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.

+ 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]...

AbstractFlat-band systems have attracted considerable interest in different branches of physics in the past decades, providing a flexible platform for studying fundamental phenomena associated with completely dispersionless bands within the whole Brillouin zone. Engineered flat-band structures have now been realized in a variety of systems, in particular, in the field of photonics. Flat-band localization, as an important phenomenon in solid-state physics, is fundamentally interesting in the exploration of exotic ground-state properties of many-body systems. However, direct observation of some flat-band phenomena is highly nontrivial in conventional condensed-matter systems because of intrinsic limitations. In this article, we briefly review recent developments on flat-band localization and the associated phenomena in various photonic lattices, including compact localized states, unconventional line states, and noncontractible loop states. We show that the photonic lattices offer a convenient platform for probing the underlying physics of flat-band systems, which may provide inspiration for exploring the fundamentals and applications of flat-band physics in other structured media from metamaterials to nanophotonic materials.

We establish experimentally a photonic super-honeycomb lattice (sHCL) by use of a cw-laser writing technique, and thereby demonstrate two distinct flatband line states that manifest as noncontractible-loop-states in an infinite flatband lattice. These localized states ("straight" and "zigzag" lines) observed in the sHCL with tailored boundaries cannot be obtained by superposition of conventional compact localized states because they represent a new topological entity in flatband systems. In fact, the zigzag-line states, unique to the sHCL, are in contradistinction with those previously observed in the Kagome and Lieb lattices. Their momentum-space spectrum emerges in the high-order Brillouin zone where the flat band touches the dispersive bands, revealing the characteristic of topologically protected bandcrossing. Our experimental results are corroborated by numerical simulations based on the coupled mode theory. This work may provide insight to Dirac-like 2D materials beyond graphene.