Coupling of Edge States and Topological Bragg Solitons Weifeng Zhang, 1 Xianfeng Chen, 1 Yaroslav V. Kartashov, 2 Vladimir V. Konotop, 3 and Fangwei Ye 1,* 1 State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 2 Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 108840, Russia 3 Departamento de Física and Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, Lisboa 1749-016, Portugal (Received 18 May 2019; published 19 December 2019) The existence of the edge states at the interface between two media with different topological properties is protected by symmetry, which makes such states robust against structural defects or disorder. We show that, if a system supports more than one topological edge state at the interface, even a weak periodic deformation may scatter one edge state into another without coupling to bulk modes. This is the Bragg scattering of the edge modes, which in a topological system is highly selective, with closed bulk and backward scattering channels, even when conditions for resonant scattering are not satisfied. When such a system bears nonlinearity, Bragg scattering enables the formation of a new type of soliton—topological Bragg solitons. We report them in a spin-orbit-coupled (SOC) Bose-Einstein condensate in a homogeneous honeycomb Zeeman lattice. An interface supporting two edge states is created by two different SOCs, with the y component of the synthetic magnetic field having opposite directions at different sides of the interface. The reported Bragg solitons are found to be stable. DOI: 10.1103/PhysRevLett.123.254103 Topological edge states are fundamental for understand- ing the physics of many phenomena including quantum [1,2], anomalous [3], and quantum spin [4,5] Hall effects, topological insulators [6], Majorana fermions [7], just to mention a few. Introduced in solid state physics, topological edge states were shown to be a universal wave phenomenon. In particular, they enable protected edge currents in photonic crystals as predicted in [8] and observed in [9–11] (see review [12]), in surface plasmon-polariton systems [13], in systems of cold atoms in optical lattices [14–16], in spin- orbit coupled Bose-Einstein condensates (SOC BECs) [17], and in optoelectronic systems, such as exciton-polariton condensates [18,19]. Topological edge states are robust against disorder [6], which distinguishes them from bulk states that can be manipulated by perturbations [20]. This makes such states promising for a variety of applications. Meanwhile, when it is necessary to selectively excite or remove edge states, or transform them in any other way, this robustness becomes a drawback. Topological edge states can be coupled by nonlinearity that gives rise to a rich set of phenomena, such as modulation instability [19] or envelope soliton propagating along the edge [17]. However, by enabling linear coupling one qualitatively enriches the physics of respective systems. Now nonlinear interactions, requiring simultaneous energy and momentum conservation laws, can be made resonant. That leads to a plethora of novel phenomena, which so far were not considered for edge states. In particular, linear mode coupling resulting from periodic modulation of the system, i.e., from Bragg scattering, in the presence of nonlinearity can lead to the formation of Bragg solitons [21,22], which are relevant for many applications [23], and whose properties qualitatively differ from those of the envelope solitons mentioned above. In this Letter, we introduce an efficient mechanism of coupling and conversion of topological edge states based on Bragg scattering by periodic modulations of an interface between topologically different media. This mechanism works when a system supports more than one topological edge state per interface. We show that even a very weak periodic perturbation of such an interface may result in periodic transitions between two edge states moving with different group velocities. Such a coupling is highly selective, with closed bulk and backward scattering chan- nels. Furthermore, we construct Bragg solitons propagating along the interface and representing a spatially localized envelope of two coupled edge states. As a case example, we address an atomic SOC BEC [24,25], characterized by a spinor order parameter Ψ ¼ ðΨ ð1Þ ; Ψ ð2Þ Þ T (T stands for the transpose). The condensate is placed in a honeycomb lattice, which can be created experimentally by applying at least three laser beams [26,27]. Almost arbitrary 2D field distributions, and thus optical potentials, can be produced by the interference of quasinondiffracting laser beams [28]. The lattice is charac- terized by inverted potential profiles for the spinor compo- nents and can be created by periodically varying Zeeman splitting (see [29] for a possibility of experimental PHYSICAL REVIEW LETTERS 123, 254103 (2019) 0031-9007=19=123(25)=254103(6) 254103-1 © 2019 American Physical Society