Topological gap solitons in Rabi Su-Schrieffer-Heeger lattices

Topological insulators are unique materials possessing forbidden topological gap and behaving similar to usual insulators in their bulk, but at the same time supporting localized in-gap states at their edges that demonstrate exceptional robustness, because they are protected by topology of the system and cannot be destroyed by local disorder or edge perturbations. Topological insulators discovered in different areas of physics, including optics, acoustics, and physics of Bose-Einstein and polariton condensates, usually employ potential landscapes (or lattices), designed such as to feature specific degeneracies in their linear modal spectra, that can be lifted by various physical effects or controllable deformations of the potential, leading to opening of the topological gap, where edge states appear. In this work, using binary Bose-Einstein condensate we propose a type of topological insulator that does not explicitly use specially designed potential landscape, but instead utilizes spatially inhomogeneous Rabi coupling between two components, in the form of a one- or two-dimensional Su-Schrieffer-Heeger structure, combined with Zeeman splitting. Such Rabi lattices reveal the appearance of topologically nontrivial phases (including higher-order ones) controlled by spatial shift of the domains with enhanced coupling between condensates within unit cells of the structure, where localized topological states appear at the edges or in the corners of truncated Rabi lattice. We also show that the properties of edge states, their spatial localization, and location of their chemical potential within topological gap can be controlled by interatomic interactions that lead to formation of gap topological edge solitons bifurcating from linear edge states. Such solitons in condensates with inhomogeneous Rabi coupling appear as very robust nonlinear topological objects that do not require any threshold norm for their formation even in two-dimensional geometries, and that can exist in stable form for both attractive and repulsive interactions. Our results demonstrate considerable enhancement of stability of solitons in topological Rabi lattices in comparison with trivial Rabi lattices. They open additional prospects for realization of topologically nontrivial phases by spatial engineering of coupling in multicomponent systems.