Flatband systems typically host "compact localized states" (CLS) due to destructive interference and macroscopic degeneracy of Bloch wave functions associated with a dispersionless energy band. Using a photonic Lieb lattice (LL), we show that conventional localized flatband states are inherently incomplete, with the missing modes manifested as extended line states which form non-contractible loops winding around the entire lattice. Experimentally, we develop a continuous-wave laser writing technique to establish a finite-sized photonic LL with speciallytailored boundaries, thereby directly observe the unusually extended flatband line states. Such unconventional line states cannot be expressed as a linear combination of the previously observed CLS but rather arise from the nontrivial real-space topology. The robustness of the line states to imperfect excitation conditions is discussed, and their potential applications are illustrated.Flatband systems, first proposed for the study of ferromagnetic ground states in multiband Hubbard models, have proven to be conceptually effective and important in condensed matter physics [1][2][3]. They are characterized by a band structure with one band being completely flat, signaling macroscopic degeneracy. One can construct CLS which remain intact during evolution due to destructive interference. Over the years, a variety of approaches have been developed to design and characterize different flatband systems [4][5][6][7][8], with lattice geometries ranging from sawtooth, stub, diamond, dice, kagome, to Lieb and perovskite lattices in general [7][8][9][10][11][12]. This is largely due to the flatband systems providing a platform for probing various fundamental phenomena that have intrigued scientists for decades, including Anderson localization [6,13, 14], nontrivial topological phases and quantum Hall states [15][16][17][18][19], and flatband superfluidity [20,21].The Lieb lattice (LL) -a face-centered square depleted lattice [ Fig. 1(a)] -is geometrically different from other two-dimensional lattices such as square and honeycomb lattices. This peculiar system possesses a single conical intersection point in its Brillouin zone (BZ), where the flatband is sandwiched between two conical Bloch bands [ Fig. 1(b)]. The flatband in the LL is protected by a chiral symmetry, and its intersection with the dispersive bands is protected by real-space topology [12,22,23]. Recently, LLs have been realized in several different settings, including Bose-Einstein condensates [4,24], surface state electrons [25,26], exciton-polaritons in micropillars [27], and waveguide arrays in photonic structures [28][29][30][31][32]. However, so far most of previous experimental studies have focused on the demonstration of the LL structures and their associated CLS, overlooking unusual features that arise in infinitely extended lattices [ Fig. 1(c)] or finite (truncated) lattices with different cutting boundaries [Figs. 1(d, e)].In this Letter, we demonstrate the CLS previously investigated in the LL are lin...

Topology, parity-time (PT) symmetry, and nonlinearity are at the origin of many fundamental phenomena in complex systems across the natural sciences, but their mutual interplay remains unexplored. We established a nonlinear non-Hermitian topological platform for active tuning of PT symmetry and topological states. We found that the loss in a topological defect potential in a non-Hermitian photonic lattice can be tuned solely by nonlinearity, enabling the transition between PT-symmetric and non–PT-symmetric regimes and the maneuvering of topological zero modes. The interaction between two apparently antagonistic effects is revealed: the sensitivity close to exceptional points and the robustness of non-Hermitian topological states. Our scheme using single-channel control of global PT symmetry and topology via local nonlinearity may provide opportunities for unconventional light manipulation and device applications.

The flourishing of topological photonics in the last decade was achieved mainly due to developments in linear topological photonic structures. However, when nonlinearity is introduced, many intriguing questions arise. For example, are there universal fingerprints of the underlying topology when modes are coupled by nonlinearity, and what can happen to topological invariants during nonlinear propagation? To explore these questions, we experimentally demonstrate nonlinearity-induced coupling of light into topologically protected edge states using a photonic platform and develop a general theoretical framework for interpreting the mode-coupling dynamics in nonlinear topological systems. Performed on laser-written photonic Su-Schrieffer-Heeger lattices, our experiments show the nonlinear coupling of light into a nontrivial edge or interface defect channel that is otherwise not permissible due to topological protection. Our theory explains all the observations well. Furthermore, we introduce the concepts of inherited and emergent nonlinear topological phenomena as well as a protocol capable of revealing the interplay of nonlinearity and topology. These concepts are applicable to other nonlinear topological systems, both in higher dimensions and beyond our photonic platform.

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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.