Nonlinear Thouless Pumps

• Nonlinear Thouless pumps are topological transport phenomena in nonlinear media where soliton excitations enable quantized displacement.
• They leverage nonlinear dynamics in systems such as ultracold atoms, photonic arrays, and liquid crystals to achieve integer and fractional quantization.
• Experimental realizations demonstrate robust, high-speed transport with tunable thresholds and disorder resilience through controlled nonlinearity.

Nonlinear Thouless pumps generalize the topological quantized pumping phenomena originally introduced by Thouless to nonlinear and interacting media, most notably systems supporting solitons and other localized excitations. In these systems, nonlinearity produces rich transport phenomenology: integer and fractional quantization of pumped displacement, robustness or breakdown of quantization depending on nonlinear thresholds, anomalous topological plateaus, and fundamentally new mechanisms such as soliton-induced band topology and nonlinear bulk-edge correspondence. Quantized nonlinear Thouless pumps have been realized and analyzed in diverse physical contexts, from ultracold atomic matter waves and photonic waveguide arrays to nematic liquid crystal platforms and nonlinear mechanical chains. Presented below is a comprehensive technical account of the key principles, regimes, and findings in the study of nonlinear Thouless pumps.

1. Paradigm: From Linear Thouless Pumping to Nonlinear Soliton Transport

In linear settings, the Thouless pump is implemented by adiabatic, cyclic modulation of a 1D lattice Hamiltonian parameter, which maps mathematically to a 2D insulator on the Brillouin zone × pump parameter torus. Under adiabatic evolution, the center-of-mass (COM) displacement per cycle of a wavepacket initialized in band $n$ is quantized to the first Chern number: $$\Delta x = C_n \cdot a,\quad C_n = \frac{1}{2\pi}\int_0^T dt \int_{BZ} dk\,\Omega_n(k,t),$$ with $\Omega_n$ the Berry curvature of the $n$th band and $a$ the lattice period. This quantization is topologically protected as long as the relevant band gap does not close (Jürgensen et al., 2021).

In nonlinear Thouless pumps, a key distinction arises: the transport and quantization are realized not via uniform band filling but through localized nonlinear excitations—typically solitons—that bifurcate from the band structure and can persist even in the presence of strong interactions (Jürgensen et al., 2021, Jürgensen et al., 2021, Jürgensen et al., 2022). Their evolution over the pump cycle can differ profoundly from the linear analogue, and the quantized transport becomes a property of the branch of nonlinear "instantaneous" solutions that can be adiabatically followed as the system evolves.

2. Soliton Solutions, Band Bifurcation, and Topological Invariants

Nonlinear Thouless pumps are generally modeled by the time-dependent discrete or continuum nonlinear Schrödinger (NLS/GP) equation,

$$i\frac{d\psi_j}{dt} = \sum_{m}H_{jm}(t)\psi_m - g|\psi_j|^2\psi_j,$$

with $H_{jm}(t)$ the periodically modulated single-particle Hamiltonian and $g$ the nonlinearity strength.

At weak $g$, instantaneous soliton solutions track the Wannier function of a single isolated band: their center-of-mass follows the evolution of that Wannier center under parameter variation (Jürgensen et al., 2021, Mostaan et al., 2021). The quantized displacement per cycle is fixed by the band Chern number, exactly as in the linear theory: $\Delta x_\mathrm{soliton} = C_\alpha,$ for a soliton bifurcating from band $\alpha$. The projection into maximally localized Wannier states rigorously connects the nonlinear soliton trajectory to the underlying linear topological invariant [(Jürgensen et al., 2021); see also (Jürgensen et al., 2022, Jürgensen et al., 2021)].

In pump cycles, this quantized drift is observed as translation by an integer number of unit cells per cycle, which is robust to details of the nonlinear branch as long as the soliton remains distinguishably localized and remains on a connected branch throughout the cycle.

3. Nonlinearity-Induced Anomalies: Breakdown, Fractional, and Anomalous Plateaux

At strong nonlinearity, the quantization of pumped displacement can break down or exhibit nontrivial fractionalization. There exist distinct nonlinear regimes (Tuloup et al., 2022, Jürgensen et al., 2022, Fu et al., 2021):

• Below a critical nonlinearity $g_c$: quantized integer pumping is observed, with soliton COM displacement per cycle matching the parent band's Chern number.
• Near and above $g_c$: loop or swallowtail structures appear in the nonlinear spectrum, generating secondary branches. The pump quantization can fail—either resulting in smaller, nonquantized (fractional) displacements, or the soliton becoming dynamically arrested with zero net pumping (Tuloup et al., 2022, Fu et al., 2021). In multi-band or strong-coupling regimes, Rabi oscillations between bands can ensue, and the COM shift after $q$ cycles is rational, producing fractional plateaux (Jürgensen et al., 2022).

Mechanistically, these transitions correspond to bifurcations—pitchforks or saddle-node in the nonlinear eigenvalue structure. Nonlinearity can "dress" the linear bands, giving rise to effective topological bands with nontrivial Chern numbers even when the parent linear bands are trivial (Tao et al., 10 Feb 2025, Tao et al., 2024). The induced Berry curvature and quantum geometric tensor of the soliton-dressed Hamiltonian underpin the fractional quantization, with the induced band Chern number dictating the average soliton shift per cycle.

4. Extensions: System Variants and Dimensional Generalizations

a. Inter-site and Vector Nonlinearities

Nonlinear Thouless pumping extends beyond onsite Kerr to include inter-site (cross-phase) nonlinearities—which can induce additional quantized jumps, anomalous plateaus, and even large engineered transport distances exceeding the linear Chern number (Jiang et al., 4 Jan 2026). The parameter space of accessible quantized plateaux can be substantially enlarged, with nonlocal or cross-interaction terms providing tunability.

In multi-component ("vector") condensates, the interplay of intercomponent and spin-dependent lattice shift parameters enables fine control of the topological phase diagram, with arrested, quantized, and reentrant pumping regimes accessible as functions of relative lattice shifts and nonlinearities (Cao et al., 2024).

b. Dissipative and Nonlocal Media

Incorporating loss and/or gain (as with the complex Ginzburg-Landau equation in temporal dissipative photonics) introduces new classes of dynamical topological phase transitions. Quantized temporal drift can emerge only above a critical dissipative threshold, and "emergent" transitions can occur where topological drift is self-induced by nonlinear reshaping, even with all external parameters fixed (Cao et al., 2024). Dissipation, therefore, can both enable and stabilize nonlinear Thouless pumping in open systems.

Nonlocal nonlinearities, in which the nonlinear response is spatially extended (as in thermal optical media or liquid crystals), elevate threshold powers for quantized topological transport and highly suppress undesirable breakdown modes observed in local Kerr media. This stabilization effect significantly increases the power window of robust transport (Ye et al., 8 Jul 2025).

c. Multi-dimensional and Anomalous Transport

Quantized nonlinear Thouless pumping generalizes to two (and higher) spatial dimensions, with the displacement vector set by the Chern numbers of separable sublattice pumps. In 2D, fractional Chern numbers govern half-period pumping in PT-symmetric systems, and the transition between "quasi-linear," single-soliton, and multi-soliton regimes is determined by the nonlinear interaction strength and initial wavepacket norm (Fu et al., 2022).

Beyond solitons, quantized transport of kinks or domain walls in nonlinear dimer chains with periodically modulated parameters exhibits "topological ratchet" behavior: quantized net displacement is realized through nonlinear boundary mode instabilities, with no single global Chern invariant controlling the motion (Bestler et al., 16 Aug 2025).

5. Robustness and Disorder, Engineering Principles, and Experimental Realization

The robustness of nonlinear Thouless pumps to disorder and inhomogeneities is markedly enhanced compared to the linear regime. The instantaneous soliton state is energetically and spectrally isolated from the continuum by a nonlinear gap $\Delta_\mathrm{gap}\sim gP$. Quantization persists up to disorder strengths $W_c\sim \alpha\,gP$, well above the linear breakdown threshold (Chaudhari et al., 12 Dec 2025, Cao et al., 2024). Nonlinearity permits faster pump protocols: the minimum pump period for adiabaticity is $T_\mathrm{min}\sim 2\pi/gP$, allowing for GHz-THz modulation rates in integrated photonic devices (Chaudhari et al., 12 Dec 2025).

This underpins proposed applications of nonlinear Thouless pumps as high-speed, robust, nonreciprocal photonic elements—delay lines, isolators, or logic devices—where disorder robustness and device simplicity are critical (Cao et al., 2024, Chaudhari et al., 12 Dec 2025).

Comprehensive experimental confirmation has been achieved in:

• Femtosecond-laser–written waveguide arrays in glass (soliton displacement and topological edge-to-edge transport) (Jürgensen et al., 2021, Jürgensen et al., 2022, Chaudhari et al., 12 Dec 2025).
• Nematic–liquid–crystal SSH arrays, where nonlinearity blockades quantized pumping at a sharply defined optical power threshold—a robust all-optical switch (Jung et al., 2022).
• Ultracold atomic gases in optical superlattices (Fu et al., 2022, Cao et al., 2024), with precision tuning of nonlinearity and transport parameters enabled by Feshbach resonances and spin-dependent lattice engineering.

6. Topological Formalism, Generalizations, and Classification

Formally, nonlinear Thouless pumps realize a broad class of quantized parameter pumps whose quantization is underwritten by nontrivial topology of parameter space and generalized "nonlinear Berry curvatures." The mathematical structure is captured by a nonlinear generalization of topological response theory, where the quantization arises from closed, integer-valued cohomology classes $\lambda_k$ on the parameter manifold, and quantized charge transport can be detected by threading synthetic fluxes or instantons (Yao, 2020).

Generalization to arbitrary dimension and gauge structure is possible. The parameter-gauge response formalism associates $k$th-order nonlinear quantized responses to Chern–Simons couplings of the pulled-back forms, subject to quantization by generalized anomaly-inflow arguments. This ties nonlinear Thouless pumping to generalized $-1$-form symmetry anomalies and the topology of the RG-flow interfaces in the parameter manifold (Yao, 2020).

7. Limitations, Open Problems, and Outlook

While nonlinear Thouless pumps extend the paradigm of topologically quantized transport into new domains, several open challenges remain:

• The breakdown of quantization due to nonlinear bifurcations, looped bands, or multi-soliton collisions poses limitations on the truly robust regime, especially for engineered transport of complex excitations (Tuloup et al., 2022, Jiang et al., 4 Jan 2026).
• Multi-soliton or interacting regimes generally lack quantization, due to cross-interaction effects and the breakdown of adiabatic soliton tracking (Jiang et al., 4 Jan 2026).
• The full classification of nonlinear topological invariants in the absence of a standard Berry curvature framework remains an open mathematical question, motivating auxiliary eigenvalue frameworks and generalized bulk-edge correspondences (Bai et al., 7 Jul 2025).
• The impact of noise, dissipation, and quantum fluctuations is nontrivial: while a certain level of noise can enhance pumping through resonance effects, higher temperatures can exponentially suppress pumped currents (Perroni et al., 2013).

Nevertheless, the unification of nonlinear dynamics and topological band theory in the context of Thouless pumps provides a versatile, robust framework for controlled quantized transport in strongly interacting and engineered nonlinear media, with significant implications for future experimental and applied advances.