Anomalous quantized nonlinear Thouless pumping

TL;DR

The paper reveals anomalous nonlinear Thouless pumping, where a soliton’s one-cycle displacement can differ from the underlying band’s Chern number, including zero, 2, or 3 unit-cell shifts for bands with $C=-1$. A general theory shows that the anomaly arises when a soliton transitions between Wannier functions via intersite soliton states, and they also demonstrate nonlinearity-induced pumping even for linearly trivial bands. By combining discrete and continuous nonlinear models, along with a supercell/topological framework, the work links soliton transport to soliton-Wannier dynamics and offers experimentally accessible platforms in photonics and cold atoms. These findings broaden the scope of topological transport in nonlinear and interacting settings and suggest new avenues for observing nonlinear, topology-driven phenomena in various 1D systems.

Abstract

It has recently been theoretically predicted and experimentally observed that a soliton resulting from nonlinearity can be pumped across an integer or fractional number of unit cells as a system parameter is slowly varied over a pump period. Nonlinear Thouless pumping is now understood as the flow of instantaneous Wannier functions, ruling out the possibility of pumping a soliton across a nonzero number of unit cells over one cycle when a corresponding Wannier function does not exhibit any flow, i.e., when the corresponding Bloch band that the soliton bifurcates from is topologically trivial. Here we surprisingly find an anomalous nonlinear Thouless pump where the displacement of a soliton over one cycle differs from the Chern number of the Bloch band from which the soliton comes. We develop a general theory showing that this anomalous behavior arises from a transition of a soliton between different Wannier functions by passing through an intersite-soliton (or dipole-soliton) state. Furthermore, we find a nonlinearity-induced integer quantized Thouless pump of a soliton, allowing a soliton to travel across one unit cell during a pump period, even when the corresponding band is topologically trivial. Our results open the door to studying nonlinearity-induced Thouless pumping of solitons.

Figures (13)

• Figure 1: Schematic illustration of how anomalous nonlinear Thouless pumping arises. The black lines represent the center-of-mass position of Wannier functions with respect to $\theta$. When a soliton follows a black line, normal nonlinear pumping appears. In contrast, if a soliton transitions between Wannier functions by passing through intersite solitons, whose center-of-mass positions are marked by diagonal crosses, anomalous nonlinear pumping occurs. For example, following the blue and red lines in a results in displacements of $0$ and $-2$, respectively, while following the green line in b results in a displacement of $-3$.
• Figure 2: Anomalous nonlinear Thouless pumping in the discrete nonlinear model.a, One-cycle trajectory of center-of-mass positions of instantaneous solitons coming from the lowest band of the linear Hamiltonian in Eq. (\ref{['DNSE']}) as a system parameter $\theta$ varies. It plots three anomalous cases corresponding to displacements of $0$ (case 1), $-2$ (case 2), and $-3$ (case 3), described by the blue ($g=-1$, $g_{12}=0$, and $m_0=1$), red ($g=1$, $g_{12}=0$, and $m_0=1$), and green lines ($g=1$, $g_{12}=0$, and $m_0=1.3$), respectively, in contrast to the normal case with a displacement of $-1$ described by the black line ($g=g_{12}=m_0=1$). The associated density distributions $|\psi_j|^2=\sum_{\sigma}|\psi_{\sigma j}|^2$ of the solitons for the three anomalous cases are plotted in b-d. e-h, Occupations $\rho_{W_l}(\theta)$ of instantaneous solitons on Wannier functions $W_l(\theta)$ of the lowest band along the Wannier centers as a function of $\theta$ for the four cases. See also the Supplementary Information Section S-3 for more information. Here, we set $N=1.45$.
• Figure 3: Transition between normal and anomalous nonlinear Thouless pumping.a-d, Center-of-mass positions of stable instantaneous solitons (black and red lines) and Wannier functions (dashed grey lines) with respect to a system parameter $\theta$ over one cycle for different $g_{12}$. The anomalous nonlinear pumping arises due to appearance of intersite soliton solutions around $\theta=\pi$ (see red lines in b). See Supplementary Information Section S-5 for stability analysis at $\theta=\pi$. Here, we set $m_0=g=1$ and $N=1.45$.
• Figure 4: Nonlinearity-induced Thouless pumping.a,b, Illustration of how the parameters $m_z(\theta)$, $J_1(\theta)$, $J_2(\theta)$ and $g_{12}(\theta)$ are varied with respect to $\theta$ in order to induce a nonlinear pumping for a topologically trivial band. c, One-cycle evolution of the density distribution $|\psi_j|^2$ of instantaneous solitons bifurcating from the lowest band. d, The evolution of center-of-mass positions of instantaneous solitons (red line) and the corresponding Wannier functions (black line) with respect to $\theta$ over one cycle. Here, we set $N=1.25$ and $g=1$.
• Figure 5: Supercell method.a, The effective potential generated by a soliton array with one soliton per supercell (three supercells are explicitly shown, each containing $L=40$ sites) as $\theta^\prime=\theta/L$ varies from $0$ to $2\pi$. The potential is periodic with period $L$. Only the potential felt by the first component is plotted, and the potential for the other component is similar. b, Energy spectra of the modulated linear Hamiltonian $H^{\textrm{sc}}$ containing the linear potentials in a with respect to momentum $k$ and $\theta^\prime$. Given that the spectra are periodic functions of $\theta^\prime$ with a period of $2\pi/L$, we only plot part of the spectrum with $\theta^\prime\in[0,2\pi/L]$. Here, we show three Bloch bands around $\mu(\theta)$ and the second band corresponds to the soliton wavefunction $\psi_{\sigma j}^{\textrm{s}}(\theta)$. c, The normalized projection of a soliton wavefunction $\psi_{\sigma j}^{\textrm{s}}$ on the second band in b, $\rho_{\textrm{sc} }(k,\theta^\prime)= \frac{L_{\textrm{sc} }}{N} |\sum_{\sigma j}\varphi_{k ,\sigma, j}(\theta^\prime)^*\psi_{\sigma j}^{\textrm{s}}(\theta^\prime)|^2,$ where $L_{\textrm{sc}}$ is the number of supercells, $\varphi_{k,\sigma,j}(\theta^\prime)$ is a Bloch state at momentum $k \in [0,2\pi/L]$ in the corresponding band of the Hamiltonian $H^{\textrm{sc}}$. Here, we consider case 2 in Fig. \ref{['fign2']} with $N=1.45$, $m_0=1$, $g_{12}=0$, and $g=1$.