Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

+ These authors made equal contribution.Topological properties of materials, as manifested in the intriguing phenomena of quantum Hall effect and topological insulators 1,2 , have attracted overwhelming transdisciplinary interest in recent years 3-7 . Topological edge states, for instance, have been realized in versatile systems including electromagnetic-waves 8-12 . Typically, topological properties are revealed in momentum space, using concepts such as Chern number and Berry phase. Here, we demonstrate a universal mapping of the topology of Dirac-like cones from momentum space to real space. We evince the mapping by exciting the cones in photonic honeycomb (pseudospin-1/2) 13,14 and Lieb (pseudospin-1) lattices 15 with vortex beams of topological charge , optimally aligned for a chosen pseudospin state , leading to direct observation of topological charge conversion that follows the rule of → (see Figs. 1a, 1b). The mapping is theoretically accounted for all initial excitation conditions with the pseudospin-orbit interaction and nontrivial Berry phases. Surprisingly, such a mapping exists even in a deformed lattice where the total angular momentum is not conserved, unveiling its topological origin. The universality of the mapping extends beyond the photonic platform and 2D lattices: equivalent topological conversion occurs for 3D Dirac-Weyl synthetic magnetic monopoles 16-18 (see Fig. 1c), which could be realized in ultracold atomic gases 19 and responsible for mechanism behind the vortex creation in electron beams traversing a magnetic monopole field 20 .The coupling of spin and orbital degrees of freedom is in many systems intertwined with the underlying topology of the space and the Berry phase 21 . For instance, in condensed matter electronic systems, study of spin-orbit interaction leads to discovery of topological insulators, which have emerged as an important field for itself. The physics of electron beams illustrates many examples where spin-orbit coupling is integrated with topology 22 . There is also a plethora of related examples in optics 23 : with real space Berry phase optical elements such as q-plates and metasurfaces, circular polarization of light (intrinsic spin) can be transformed to an optical vortex carrying orbital angular momentum (OAM) 24-26 ; for light propagating along a coiled ray trajectory, the dynamics is governed by the action of the monopole in Berry curvature, leading to the spin-Hall effect of light 27 . Interestingly, an analogous topological transport of sound waves was recently observed, thanks to the spin-redirection geometric phase 28 .When discussing spin in optical systems, it is light polarization or photon spin that is usually considered as the spin degree of freedom 23,29 . Similarly, in electronic systems it is the intrinsic electron spin 1,2 . However, for light (electrons) propagating in structured photonic media (crystalline lattices) with inherent degrees of freedom, the concept of pseudospin independent of any intrinsic particle property emerges [13][14][15][30]...

We demonstrate universal conversion of topological singularities in momentum-space (Dirac points) to topological defects in real-space (optical vortices). We show that this conversion persists even in stretched lattices, topologically protected by a quantized Berry phase.

We report universal mapping of topological Dirac-like points in momentum-space to optical vortices in real-space. This conversion persists even in stretched lattices, topologically protected by a quantized Berry phase.

We demonstrate nonlinear control of corner modes in a photonic second-order topological insulator, representing topological bound-states in the continuum which are coupled with edge states at low nonlinearity but driven out of the continuum at high nonlinearity.