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A hallmark of symmetry-protected topological phases are topological boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys such a topological phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker sub-symmetry requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. Here we demonstrate that the boundary states are protected by only the sub-symmetry, using Su–Schrieffer–Heeger and breathing kagome lattice models, even though the overall topological invariant and the associated topological phase can be destroyed by sub-symmetry-preserving perturbations. By precisely controlling symmetry breaking in photonic lattices, we experimentally demonstrate such sub-symmetry protection of topological states. Furthermore, we introduce a long-range hopping symmetry in breathing kagome lattices, which resolves a debate on the higher-order topological nature of their corner states. Our results apply beyond photonics and could be used to explore the properties of symmetry-protected topological phases in the absence of full symmetry in different physical contexts.

The orbital degrees of freedom play a pivotal role in understanding fundamental phenomena in solid-state materials as well as exotic quantum states of matter including orbital superfluidity and topological semimetals. Despite tremendous efforts in engineering synthetic cold-atom, as well as electronic and photonic lattices to explore orbital physics, thus far high orbitals in an important class of materials, namely, higher-order topological insulators (HOTIs), have not been realized. Here, we demonstrate $$p$$ p -orbital corner states in a photonic HOTI, unveiling their underlying topological invariant, symmetry protection, and nonlinearity-induced dynamical rotation. In a Kagome-type HOTI, we find that the topological protection of $$p$$ p -orbital corner states demands an orbital-hopping symmetry in addition to generalized chiral symmetry. Due to orbital hybridization, nontrivial topology of the $$p$$ p -orbital HOTI is “hidden” if bulk polarization is used as the topological invariant, but well manifested by the generalized winding number. Our work opens a pathway for the exploration of intriguing orbital phenomena mediated by higher-band topology applicable to a broad spectrum of systems.

Topological singularities (TSs) in momentum space give rise to intriguing fundamental phenomena as well as unusual material properties, attracting a great deal of interest in the past decade. Recently, we have demonstrated universal momentum-to-real-space mapping of TSs and pseudospin angular momentum conversion using photonic honeycomb (graphene-like) and Lieb lattices. Such mapping arises from the Berry phase encircling the Dirac or Dirac-like cones, and is thus of topological origin. In this paper, we briefly present previous observations of topological charge conversion, and then we present our first theoretical analysis and experimental demonstration of TS mapping in a new T-graphene lattice. Unlike other lattices, there are two coexisting but distinct TSs located at different high-symmetry points in the first Brillouin zone of T-graphene, which enables controlled topological charge conversion in the same lattice. We show active manipulation of the TS mapping, turning the two TSs into vortices of different helicities, or one into a high-order vortex but the other into a quadrupole. Such TS manipulation and pseudospin-to-orbital conversion may find applications in optical communications and quantum information, and may bring insight into the study of other Dirac-like structures with multiple TSs beyond the 2D photonic platform., 19 pages, 6 figures

Square-root higher-order topological insulators (HOTIs) are recently discovered new topological phases, with intriguing topological properties inherited from a parent lattice Hamiltonian. Different...

Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize phases of a generalized Aubry-Andr\'{e}-Harper model. The persistent entropy and mean squared lifetime of features obtained using persistent homology behave similarly to conventional measures (Shannon entropy and inverse participation ratio) and can distinguish localized, extended, and crticial phases. However, we find that the persistent entropy also clearly distinguishes ordered from disordered regimes of the model. The persistent homology approach can be applied to both the energy eigenstates and the wavepacket propagation dynamics., Comment: Published version. 8 pages, 9 figures

Flat-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 demonstrate orbital corner states in photonic higher-order topological insulators (HOTIs), comparing the p-band characteristics between two-dimensional Su-Schrieffer-Heeger (SSH) and breathing Kagome lattices. The orbital corner excitations are explored in topological nontrivial and trivial regimes.

We theoretically and experimentally demonstrate the concept of sub-symmetry in symmetry-protected topological systems, wherein the original symmetry is partially broken so bulk topological invariant no longer exists, but some edge states are still topologically protected.

We demonstrate nontrivial higher-order topology in photonic honeycomb lattices with different corner/edge geometries established by cw-laser-writing technique, unveiling two distinct corner states that are expected to represent features of higher-order topological Dirac semimetal.

We experimentally demonstrate photonic higher-order topological orbital corner states in laser-written breathing Kagome lattices, unveiling their charaterisitic dipole-like mode distribution and dynamical rotation in presence of optical nonlinearity.

We demonstrate corner states in trivial and nontrivial 2D SSH lattices diagonally stretched to exhibit both HOTI and higher-order semimetal-like features and examine the underlying topological mechanisms from the calculated spectral phase density.

We report higher-order orbital corner states in two-dimensional Su-Schrieffer-Heeger (SSH) photonic lattices, representing an orbital HOTI model. With appropriately shaped input beams, we explore orbital corner excitations in topologically nontrivial and trivial regimes.

We demonstrate a new type of edge states in photonic graphene, residing across entire Brillouin zone and topologically protected by the winding number. Compact localized edge states associated with flat dispersion are also observed.

We propose a scheme to realize parity-time-symmetric flatband systems by introducing on-site gain/loss to rhombic lattices with staggered couplings. Such non-Hermitian lattices possess chiral-symmetry-protected third-order exceptional points, exhibiting an ∼є1/3 eigenvalue separation.

We design synthetic modal dimension (SMD) by deep learning and demonstrate direct mode control in such SMD through engineered photonic lattices, including mode confinement and conversion which are otherwise difficult to achieve in real-space dimensions.

We experimentally demonstrate higher-order topological charge conversion, exhibiting a vortex pair of identical helicities that can be reversed altogether by selectively mapping different momentum-space topological singularities in photonic T-graphene lattices with integer pseudospin-1 Dirac-like cones.

We realize fractal-like photonic lattices, thereby demonstrating noncontractible loop states in a Corbino-shaped geometry as well as distinct compact localized states in a triangle-shaped geometry that possess both point (singular) and plane (nonsingular) flatband degeneracies.

We demonstrate nonlinear generation and topologically tuned confinement of THz waves in a wedge-shaped Su-Schrieffer-Heeger lattice laser-written in a lithium-niobate chip, examining THz defect modes from band structure and analyzing their robustness to chiral perturbation.

We discuss our recent demonstrations of nonlinear control of topological states in cw-laser-written photonic lattices of different symmetries, including manipulation of edge and corner states in higher-order topological insulators and dynamically emerging nonlinear topological phenomena.

The orbital degrees of freedom play a pivotal role in understanding fundamental phenomena in solid-state materials as well as exotic quantum states of matter including orbital superfluidity and topological semimetals. Despite tremendous efforts in engineering synthetic cold-atom, electronic and photonic lattices to explore orbital physics, thus far high orbitals in an important class of materials, namely, the higher-order topological insulators (HOTIs), have not been realized. Here, we demonstrate p-orbital corner states in a photonic HOTI, unveiling their underlying topological invariant, symmetry protection, and nonlinearity-induced dynamical rotation. In a Kagome-type HOTI, we find that topological protection of the p-orbital corner states demands an orbital-hopping symmetry, in addition to the generalized chiral symmetry. Due to orbital hybridization, the nontrivial topology of the p-orbital HOTI is hidden if bulk polarization is used as the topological invariant, but well manifested by the generalized winding number. Our work opens a pathway for the exploration of intriguing orbital phenomena mediated by higher band topology applicable to a broad spectrum of systems., Comment: 11 pages, 4 figures

Higher-order exceptional points (EPs), which appear as multifold degeneracies in the spectra of non-Hermitian systems, are garnering extensive attention in various multidisciplinary fields. However, constructing higher-order EPs still remains a challenge due to the strict requirement of the system symmetries. Here we demonstrate that higher-order EPs can be judiciously fabricated in parity–time ( PT )-symmetric staggered rhombic lattices by introducing not only on-site gain/loss but also non-Hermitian couplings. Zero-energy flatbands persist and symmetry-protected third-order EPs (EP3s) arise in these systems owing to the non-Hermitian chiral/sublattice symmetry, but distinct phase transitions and propagation dynamics occur. Specifically, the EP3 arises at the Brillouin zone (BZ) boundary in the presence of on-site gain/loss. The single-site excitations display an exponential power increase in the PT -broken phase. Meanwhile, a nearly flatband sustains when a small lattice perturbation is applied. For the lattices with non-Hermitian couplings, however, the EP3 appears at the BZ center. Quite remarkably, our analysis unveils a dynamical delocalization-localization transition for the excitation of the dispersive bands and a quartic power increase beyond the EP3. Our scheme provides a new platform toward the investigation of the higher-order EPs and can be further extended to the study of topological phase transitions or nonlinear processes associated with higher-order EPs.

Type-II Dirac/Weyl points, although impermissible in particle physics due to Lorentz covariance, were uncovered in condensed matter physics, driven by fundamental interest and intriguing applications of topological materials. Recently, there has been a surge of exploration of such generic points using various engineered platforms including photonic crystals, waveguide arrays, metasurfaces, magnetized plasma and polariton micropillars, aiming towards relativistic quantum emulation and understanding of exotic topological phenomena. Such endeavors, however, have focused mainly on linear topological states in real or synthetic Dirac/Weyl materials. Here, we demonstrate nonlinear valley Hall edge states (VHESs) in laser-writing anisotropic photonic lattices hosting type-II Dirac points. These self-trapped VHESs, manifested as topological gap quasi-solitons, are fundamentally distinct from their linear counterparts in type-I lattices and from all previously found topological solitons. Our finding may provide a route for understanding nonlinear phenomena in Lorentz-violating topological systems and for developing advanced light sources from topological insulator lasers., Comment: 15 pages, 4 figures

In this work, we study topological edge and corner states in two-dimensional (2D) Su-Schrieffer-Heeger lattices from designer surface plasmon crystals (DSPCs), where the vertical confinement of the designer surface plasmons enables signal detection without the need of additional covers for the sample. In particular, the formation of higher-order topological insulator can be determined by the two-dimensional Zak phase, and the zero-dimensional subwavelength corner states are found in the designed DSPCs at the terahertz (THz) frequency band together with the edge states. Moreover, the corner state frequency can be tuned by modifying the defect strength, i.e., the location or diameter of the corner pillars. This work may provide a new approach for confining THz waves in DSPCs, which is promising for the development of THz topological photonic integrated devices with high compactness, robustness and tunability.

Topological photonics is nowadays one of the most active areas in optics, yet thus far research efforts in this area are mainly focused on investigating linear topological photonic structures [1] , where dynamics are usually described through eigenmode expansions. The presence of nonlinearity typically couples the eigenmodes and gives rise to nonlinear phenomena ranging from solitons to extreme waves and chaos. Recently, there has been a surge of interest in nonlinear topological photonics, demonstrating that nonlinearity can induce topological phase transitions, coupling into topologically protected edge states, as well as nontrivial topological gap solitons and topological insulators [2] - [7] . However, how nonlinearity affects nontrivial topological systems and couples eigenmodes in nonlinear topological systems still remains a largely unexplored territory. Here, we report dynamical topological phase transitions entirely driven by nonlinearity, achieved in colliding soliton lattices. By forming two one-dimensional soliton sublattices and kicking them initially in opposite directions, soliton interaction forms a paradigm model of topological physics, namely, the Su-Schrieffer-Heeger (SSH) lattice [8] - [10] . During nonlinear propagation, such a soliton SSH-like lattice repetitively evolves from a topological trivial to a nontrivial regime (featured by the Zak phase [8] ), exhibiting dynamical topological phase transitions.

In the past decade, topological photonics has proved to be one of the most active research fields with interdisciplinary interest [1] . In particular, the Su-Schrieffer-Heeger (SSH) model [2] , [3] representing a prototypical one-dimensional (1D) topological system has received a great amount of attention due to the simplicity of its energy band structure exhibiting nontrivial chiral topological modes. In its basic 1D version, the two-level bandgap depends on the absolute difference between intra- and inter-cell coupling, singling out the band topology associated with the Zak phase [2] . Two-dimensional (2D) SSH configurations can typically support zero-dimensional corner states regardless of their physical dimensions, making it a useful candidate as a testbench for understanding high-order topological insulators (HOTIs) [4] – [7] . Thus far, most endeavors with such SSH systems have been focused on the fundamental topological states. Here, we report high-order topological states in finite SSH photonic lattices, with emphasis on orbital (or p-) states in both 1D and 2D systems. Interest arises from the fact that such modes present robust and stable proprieties [8] – [9] .

We demonstrate nonlinear generation of terahertz topological edge states in an SSH lattice engineered on a LiNbO 3 chip, manifested directly in the bandgap from the dispersion relation and further verified by the characteristic electric field distribution.

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 a 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 demonstrate the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states which, serendipitously, are also BICs in a laser-written second-order topological lattice. We further demonstrate nonlinear coupling with edge states at a low nonlinearity, transitioning to solitons at a high nonlinearity. Theoretically, we calculate the analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by both focusing and defocusing nonlinearities. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices., Comments are welcome

We demonstrate dynamical topological phase transitions entirely driven by nonlinearity, which constitute an example of emergent nonlinear topological phenomena. These transitions in our system occur due to soliton interactions forming Su-Schrieffer-Heeger lattices.

We report on hollow beam generation in an otherwise uniform low-power infrared beam co-propagating with a green beam in plasmonic nanosuspensions. Experimentally observed beam dynamics are numerically corroborated based on nonlinear cross-phase modulation.

Effects of nonlinearity on topologically protected linear modes in photonic SSH lattices are studied theoretically and experimentally. Nonlinearity turns these linear modes into weakly nonlinear topological gap solitons that cannot sustain high powers.

The toolbox quantities used for manipulating the flow of light include typically amplitude, phase, and polarization. Pseudospins, such as those arising from valley degrees of freedom in photonic structures, have recently emerged as an excellent candidate for this toolbox, in parallel with rapid development of spintronics and valleytronics in condensed-matter physics. Here, by employing symmetry-broken honeycomb photonic lattices, valley-dependent wavepacket self-rotation manifested in spiraling intensity patterns is demonstrated, which occurs without any initial orbital angular momentum. Theoretically, it is shown that such wavepacket self-rotation is induced by the Berry phase and results in Zitterbewegung oscillations. The frequency of Zitterbewegung is proportional to the gap size, while the helicity of self-rotation is valley-dependent, i.e., correlated with the Berry curvature. These results lead to new understanding of the venerable Zitterbewegung phenomenon from the perspective of topology and are readily applicable on other platforms such as 2D Dirac materials and ultracold atoms.

We investigate nonlinear effects on corner modes in a two-dimensional Su-Schrieffer-Heeger-type photonic platform. Nonlinearity induces transition between corner and edge states in topologically nontrivial lattices, in contrast to facilitating bulk modes in the trivial counterparts.

We demonstrate that optical nonlinearity can effectively modulate the loss of a topological defect waveguide in a non-Hermitian photonic lattice, leading to switching between PT and non-PT-symmetric regimes and control of topological zero modes.

Compact terahertz (THz) functional devices are greatly sought after for high-speed wireless communication, biochemical sensing, and non-destructive inspection. However, controlled THz generation, along with transport and detection, has remained a challenge especially for chip-scale devices due to low-coupling efficiency and unavoidable absorption losses. Here, based on the topological protection of electromagnetic waves, we demonstrate nonlinear generation and topologically tuned confinement of THz waves in an engineered lithium niobate chip forming a wedge-shaped Su–Schrieffer–Heeger lattice. Experimentally measured band structures provide direct visualization of the THz localization in the momentum space, while robustness of the confined mode against chiral perturbations is also analyzed and compared for both topologically trivial and nontrivial regimes. Such topological control of THz waves may bring about new possibilities in the realization of THz integrated circuits, promising for advanced photonic applications.