Fractional Thouless pumping of solitons: a unique manifestation of bulk-edge correspondence of nonlinear eigenvalue problems

TL;DR

Problem addressed: Can eigenvalue nonlinearity generate observable bulk-edge phenomena not present in linear theory? Approach: use the auxiliary-eigenvalue framework for the nonlinear extended Rice-Mele model with Kerr nonlinearity $g$ and next-nearest-neighbor couplings $t_a,t_b$, compute linear and nonlinear Chern numbers and analyze soliton pumping via both dynamical and instantaneous methods. Key contributions: demonstrate fractional Chern numbers such as $\\mathcal{C}=-\\frac{1}{2}$ at special parameter points, realize fractional nonlinear Thouless pumping of solitons controlled by $g$ and $t_a,t_b$, and establish anomalous bulk-edge inheritance through the auxiliary spectrum. Significance: reveals a novel nonlinear bulk-edge phenomenon with potential applications to photonic lattices and ultracold atoms, enabling tunable edge transport in nonlinear topological systems.

Abstract

Recent foundational studies have established the bulk-edge correspondence for nonlinear eigenvalue problems using auxiliary eigenvalues $\hat{H}Ψ=ωS(ω)Ψ$, spanning both linear [T. Isobe et al., Phys. Rev. Lett. 132, 126601 (2024)] and nonlinear [Chenxi Bai and Zhaoxin Liang, Phys. Rev. A. 111, 042201 (2025)] Hamiltionians. This progress prompts a fundamental question: Can eigenvalue nonlinearity generate observable physical phenomena absent in conventional approaches ($\hat{H}Ψ=EΨ$)? In this work, we address this question by demonstrating the first uniquely nonlinear manifestation of the bulk-edge correspondence: fractional Thouless pumping of solitons. Through systematic investigation of nonlinear Thouless pumping in an extended Rice-Mele model incorporating next-nearest-neighbor (NNN) couplings, we uncover that NNN interaction parameters can induce fractional topological phases|even in the presence of quantized topological invariants as predicted by conventional linear approaches. Crucially, these fractional phases are naturally explained within the auxiliary eigenvalue framework, directly linking nonlinear spectral characteristics to the bulk-boundary correspondence. Our findings reveal novel emergent phenomena arising from the interplay between nonlinearity and NNN couplings, providing key insights for the design of topological insulators and the controlled manipulation of quantum edge states in nonlinear regimes.

Figures (8)

• Figure 1: (a) Schematic diagram of the nonlinear extended RM model. The dotted box indicates the unit cell composed of sublattices A and B. (b) Chern number $\mathcal{C}$ topological phase diagram of the linear extended RM Hamiltonian (\ref{['H']}) with $g=0$ under $H_{\text{ERM}}\Psi=E\Psi$. (c) Chern number $\mathcal{C}$ topological phase diagram of the linear extended RM Hamiltonian (\ref{['H']}) with $g=0$ under $H_{\text{ERM}}\Psi=\omega S(\omega)\Psi$. The red dot represent the fractional Chern number of $\mathcal{C}=-\frac{1}{2}$ in specific NNN coupling parameter regimes of $t_a$ and $t_b$. (b,c) The parameters are given as $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, and $\omega_{\text{d}}=10^{-3}$
• Figure 2: (a1)-(d1): Band structures of the nonlinear eigenvalues for the nonlinear extended RM Hamiltonian (\ref{['H']}), (a2)-(d2): The position expectation value $\langle X\rangle$ of the soliton as a function of time over a single period. The parameters are fixed as $J=1$, $\delta=0.5$, $\Delta=1$, $t_a=0.4$, $t_b=-0.1$, $T=2000\pi$, and $\omega_{\text{d}}=10^{-3}$. (a1)-(d1) [(a2)-(d2)]: The interaction strengths are set to $g=0$, $g=1$, $g=5$ and $g=10$, respectively.
• Figure 3: NNN tunneling control of topological transport: (a,c) Nonlinear eigenvalue spectra ($\mu$); (b,d,e) Quantized soliton dynamics. Fixed parameters: $J=1$, $\delta=0.5$, $\Delta=1$, $g=1$, $T=2000\pi$, $\omega_{\text{d}}=10^{-3}$. Tunneling configurations: (a,b) $t_a=0.4$, $t_b=-0.1$; (c,d) $t_a=0.5$, $t_b=-0.5$.
• Figure 4: NNN tunneling modulation of topological phases: (a,c) Nonlinear eigenvalue spectra ($\mu$); (b,d,e) Quantized soliton transport dynamics. Fixed parameters: $J=1$, $\delta=0.5$, $\Delta=1$, $g=6.215$, $T=2000\pi$, $\omega_{\text{d}}=10^{-3}$. Parameter configurations: (a,b) $t_a=0.4$, $t_b=-0.1$; (c,d) $t_a=0.5$, $t_b=-0.5$.
• Figure 5: Eigenvalue's nonlinearity of nonlinear extended RM Hamiltonian and nonlinear Thouless pumping of soliton. (a1)-(d1): The auxiliary $\lambda$-spectrum for the nonlinear extended RM Hamiltonian (\ref{['H']}). (a2)-(d2): The anticipated position of the nonlinear excitation of soliton as a function of time over one period. The parameters are fixed at $J=1$, $\delta=0.5$, $\Delta=1$, $t_a=0.4$, $t_b=-0.1$, $T=2000\pi$, and $\omega=\omega_{\text{d}}=10^{-3}$. Specifically, (a1)-(d1) and their corresponding (a2)-(d2) panels represent interaction strengths of $g=0$, $g=1$, $g=5$ and $g=10$, respectively.