Atomic-Limit Mott Insulator in [4]Triangulene Frameworks.

Triangulene, one unique class of zigzag-edged triangular graphene molecules, has attracted tremendous research interest. In this work, as an ultimate phase of the Mott insulator, we present the realization of the atomic-limit Mott insulator in experimentally synthesized [4]triangulene frameworks ([4]-TGFs) from first-principles calculations. The frontier molecular orbitals of the nonmagnetic [4]triangulene consist of three coupled corner modes. After the isolated [4]triangulene is assembled into [4]-TGF, one special enantiomorphic flat band is created through the coupling of these corner modes, which is identified to be a second-order topological insulator with half-filled topological corner states at the Fermi level. Moreover, [4]-TGF prefers an antiferromagnetic ground state under Hubbard interactions, which further splits these metallic zero-energy states into an atomic-limit Mott insulator with spin-polarized corners. Since the fractional filling of topological corner states is a smoking-gun signature of higher-order topology, our results demonstrate a universal approach to explore the atomic-limit Mott insulators in higher-order topological materials.

Magnetic Real Chern Insulator in 2D Metal-Organic Frameworks.

Real Chern insulators have attracted great interest, but so far, their material realization is limited to nonmagnetic crystals and systems without spin-orbit coupling. Here, we reveal the magnetic real Chern insulator (MRCI) state in a recently synthesized metal-organic framework material Co3(HITP)2. Its ground state with in-plane ferromagnetic ordering hosts a nontrivial real Chern number, enabled by the C2zT symmetry and robustness against spin-orbit coupling. Distinct from previous nonmagnetic examples, the topological corner zero modes of MRCIs are spin-polarized. Furthermore, under small tensile strains, the material undergoes a topological phase transition from the MRCI to a magnetic double-Weyl semimetal phase, via a pseudospin-1 critical state. Similar physics can also be found in closely related materials Mn3(HITP)2 and Fe3(HITP)2, which also exist. Possible experimental detections and implications of an emerging magnetic flat band in the system are discussed.

Engineering Second-Order Corner States in 2D Multiferroics.

The understanding and manipulate of the second-order corner states are central to both fundamental physics and future topotronics applications. Despite the fact that numerous second-order topological insulators (SOTIs) are achieved, the efficient engineering in a given material remains elusive. Here, the emergence of 2D multiferroics SOTIs in SbAs and BP5 monolayers is theoretically demonstrated, and an efficient and straightforward way for engineering the nontrivial corner states by ferroelasticity and ferroelectricity is remarkably proposed. With ferroelectric polarization of SbAs and BP5 monolayers, the nontrivial corner states emerge in the mirror symmetric corners and are perpendicular to orientations of the in-plane spontaneous polarization. And remarkably the spatial distribution of the corner states can be effectively tuned by a ferroelastic switching. At the intermediate states of both ferroelectric and ferroelastic switchings, the corner states disappear. These finding not only combines exotic SOTIs with multiferroics but also pave the way for experimental discovery of 2D tunable SOTIs.

Nonlinear Imaging of Nanoscale Topological Corner States.

Topological states of light represent counterintuitive optical modes localized at boundaries of finite-size optical structures that originate from the properties of the bulk. Being defined by bulk properties, such boundary states are insensitive to certain types of perturbations, thus naturally enhancing robustness of photonic circuitries. Conventionally, the N-dimensional bulk modes correspond to (N - 1)-dimensional boundary states. The higher-order bulk-boundary correspondence relates N-dimensional bulk to boundary states with dimensionality reduced by more than 1. A special interest lies in miniaturization of such higher-order topological states to the nanoscale. Here, we realize nanoscale topological corner states in metasurfaces with C6-symmetric honeycomb lattices. We directly observe nanoscale topology-empowered edge and corner localizations of light and enhancement of light-matter interactions via a nonlinear imaging technique. Control of light at the nanoscale empowered by topology may facilitate miniaturization and on-chip integration of classical and quantum photonic devices.

Acoustic Higher-Order Topological Insulators Induced by Orbital-Interactions.

The discovery of higher-order topological insulator metamaterials, in analogy with their condensed-matter counterparts, has enabled various breakthroughs in photonics, mechanics, and acoustics. A common way of inducing higher-order topological wave phenomena is through pseudo-spins, which mimic the electron spins as a symmetry-breaking degree of freedom. Here, this work exploits degenerate orbitals in acoustic resonant cavities to demonstrate versatile, orbital-selective, higher-order topological corner states. Type-II corner states are theoretically investigated and experimentally demonstrated based on tailored orbital interactions, without the need for long-range hoppings that has so far served as a key ingredient for Type-II corner states in single-orbital systems. Due to the orthogonal nature of the degenerate p orbitals, this work also introduces a universal strategy to realize orbital-dependent edge modes, featuring high-Q edge states identified in bulk bands. These findings provide an understanding of the interplay between acoustic orbitals and topology, shedding light on orbital-related topological wave physics, as well as its applications for acoustic sensing and trapping.

Higher-order topological states in T-graphene and their realization in photonic crystals.

Higher-order topological states extend the power of nontrivial topological states beyond the bulk-edge correspondence. Here we study the higher-order topological states (corner states) in an open-boundary two-dimensional T-graphene lattice. Unlike the common zero-energy corner states, our findings reveal non-zero energy corner states in such lattice systems, and the energy could be controlled by modifying the hopping parameters. Moreover, the corner states could be transferred away from the lattice corners by designing the position-specific vacancy defects. The strong robustness of the corner states is also demonstrated against the uniaxial strain and vacancy defects, respectively. A plasmonic crystal is constructed to testify to the theory, in which the corner states are realized in optical modes and their higher-order topological properties are verified. Our results open the avenue of corner-states engineering, which holds significant physical implications of higher-order topological states for the design of photonic and electronic devices with specialized functionalities.

Third-order square-root topological insulators on decorated diamond sonic crystals.

The square-root operation can generate novel topological phases, whose nontrivial topological properties are inherited from the parent Hamiltonian. Here we report the acoustic realization of third-order square-root topological insulators by adding additional resonators between the site resonators of original diamond lattice. Due to the square-root operation, multiple acoustic localized modes appear in doubled bulk gaps. The bulk polarizations of the tight-binding models are employed to reveal the topological feature of the higher-order topological states. By tuning the coupling strength, we find the emergence of third-order topological corner states in doubled bulk gaps on tetrahedron-like and rhombohedron-like sonic crystals, respectively. The shape dependence of square-root corner states provides an extra degree of freedom for flexible manipulation on the sound localization. Furthermore, the robustness of the corner states in three-dimensional (3D) square-root topological insulator is well elucidated by introducing random disorders into the irrelevant bulk region of the proposed 3D lattices. This work extends square-root higher-order topological states into 3D system, and may find possible applications in selective acoustic sensors.

Majorana corner states in an attractive quantum spin Hall insulator with opposite in-plane Zeeman energy at two sublattice sites.

Higher-order topological superconductors and superfluids (SFs) host lower-dimensional Majorana corner and hinge states since novel topology exhibitions on boundaries. While such topological nontrivial phases have been explored extensively, more possible schemes are necessary for engineering Majorana states. In this paper we propose Majorana corner states could be realized in a two-dimensional attractive quantum spin-Hall insulator with opposite in-plane Zeeman energy at two sublattice sites. The appropriate Zeeman field leads to the opposite Dirac mass for adjacent edges of a square sample, and naturally induce Majorana corner states. This topological phase can be characterized by Majorana edge polarizations, and it is robust against perturbations on random potentials and random phase fluctuations as long as the edge gap remains open. Our work provides a new possibility to realize a second-order topological SF in two dimensions and engineer Majorana corner states.

Orbital Shift-Induced Boundary Obstructed Topological Materials with a Large Energy Gap.

Boundary obstructed topological phases caused by Wannier orbital shift between ordinary atomic sites are proposed, which, however, cannot be indicated by symmetry eigenvalues at high symmetry momenta (symmetry indicators) in bulk. On the open boundary, Wannier charge centers can shift to different atoms from those in bulk, leading to in-gap surface states, higher-order hinge states or corner states. To demonstrate such orbital shift-induced boundary obstructed topological insulators, eight material candidates are predicted, all of which are overlooked in the present topological databases. Metallic surface states, hinge states, or corner states cover the large bulk energy gap (e.g., more than 1 eV in TlGaTe2 ) at related boundary, which are ready for experimental detection. Additionally, these materials are also fragile topological insulators with hourglass-like surface states.

Out of equilibrium chiral higher order topological insulator on aπ-flux square lattice.

One of the hallmarks of bulk topology is the existence of robust boundary localized states. For instance, a conventionalddimensional topological system hostsd- 1 dimensional surface modes, which are protected by non-spatial symmetries. Recently, this idea has been extended to higher order topological phases with boundary modes that are localized in lower dimensions such as in the corners or in one dimensional hinges of the system. In this work, we demonstrate that a higher order topological phase can be engineered in a nonequilibrium state when the time-independent model does not possess any symmetry protected topological states. The higher order topology is protected by an emerging chiral symmetry, which is generated through the Floquet driving. Using both the exact numerical method and an effective high-frequency Hamiltonian obtained from the Brillouin-Wigner (BW) perturbation theory, we verify the emerging topological phase on aπ-flux square lattice. We show that the localized corner modes in our model are robust against a chiral symmetry preserving perturbation and can be classified as 'extrinsic' higher order topological phase. Finally, we identify a two dimensional topological invariant from the winding number of the corresponding sublattice symmetric one dimensional model. The latter model belongs to class AIII of ten-fold symmetry classification of topological matter.

Gapped topological kink states and topological corner states in honeycomb lattice.

Based on the tight-binding calculations on honeycomb lattice and photonic experimental visualization on artificial graphene (AG), we report the domain-wall-induced gapped topological kink states and topological corner states. In honeycomb lattice, domain walls (DWs) with gapless topological kink states could be induced either by sublattice symmetry breaking or by lattice deformation. We find that the coexistence of these two mechanisms will induce DWs with gapped topological kink states. Significantly, the intersection of these two types of DWs gives rise to topological corner state localized at the crossing point. Through the manipulation of the DWs, we show AG with honeycomb lattice structure not only a versatile platform supporting multiple topological corner modes in a controlled manner, but also possessing promising applications such as fabricating topological quantum dots composed of gapped topological kink states and topological corner states.

Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators.

Higher-order topological insulators (HOTIs) belong to a new class of materials with unusual topological phases. They have garnered considerable attention due to their capabilities in confining energy at the hinges and corners, which is entirely protected by the topology, and have thus become attractive structures for acoustic wave studies and control. However, for most practical applications at audible and low frequencies, compact and subwavelength implementations are desirable in addition to providing robust guiding of sound beyond a single-frequency operation. Here, a holey HOTI capable of sustaining deeply confined corner states 50 times smaller than the wavelength is proposed. A remarkable resilience of these surface-confined acoustic states against defects is experimentally observed, and topologically protected sound is demonstrated in three different frequency regimes. Concerning this matter, the findings will thus have the capability to push forward exciting applications for robust acoustic imaging way beyond the diffraction limit.