Nonreciprocity Induced Fractional Nonlinear Thouless Pumping

TL;DR

This work investigates topological transport in a non-Hermitian, nonlinear Rice-Mele lattice. It uses the auxiliary eigenvalue equation $H\\Psi=\\omega S(\\omega)\\Psi$ to connect nonlinear spectral features with bulk-edge physics and computes Chern numbers in both linear and nonlinear settings. The key finding is that non-Hermiticity can induce fractional Thouless pumping, with a nonlinear Chern number $\\mathcal{C}=\\frac{1}{2}$, when strong nonlinearity is present. These results reveal a generalized bulk-boundary correspondence and potential routes to engineer edge-state transport in photonic and ultracold-atom platforms.

Abstract

Recent interest has surged in eigenvalue's nonlinearity-based topological transport governed by the equation of auxiliary eigenvalues $HΨ=ωS(ω)Ψ$ [T. Isobe et al., Phys. Rev. Lett. 132, 126601 (2024); C. Bai and Z. Liang, 111, 042201 (2025); Phys. Rev. A 112, 052207 (2025)] rather than the conventional Schrodinger equation $HΨ=EΨ$ in conservative settings, yet non-Hermitian generalizations remain uncharted. In this work, we are motivated to investigate the nonlinear Thouless pumping in a non-Hermitian and nonlinear Rice-Mele model. In particular, we uncover how non-Hermiticity 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 framework of the equation of auxiliary eigenvalues, directly linking nonlinear spectral characteristics to the bulk-boundary correspondence. Our findings reveal novel emergent phenomena arising from the interplay between nonlinearity and non-Hermiticity, providing key insights for the design of topological insulators and the controlled manipulation of quantum edge states in the real world.

Figures (5)

• Figure 1: (a) Schematic of the nonlinear non-Hermitian RM model. The dashed box indicates a unit cell consisting of sublattice A and B. (b) Chern number $\mathcal{C}$ as a function of $\gamma$ for $g=0$. The Chern numbers for the linear eigenvalue problem $H\Psi=E\Psi$, computed analytically and numerically, are shown by the red star and black hollow circle, respectively. The blue solid circle represents the Chern number for the nonlinear problem $H\Psi=\omega S(\omega)\Psi$, obtained numerically. The relevant parameters are given as: $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, $\omega=\omega_d=10^{-3}$.
• Figure 2: (a) The phase diagram of the soliton centroid transport variation calculated under the scanned parameters $\left(g,\gamma\right)$. (b) The centroid transport conditions corresponding to the four points marked in Fig. 2(a). The relevant parameters are given as: $A\left(2.6,0.2\right)$, $B\left(5.7,-1.5\right)$, $C\left(7.0,1.0\right)$, $D\left(9.0,-1.0\right)$. Fixed parameters: $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, $\omega=10^{-3}$.
• Figure 3: (a1)-(d1): The real energy spectrum structure corresponding to the nonlinear non-Hermitian RM Hamiltonian; (a2)-(d2): The complex energy spectrum structure corresponding to the nonlinear non-Hermitian RM Hamiltonian; (a3)-(d3) The variation of the soliton centroid position expectation value $\langle X \rangle$ with time over a period.The relevant parameter values are: $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, $\omega=10^{-3}$. In(a1)-(d1)[(a2)-(d2),(a3)-(d3)], the interaction strength and the non-Hermitian strength are taken as $g=2.6$, $\gamma=0.2$; $g=5.7$, $\gamma=-1.5$; $g=7.0$, $\gamma=1.0$; and $g=9.0$, $\gamma=-1.0$, respectively.
• Figure 4: Non-Hermitian control of topological transport: (a,b) Real Part of the Non-Hermitian Nonlinear Eigenvalue Spectrum; (c,d,e) Quantized Soliton Dynamics. Fixed parameters: $g=5$, $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, $\omega=10^{-3}$.
• Figure 5: Non-Hermitian modulation of topological transport: (a,b) Real Part of the Non-Hermitian Nonlinear Eigenvalue Spectrum; (c,d,e) Quantized Soliton Dynamics. Fixed parameters: $g=8$, $J=1$, $\delta=0.5$, $\Delta=1$, $T=2000\pi$, $\omega=10^{-3}$.