🌀 nonlinear sciences

Nonreciprocity Induced Fractional Nonlinear Thouless Pumping

This paper investigates nonlinear Thouless pumping in a non-Hermitian Rice-Mele model, revealing that the interplay between nonlinearity and non-Hermiticity induces fractional topological phases that are naturally explained through the equation of auxiliary eigenvalues, thereby linking nonlinear spectral characteristics to bulk-boundary correspondence.

Original authors: Yanqi Zheng, Kun Pu, Ligging Ren, Chenxi Bai, Zhaoxin Liang

Published 2026-05-22

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CC0 1.0

Original authors: Yanqi Zheng, Kun Pu, Ligging Ren, Chenxi Bai, Zhaoxin Liang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a line of people (particles) walking across a series of stepping stones. In the world of standard physics, if you rhythmically shift the stones back and forth in a perfect cycle, the people will move a specific, whole number of steps forward every time you finish a cycle. This is called Thouless pumping. It's like a perfectly choreographed dance where the music (the changing parameters) forces the dancers to move exactly one, two, or three steps, never a fraction of a step.

This paper explores what happens when you introduce two "twists" to this dance: Nonlinearity and Non-Hermiticity.

The Two Twists

The "Self-Interested" Dancer (Nonlinearity):
In a normal dance, everyone moves independently. But in a nonlinear system, the dancers start reacting to each other. If a dancer gets crowded, they might push harder or change their rhythm based on how many neighbors they have. In physics, this is like particles that interact with one another, forming tight-knit groups called solitons (think of them as a single, cohesive wave of people moving together).
The "One-Way Street" (Non-Hermiticity):
Standard physics usually assumes a balanced world: if you can walk from Stone A to Stone B, you can walk back from B to A with equal ease. This is a "Hermitian" world.
Non-Hermiticity breaks this balance. Imagine Stone A has a strong wind pushing you toward Stone B, but Stone B has no wind pushing you back. Or, imagine Stone A is a "boost pad" that makes you move faster, while Stone B is a "brake pad." This creates a one-way street where movement is easier in one direction than the other.

The Big Discovery: Fractional Steps

The researchers combined these two twists in a specific model (the Rice-Mele model) and discovered something surprising: The dancers started taking "half-steps."

In the old rules of physics, the dancers could only move whole numbers of steps (1, 2, 3). But when they added the "one-way street" (Non-Hermiticity) to the "self-interacting" dancers (Nonlinearity), the system allowed the group to move exactly 1.5 steps (or 0.5 steps) per cycle.

The Analogy: Imagine a conveyor belt that usually moves you exactly 10 feet forward. If you add a specific type of friction (nonlinearity) and a wind blowing from behind (non-Hermiticity), the belt suddenly starts moving you exactly 5.5 feet. It's a "fractional" move that shouldn't be possible under the old rules.

How They Explained It: The "Shadow" Equation

Usually, physicists use a standard equation (the Schrödinger equation) to predict how these dancers move. But this equation failed to predict the "half-steps" in this new, messy environment.

The authors used a special tool called the "Auxiliary Eigenvalue Equation."

The Metaphor: Think of the standard equation as a flat map of the dance floor. It works great for a simple dance. But for this complex, windy, self-interacting dance, the flat map is useless.
Instead, they used a "3D holographic map" (the auxiliary equation). This new map accounts for the fact that the "music" (the energy/frequency) changes depending on how the dancers are moving.
This new map revealed that the "topology" (the shape of the dance floor's hidden geometry) had changed. The "half-steps" were actually a natural result of this new geometry, which only appears when the "one-way street" and the "self-interaction" work together.

The Key Takeaways

It's a Team Effort: You can't get these "half-steps" with just one twist. If you only have self-interacting dancers (nonlinearity) but a balanced world, they still take whole steps. If you only have a one-way street (non-Hermiticity) but no interaction, they also take whole steps. You need both to break the rules and create fractional movement.
The "Skin" Effect: The paper also notes that under certain conditions, the "one-way street" pushes all the dancers to the very edge of the line (the boundary), leaving the middle empty. This is called the "Non-Hermitian Skin Effect." However, the "half-step" phenomenon happens in a specific sweet spot where the dancers are still moving across the floor, just in fractional increments.
Where this applies: The authors suggest this could be seen in photonic waveguides (light traveling through special glass fibers) and cold-atom systems (ultra-cold gases in a lab). They are not claiming this works in human biology or medicine; they are strictly talking about light and atoms in controlled physics experiments.

In Summary

This paper shows that by mixing "self-interacting" particles with "one-way" physics, we can break the traditional rule that topological transport must be a whole number. We can now engineer systems where particles move in "half-steps," a phenomenon that can be predicted and understood using a new mathematical "holographic map" (the auxiliary eigenvalue equation).

Technical Summary: Nonreciprocity-induced fractional nonlinear Thouless pumping

Problem Statement
While topological transport in linear quantum systems is well-established through the paradigm of Thouless pumping (governed by the TKNN relation and quantized by integer Chern numbers), the interplay between nonlinearity and non-Hermiticity remains largely unexplored. Recent advances have established a link between bulk-edge correspondence and eigenvalue nonlinearity in Hermitian systems using an auxiliary eigenvalue equation, $H\Psi = \omega S(\omega)\Psi$ . However, the non-Hermitian generalization of this framework is uncharted. Specifically, it is unclear how non-Hermitian parameters (such as nonreciprocal couplings) interact with nonlinearities to reshape topological transport mechanisms, particularly whether they can induce fractional topological phases that deviate from conventional integer quantization.

Methodology
The authors investigate a one-dimensional non-Hermitian nonlinear Rice-Mele (RM) model characterized by:

Nonlinearity ( $g$ ): A Kerr-type interaction term.
Non-Hermiticity ( $\gamma$ ): A nonreciprocal hopping term.
Adiabatic Modulation: Time-dependent intra-cell and inter-cell hopping strengths and an on-site potential.

The study employs a dual approach:

Spectral Analysis: The authors calculate topological invariants (Chern numbers) under two distinct eigenvalue conditions:
- Linear condition: The conventional Schrödinger equation $H\Psi = E\Psi$ .
- Nonlinear condition: The auxiliary eigenvalue equation $H\Psi = \omega S(\omega)\Psi$ , where the eigenvalue $\omega$ appears in the operator via an overlap matrix $S(\omega)$ . The Chern numbers are computed using a biorthogonal basis to account for non-Hermiticity.
Dynamical Simulation: The authors numerically solve the discrete nonlinear non-Hermitian Schrödinger equations (derived from the Hamiltonian) to track the time evolution of soliton wave packets. They monitor the soliton's center-of-mass displacement ( $\langle X \rangle$ ) over a pumping cycle to observe quantized or fractional transport.

Key Contributions and Results

Fractional Topological Phases via Nonreciprocity: The study reveals that while the linear Chern number remains quantized at $C=1$ regardless of the non-Hermitian parameter $\gamma$ (in the absence of nonlinearity), the introduction of nonlinearity combined with non-Hermiticity induces a transition to fractional topological phases. Specifically, in certain parameter regimes, the Chern number calculated via the auxiliary eigenvalue framework stabilizes at $C = 1/2$ .
Mechanism of Fractional Pumping: The fractional pumping is not solely a result of strong nonlinearity coupling solitons to multi-band Wannier functions (a mechanism observed in Hermitian systems). Instead, the authors demonstrate that fractional Thouless pumping emerges exclusively when nonreciprocity ( $\gamma \neq 0$ ) is present. The interplay between the non-Hermitian skin effect and nonlinear band engineering reconstructs the effective Bloch bands, leading to fractional quantization.
Bulk-Boundary Correspondence in Non-Hermitian Nonlinear Systems: The work establishes a generalized bulk-boundary correspondence where the fractional Chern number ( $C=1/2$ ) directly predicts the fractional transport of solitons (displacement of one lattice site per cycle, $\Delta \langle X \rangle = 1$ ). This contrasts with the linear or purely Hermitian nonlinear cases, which exhibit integer quantization ( $\Delta \langle X \rangle = 2$ ) or suppressed transport.
Phase Diagrams: The authors map out a phase diagram in the $(g, \gamma)$ $(g, γ)$ plane, identifying distinct regimes:
- Integer Quantized Regime: Occurs at weak nonlinearity or specific non-Hermitian strengths where transport remains robust ( $C=\pm 1$ ).
- Fractional Regime: Occurs at strong nonlinearity and specific non-Hermitian strengths where transport becomes fractional ( $C=\pm 1/2$ ).
- Skin Effect Regime: At high non-Hermiticity, solitons accumulate at boundaries, suppressing bulk transport.

Significance and Claims
The paper claims to provide a key insight into the design of topological insulators and the manipulation of quantum edge states in real-world, non-equilibrium open systems. By utilizing the auxiliary eigenvalue framework, the authors bridge the gap between nonlinear spectral characteristics and bulk-boundary correspondence in non-Hermitian settings.

The authors assert that their findings predict the realization of fractional non-Hermitian nonlinear Thouless pumping in existing experimental platforms, specifically photonic waveguides and cold-atom systems. They emphasize that this work, alongside recent studies on Hermitian nonlinear eigenvalue problems, establishes a comprehensive description of topological transport driven by eigenvalue nonlinearity in both conservative and nonconservative settings. The results offer a theoretical foundation for engineering exotic topological phases where interactions, topology, and dissipation synergistically enable novel nonequilibrium phenomena.

The Two Twists

The Big Discovery: Fractional Steps

How They Explained It: The "Shadow" Equation

The Key Takeaways

In Summary

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