys.82, 1225 (2010)

    2010

  6. [13]

    Busch and J

    T. Busch and J. R. Anglin, Motion of dark solitons in trapped bose-einstein condensates, Phys. Rev. Lett.84, 2298 (2000)

    2000

  7. [14]

    Liu, S.-C

    J. Liu, S.-C. Li, L.-B. Fu, and D.-F. Ye,Nonlinear Adia- batic Evolution of Quantum Systems (Springer, 2018)

    2018

  8. [15]

    E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130 (1963)

    1963

  9. [16]

    R. M. May, Simple mathematical models with very com- plicated dynamics, Nature261, 459 (1976)

    1976

  10. [17]

    M. C. Gutzwiller, Chaos in classical and quantum me- chanics (Springer, 1990)

    1990

  11. [18]

    Lumer, Y

    Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Self-localized states in photonic topological insulators, Phys. Rev. Lett.111, 243905 (2013)

    2013

  12. [19]

    Mukherjee and M

    S. Mukherjee and M. C. Rechtsman, Observation of flo- quet solitons in a topological bandgap, Science368, 856 (2020)

    2020

  13. [20]

    M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma, Linear and nonlinear traveling edge waves in optical honeycomb lat- tices, Phys. Rev. A90, 023813 (2014)

    2014

  14. [21]

    Leykam and Y

    D. Leykam and Y. D. Chong, Edge solitons in nonlinear- photonic topological insulators, Phys. Rev. Lett. 117, 143901 (2016)

    2016

  15. [22]

    Mukherjee and M

    S. Mukherjee and M. C. Rechtsman, Observation of uni- directional solitonlike edge states in nonlinear Floquet topological insulators, Phys. Rev. X11, 041057 (2021)

    2021

  16. [23]

    Y.-L. Tao, N. Dai, Y.-B. Yang, Q.-B. Zeng, and Y. Xu, Hinge solitons in three-dimensional second-order topo- logical insulators, New J. Phys.22, 103058 (2020)

    2020

  17. [24]

    L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Kono- top, L. Torner, D. Bauer, and A. Szameit, Nonlinearity- induced photonic topological insulator, Science370, 701 (2020)

    2020

  18. [25]

    K. Sone, M. Ezawa, Y. Ashida, N. Yoshioka, and T. Sagawa, Nonlinearity-induced topological phase tran- sition characterized by the nonlinear Chern number, Nat. Phys. 20, 1164 (2024)

    2024

  19. [26]

    Jürgensen, S

    M. Jürgensen, S. Mukherjee, and M. C. Rechtsman, Quantized nonlinear Thouless pumping, Nature596, 63 (2021)

    2021

  20. [27]

    Jürgensen and M

    M. Jürgensen and M. C. Rechtsman, Chern number gov- erns soliton motion in nonlinear thouless pumps, Phys. Rev. Lett. 128, 113901 (2022)

    2022

  21. [28]

    Q. Fu, P. Wang, Y. V. Kartashov, V. V. Konotop, and F. Ye, Nonlinear Thouless pumping: solitons and trans- port breakdown, Phys. Rev. Lett.128, 154101 (2022)

    2022

  22. [29]

    Q. Fu, P. Wang, Y. V. Kartashov, V. V. Konotop, and F. Ye, Two-dimensional nonlinear Thouless pumping of matter waves, Phys. Rev. Lett.129, 183901 (2022)

    2022

  23. [30]

    Mostaan, F

    N. Mostaan, F. Grusdt, and N. Goldman, Quantized topological pumping of solitons in nonlinear photonics and ultracold atomic mixtures, Nat. Commun.13, 5997 (2022)

    2022

  24. [31]

    Tuloup, R

    T. Tuloup, R. W. Bomantara, and J. Gong, Breakdown of quantization in nonlinear Thouless pumping, New J. Phys. 25, 083048 (2023)

    2023

  25. [32]

    Citro and M

    R. Citro and M. Aidelsburger, Thouless pumping and topology, Nat. Rev. Phys.5, 87 (2023)

    2023

  26. [33]

    X. Hu, Z. Li, A.-X. Chen, and X. Luo, Pumping of mat- ter wave solitons in one-dimensional optical superlattices, New J. Phys.26, 123006 (2024)

    2024

  27. [34]

    H. Lyu, Y. Zhang, and T. Busch, Thouless pumping and trapping of two-component gap solitons, Phys. Rev. Re- search 6, L042010 (2024)

    2024

  28. [35]

    Szameit and M

    A. Szameit and M. C. Rechtsman, Discrete nonlinear topological photonics, Nat. Phys.20, 905 (2024)

    2024

  29. [36]

    D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983)

    1983

  30. [37]

    Niu and D

    Q. Niu and D. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many- body interaction, J. Phys. A Math. Gen.17, 2453 (1984)

    1984

  31. [39]

    Tao, J.-H

    Y.-L. Tao, J.-H. Wang, and Y. Xu, Nonlinearity-induced thouless pumping of solitons (2024), arXiv:2409.19515

    2024 arXiv

  32. [40]

    P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A68, 874 (1955)

    1955

  33. [41]

    Aubry and G

    S. Aubry and G. André, Analyticity breaking and Ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc. 3, 18 (1980)

    1980

  34. [42]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil- berberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett.109, 106402 (2012)

    2012

  35. [43]

    L.-J. Lang, X. Cai, and S. Chen, Edge states and topo- logical phases in one-dimensional optical superlattices, Phys. Rev. Lett.108, 220401 (2012)

    2012

  36. [44]

    F. Liu, S. Ghosh, and Y. D. Chong, Localization and adiabatic pumping in a generalized aubry-andré-harper model, Phys. Rev. B91, 014108 (2015)

    2015

  37. [45]

    Y. Ke, X. Qin, F. Mei, H. Zhong, Y. S. Kivshar, and C. Lee, Topological phase transitions and Thouless pumping of light in photonic waveguide arrays, Laser Photon. Rev. 10, 995 (2016)

    2016

  38. [46]

    Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys.61, 763 (1989)

    1989

  39. [47]

    Y. B. Band and M. Trippenbach, Bose-Einstein conden- sates in time-dependent light potentials: Adiabatic and nonadiabatic behavior of nonlinear wave equations, Phys. Rev. A 65, 053602 (2002)

    2002

  40. [48]

    Y. B. Band, B. Malomed, and M. Trippenbach, Adia- baticity in nonlinear quantum dynamics: Bose-Einstein condensate in a time-varying box, Phys. Rev. A 65, 033607 (2002)

    2002

  41. [49]

    J. Liu, B. Wu, and Q. Niu, Nonlinear evolution of quan- tum states in the adiabatic regime, Phys. Rev. Lett.90, 170404 (2003)

    2003

  42. [50]

    X x x|ψjp+2xr−x(θ)|2 # /N (S5) =

    B. Wu, J. Liu, and Q. Niu, Geometric phase for adiabatic evolutions of general quantum states, Phys. Rev. Lett. 94, 140402 (2005). In the Supplemental Material, we will prove that the parity-time symmetry can protect the half nonlinear Thouless pumping in Section S-1, and prov...

    2005

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