Nonlinearity-Induced Thouless Pumping in Quasiperiodic Lattices

Abstract

Nonlinear Thouless pumping has been established in periodic lattices; its counterpart in quasiperiodic lattices remains unexplored. Here, we show a nonlinear topological pumping of gap solitons in quasiperiodic lattices where the local nonlinear self-consistent potentials lead to a lattice potential reconstruction; as a result, an emergent topological structure induced by this local reconstruction governs the dynamics of the gap solitons. This enables solitons to adiabatically occupy a single topological band, realizing quasi-quantized Thouless pumping. In addition, the intrinsic lattice perturbations disrupt this band occupation, which drives solitons into a non-quantized drifting regime. However, even in this regime, we also find that the soliton transport is constrained by the topological properties of a critical rational approximant. Tuning nonlinearity or lattice scaling reveals a controllable switching among topological pumping, drifting, and localization. Our work uncovers a mechanism for nonlinearity-induced topological behavior in complex lattice potentials.

Figures (6)

• Figure 1: Gap soliton pumping in a quasiperiodic superlattice. (a)--(b) For $\alpha=\frac{\sqrt{5}-1}{2}$: (a) Temporal evolution of the soliton density. (b) Soliton center-of-mass trajectories for successive rational approximants. These trajectories converge beyond the fifth-order approximant. The red dashed line is the trajectory for the fifth-order approximant including the perturbation $W_n^{\infty}$. (c)--(d) For $\alpha=\sqrt{3}$: (c) Temporal evolution of the soliton density. (d) Soliton center-of-mass trajectory (solid line) compared with the instantaneous Wannier center of a sliding lattice (dashed line), showing adiabatic following. Other parameters: $p_1=p_2=25$, $N=0.2$, $v=0.1$, $T=10\pi$.
• Figure 2: (a)--(c) Band occupation case of the adiabatically pumped gap soliton, expanded on the instantaneous Wannier basis derived from: (a) a linear superlattice (with $\alpha=5/8$), (b) a rational-approximation lattice at the critical order $\alpha_5$ (where $\alpha=\frac{\sqrt{5}-1}{2}$), and (c) a sliding single lattice (with incommensurate parameter $\alpha=\sqrt{3}$). (d)--(f) The shapes of the propagated solitons at $t=0$ and after each quarter adiabatic period ($T/4$) for (d) $\alpha=5/8$, (e) $\alpha=\frac{\sqrt{5}-1}{2}$ and (f) $\alpha=\sqrt{3}$. The lines at the bottom represent the lattice potential at $t=0$ and after each quarter adiabatic period. (g) The relationship between the center displacement of soliton in an adiabatic cycle and the parameter $\alpha$. Other parameters: $p_1=p_2=25$, $N=0.2$, $v=0.1$, $T=10\pi$.
• Figure 3: Nonlinearity-induced quasi-quantized pumping and trapping in a quasiperiodic superlattice. Upper panels ($N=7$): (a), (b) Temporal evolution of the soliton density and profile under adiabatic driving, showing directed transport. (c) Projection weights of the soliton wave function onto each band in the maximally localized Wannier basis of the sliding lattice. (d) Soliton center-of-mass trajectory (solid line) and the Wannier center trajectory (dashed line). Lower panels ($N=20$): (e), (f) Temporal evolution of the soliton density and profile under adiabatic driving, showing trapping. (g), (h) With a fixed long-period lattice and a sliding short-period lattice, the soliton still exhibits quasi-quantized pumping. Comparing the upper and lower panels shows that the nonlinear strength controls the transition between quasi-quantized pumping and localization. Other parameters: $p_1=p_2=15$, $\alpha=\frac{\sqrt{5}-1}{2}$, $v=0.1$, $T=10\pi$.
• Figure S1: We numerically computed the temporal variation of $\Delta$ under different conditions of nonlinearity strength and lattice period. (a) $N=0.2$, $\alpha=2/3$. (b) $N=7$, $\alpha=2/3$. (c) $N=0.2$, $\alpha=\sqrt{3}$. (d) $N=7$, $\alpha=\frac{\sqrt{5}-1}{2}$. The other parameters are $p_1=p_2=25$, $v=0.1$, $T=10\pi$.
• Figure S2: Soliton dynamics under rational approximations of two irrational ratios, $\alpha=\sqrt{3}/3$ and $\alpha=\sqrt{5}/5$. The upper panels (a)-(d) correspond to $\alpha=\sqrt{3}/3$, whose continued fraction expansion is $[0;1,1,2,1,2,...]$. (a) Center-of-mass displacement of the soliton over three adiabatic cycles for successive rational approximants. (b) Identification of the critical approximant order $n_c=4$, which separates regimes of quantized transport and non-quantized drift. (c) Band occupancy of the soliton exhibiting quantized transport within the critical approximant Hamiltonian $H_{n_c}$. (d) Band occupancy of the soliton in the $n_c+1$ approximant, plotted in the band basis of $H_{n_c}$. The lower panels (e)-(h) present corresponding results for (continued fraction: $[0;2,4,4,...]$). (e) Center-of-mass displacement for its rational approximants. (f) The critical order is $n_c=2$. (g) Band occupancy under quantized transport for the $n_c$ approximant. (h) Band occupancy for the $n_c+1$ approximant, plotted in the band basis of $H_{n_c}$. Other parameters are $p_1=p_2=25$, $N=0.2$, $v=0.1$, $T=10\pi$.