Quantum theory of fractional topological pumping of lattice solitons

TL;DR

The paper tackles the problem of quantized topological transport for self-bound many-particle states in interacting lattice systems. By introducing a quantum COM-based effective Hamiltonian and a corresponding soliton band structure $E_\mu(K)$, the authors classify topological phases via a generalized symmetry framework and define an invariant (an effective single-particle Chern number or Wilson loop) that governs soliton transport. They derive analytic expressions in the strong-interaction limit (triplon model) and validate them with numerics, showing how increasing $U$ causes COM-band merging and fractional transport, and identifying conditions under which transport quantization can fail due to degeneracies with extended states. The results connect topological pumping in nonlinear photonic lattices to a robust, fully quantum description and offer predictive insight for experiments in photonics and cold-atom platforms with bound soliton-like states.

Abstract

One of the hallmarks of topological systems is the robust quantization of particle transport. It is the origin of the integer-valued quantum Hall conductivity and a potential tool for quantum information technology. Recent experiments on topological pumps constructed by using arrays of photonic waveguides and described by the (lattice-translational invariant) Aubry-André-Harper (AAH) model, have demonstrated both integer and fractional transport of lattice solitons. In these systems, a background medium mediates interactions between photons via a Kerr nonlinearity and leads to the formation of self-bound multi-photon states. Upon increasing the interaction strength a sequence of transitions was observed from a phase with integer transport in a pump cycle through different phases of fractional transport to a phase with no transport. We here present a quantum description of topological pumps of self-bound many-particle states in terms of an effective Hamiltonian of their center-of-mass (COM) motion, which allows to introduce an effective band structure $E_μ(K)$ with $K$ being the COM momentum, and to classify topological phases in terms of generalized symmetries. We provide an explicit analytic expression of the effective Hamiltonian for few particles in the strong interaction limit and present numerical results in the more general case. We identify a topological invariant, an effective single-particle Chern number, which fully governs the soliton transport. Increasing the interaction strength in the AAH model leads to a successive merging of COM bands, which is the origin of the observed sequence of topological phase transitions and also the potential breakdown of topological quantization for some interaction strength.

Figures (8)

• Figure 1: (a) and (b) 1D Aubry-André-Harper model with modulated hopping rates $J_1(t) ... J_5(t)$ and on-site interactions $U$. (c) Motion of center of mass of a soliton $\Delta X$ in a pump cylce. Upon increasing the interaction strength there are multiple transitions between phases of integer, fractional and eventually absent transport, in some cases intersected by a small interval of non-quantized transport (grey area).
• Figure 2: Overlap of eigensolution of the DNLSE, Eq. \ref{['eq:DNLSE']}, for the AAH model with phase offset $k=2$ and unit-cell size $p=5$ with single-particle Bloch bands for different interaction strength $U/J$. Data points for the same interaction strength correspond to different times in the pumping cycle.
• Figure 3: Instantaneous soliton energies for 10 particles for attractive interaction $U/J=0.1$ with phase-offset $k=1$ in (a) and with phase-offset $k=2$ in (c), and unit-cell size $p=5$. Note that the soliton bands are almost flat and the variation of $E(K)$ with $K$ is less than the width of the lines. The corresponding COM-movements in a pump cycle of the red and blue marked energies are shown in (b) and (d). The energies and COM-positions are obtained with the three-site soliton ansatz.
• Figure 4: Logarithm of the absolute value of the Berry curvature $\ln|{\cal F}(K,t)|$ in the lowest energy solution for $N=3$ particles, a $p=5$ unit-cell and a $k=1$ phase-offset. Integrating Eq. \ref{['eq:COM-Chern-number']} with this Berry curvature yields a Chern number $C=1$. The sharp peaks, where $\ln|{\cal F}(K,t)|$ is on the order of 10, are numerical relicts but are effectively a set of measure zero for the integration.
• Figure 5: Merging of soliton energies obtained from exact diagonalization with increasing interaction strength $U$ for $N=3$ particles, a $p=5$ unit-cell and $k=2$ phase-offset. The number of unit cells is $L=6$. A soliton originally prepared at $t=0$ in one of the two bands will remain in this band if the changes are adiabatic and the soliton energies do not touch (Figs. a) and b)). Once the energies touch in Dirac-like cones, the solutions switch bands at every crossing (red curve in Fig. c). The width of the lines is larger than the width of the soliton bands in $K$ space.