Topological States Enabled by Non-local Nonlinearity in Synthetic Dimensions

TL;DR

This paper explores how nonlocal nonlinearity in a synthetic topological lattice induces topological phases, transitions, and phenomena like fractional windings and double-period oscillations, expanding the understanding of nonlinear topological physics.

Contribution

It introduces a model with all-to-all nonlocal interactions in a synthetic topological lattice, revealing new topological phases and transitions driven by nonlinearity.

Findings

Nonlocal nonlinearity maintains chiral symmetry and induces quantized nonlinear winding.

Topological transitions occur with swallowtail band structures and band swapping.

Emergence of fractional windings and double-period Bloch oscillations from trivial systems.

Abstract

The interplay between topology and nonlinearity represents a central challenge in modern physics. Here, we investigate this interplay by considering a synthetic Su-Schrieffer-Heeger lattice with all-to-all nonlocal interactions. We find that the distinctive nonlinearity maintains an effective chiral symmetry and leads to a quantized nonlinear winding and Berry phase, as corroborated by the developed Bogoliubov nonlinear adiabatic theory. Increasing nonlinearity drives a sequence of topological transitions signaled by the appearance of characteristic swallowtail band structures at intermediate interaction strengths and band swapping in the strong nonlinear regime. The band swapping results in quantized fractional windings and double-period Bloch oscillations that are closely related to discrete time crystals. Remarkably, even starting from a topologically trivial linear system, nonlocal nonlinearity can induce an emergent topological phase with fractional windings. Experimentally, our model can be realized using photons in a degenerate optical cavity with Rydberg-mediated interactions. Our results establish a rigorous framework and pave the way for exploring nonlinear topological phenomena and their applications in synthetic quantum platforms.