Real-space Topology Research Articles

Engineering Topological Spin Hall Effect in 2D Multiferroic Material.

Topological spin Hall effect (TSHE), promoted by coupling between noncoplanar spins and real-space topology, is a significant phenomenon in condensed matter physics. However, the control of TSHE characteristics is missing due to its intrinsic robustness, and such fundamental difficulty prevents it from being used for spintronics up to now. Here, a rational design approach is demonstrated to engineer TSHE in a controllable and reversible fashion. Through symmetry and model analysis, it is unveiled that antiferromagnetic topological charge, as well as Lorentz forces, acted on conduction electrons, can be coupled with Dzyaloshinskii-Moriya interaction chirality for antiferromagnetic bimerons in 2D multiferroic materials. Such coupling guarantees the ferroelectric control of TSHE. Using first-principles calculations and atomic spin model simulations, the validity of this mechanism is further demonstrated in multiferroic monolayer CuCr2Se4 with experimental feasibility. The alter-chirality of the Dzyaloshinskii-Moriya interaction is found to play a crucial role in realizing this mechanism. This results extend TSHE to be used in spintronics and open a new direction for spintronics research.

Polarization textures in crystal supercells with topological bands

Two-dimensional materials are a highly tunable platform for studying the momentum space topology of the electronic wavefunctions and real space topology in terms of skyrmions, merons, and vortices of an order parameter. Such textures for electronic polarization can exist in moiré heterostructures. A quantum-mechanical definition of local polarization textures in insulating supercells was recently proposed. Here, we propose a definition for local polarization that is also valid for systems with topologically nontrivial bands. We introduce semilocal hybrid polarizations, which are valid even when the Wannier functions in a system cannot be made exponentially localized in all dimensions. We use this definition to explicitly show that nontrivial real-space polarization textures can exist in topologically nontrivial systems with nonzero Chern number under (1) an external superlattice potential, and (2) under a stacking-induced moiré potential. In the latter, we find that while the magnitude of the local polarization decreases discontinuously across a topological phase transition from trivial to topologically nontrivial, the polarization does not completely vanish. Our findings suggest that band topology and real-space polar topology may coexist in real materials. Published by the American Physical Society 2024

Near-Field Spin Chern Number Quantized by Real-Space Topology of Optical Structures.

The Chern number has been widely used to describe the topological properties of periodic structures in momentum space. Here, we introduce a real-space spin Chern number for the optical near fields of finite-sized structures. This new spin Chern number is intrinsically quantized and equal to the structure's Euler characteristic. The relationship is robust against continuous deformation of the structure's geometry and is irrelevant to the specific material constituents or external excitation. Our Letter enriches topological physics by extending the Chern number to real space, opening exciting possibilities for exploring the real-space topological properties of light.

Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism.

Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

Bloch Point Quadrupole Constituting Hybrid Topological Strings Revealed with Electron Holographic Vector Field Tomography.

Topological magnetic (anti)skyrmions are robust string-like objects heralded as potential components in next-generation topological spintronics devices due to their low-energy manipulability via stimuli such as magnetic fields, heat, and electric/thermal current. While these 2D topological objects are widely studied, intrinsically 3D electron-spin real-space topology remains less explored despite its prevalence in bulky magnets. 2D-imaging studies reveal peculiar vortex-like contrast in the core regions of spin textures present in antiskyrmion-hosting thin plate magnets with S4 crystal symmetry, suggesting a more complex 3D real-space structure than the 2D model suggests. Here, holographic vector field electron tomography captures the 3D structure of antiskyrmions in a single-crystal, precision-doped (Fe0.63Ni0.3Pd0.07)3P (FNPP) lamellae at room temperature and zero field. These measurements reveal hybrid string-like solitons composed of skyrmions with topological number W = -1 on the lamellae's surfaces and an antiskyrmion (W = + 1) connecting them. High-resolution images uncover a Bloch point quadrupole (four magnetic (anti)monopoles that are undetectable in 2D imaging) which enables the observed lengthwise topological transitions. Numerical calculations corroborate the stability of hybrid strings over their conventional (anti)skyrmion counterparts. Hybrid strings result in topological tuning, a tunable topological Hall effect, and the suppression of skyrmion Hall motion, disrupting existing paradigms within spintronics.

Correspondence between real-space topology and spectral flows at disclinations

Correspondence between real-space topology and spectral flows at disclinations

Noncontractible loop states from a partially flat band in a photonic borophene lattice

Abstract Flat band systems are commonly associated with compact localized states (CLSs) that arise from the macroscopic degeneracy of eigenstates at the flat band energy. However, in the case of singular flat bands, conventional localized flat band states are incomplete, leading to the existence of noncontractible loop states (NLSs) with nontrivial real-space topology. In this study, we experimentally and analytically demonstrate the existence of NLSs in a 2D photonic borophene lattice without a CLS counterpart, owing to a band that is flat only along high-symmetry lines and dispersive along others. Our findings challenge the conventional notion that NLSs are necessarily linked to robust boundary modes due to a bulk-boundary correspondence. Protected by the band flatness that originates from band touching, NLSs play a significant role in investigating the fundamental physics of flat band systems.

Non-singular and singular flat bands in tunable phononic metamaterials

Dispersionless flat bands can be classified into two types: (1) non-singular flat bands whose eigenmodes are completely characterized by compact localized states, and (2) singular flat bands that have a discontinuity in their Bloch eigenfunctions at a band touching point with an adjacent dispersive band, thereby requiring additional extended states to span their eigenmode space. In this study, we design and numerically demonstrate two-dimensional thin-plate phononic metamaterials in which tunable flat bands of both kinds can be achieved. Non-singular flat bands are achieved by fine tuning the ratio of the global tension and the bending stiffness in triangular and honeycomb lattices of plate resonators. A singular flat band arises in a kagome lattice due to the underlying lattice geometry, which can be made degenerate with two additional flat bands by tuning the plate tension. A discrete model of the continuum thin-plate system reveals the interplay of geometric and mechanical factors in determining the existence of flat bands of both types. The singular nature of the kagome lattice flat band is established via a metric called the Hilbert-Schmidt distance calculated between a pair of eigenstates infinitesimally close to the quadratic band touching point. We also simulate a phononic manifestation of a robust boundary mode arising from the singular flat band and protected by real-space topology in a finite system. Our theoretical and computational study establishes a framework for exploring flat-band physics in a tunable classical system, and for designing phononic metamaterials with potentially useful sound manipulation capabilities.

Defect-controlled topological edge states in the curved acoustic lattices

The concept of disclinations and dislocations, which are related to the real-space topology, have been widely studied in crystal materials. In two-dimensional (2D) materials, disclinations lead to the warping and deformation of the hosting material, yielding non-Euclidean geometries. However, such geometries have not been linked to the effects of topological edge states. Here we construct the curved waveguide paths by disclinations in the C 3 symmetric valley acoustic crystals with the non-Euclidean geometries. We then discuss the topological edge states as well as the curved waveguide modes in different defect-constructed structures. Simulation results show that the topological waveguide modes can be controlled not only by the alignment of topologically different valley acoustic crystals, but also by the edge dislocation. Our work may provide a route to study new exotic phenomena related to topological defects in non-Euclidean geometries and metamaterials.

Chiral broken symmetry descendants of the kagome lattice chiral spin liquid

The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids (CSLs) on the kagome lattice. Such CSLs are closely tied to bosonic fractional quantum Hall states with anyonic quasiparticles; however, their connection to nearby ordered states has remained a mystery. Here, we use spin-wave theory, parton Gutzwiller wave functions, and exact diagonalization to show that two distinct magnetic orders with uniform scalar chirality---the XYZ umbrella state and the octahedral spin crystal--emerge as competing orders in close proximity to the CSL. In this letter, we highlight the intimate link between a topologically ordered liquid and broken symmetry states with nontrivial real-space topology.

Continuously tunable topological defects and topological edge states in dielectric photonic crystals

Topological defects in solid-state materials are crystallographic imperfections that local perturbations cannot remove. Owing to their nontrivial real-space topology, topological defects such as dislocations and disclinations could trap anomalous states associated with nontrivial momentum-space topology. The real-space topology of dislocations and disclinations can be characterized by the Burgers vector $\mathbf{B}$, which is usually a fixed fraction and integer of the lattice constant in solid-state materials. Here we show that in a dielectric photonic crystal---an artificial crystalline structure---it is possible to tune $\mathbf{B}$ continuously as a function of the dielectric constant of dislocations. Through this unprecedented tunability of $\mathbf{B}$, we achieve proper controls of topological interfacial states, i.e., reversal of their helicities. Based on this fact, we propose a topological optical switch controlled by the dielectric constant of the tunable dislocation. Our results shed light on the interplay of real and reciprocal space topologies and offer a scheme to implement scalable and tunable robust topological waveguides in dielectric photonic crystals.

Experimental Realization of a Skyrmion Circulator.

Magnetic skyrmions are mobile topological spin textures that can be manipulated by different means. Their applications have been frequently discussed in the context of information carriers for racetrack memory devices, which on the other hand, exhibit a skyrmion Hall effect as a result of the nontrivial real-space topology. While the skyrmion Hall effect is believed to be detrimental for constructing racetrack devices, we show here that it can be implemented for realizing a three-terminal skyrmion circulator. In analogy to the microwave circulator, nonreciprocal transportation and circulation of skyrmions are studied both numerically and experimentally. In particular, successful control of the circulating direction of being either clockwise or counterclockwise is demonstrated, simply by changing the sign of the topological charge. Our studies suggest that the topological property of skyrmions can be incorporated for enabling novel spintronic functionalities; the skyrmion circulator is just one example.

Flat bands and band-touching from real-space topology in hyperbolic lattices

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetry. We show that two characteristic features of the energy spectrum of those models, namely the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain new insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.

Scanning Probe Microscopy Investigation of Topological Defects

Symmetry lowering phase transitions in ferroelectrics, magnets, and materials with various other forms of inherent order lead to the formation of topological defects. Their non-trivial real-space topology is characterized by a topological charge, which represents the topological invariant. The study of topological defects in such materials has seen increased interest over the last decade. Among the methods used for their study, scanning probe microscopy (SPM) with its many variants has provided valuable new insight into these structures at the nanoscale. In this perspective, various approaches are discussed, and different techniques are compared with regard to their ability to investigate topological defect properties.

Manipulation of acoustic vortex with topological dislocation states

Higher-order topological insulators as an exotic type of topological phases harboring fascinating topological corner or hinge states have attracted extensive attention recently. Dislocations are crystallinity-breaking defects in lattices that cannot be removed by local deformations due to nontrivial real-space topology. It is recently realized that dislocations can be used as a probe for higher-order topology. In this work, we propose a scheme to obtain acoustic dislocation states by introducing screw dislocations into higher-order topological insulators in a Kagome lattice. The topological dislocation states carry nonzero orbital angular momentum, which are locked to their propagation direction. We show that the screw dislocation states exist for both the tight binding model and the waveguide model as long as the system symmetry is preserved. By delicately designing the dislocation core, the dislocation states with selective angular momentum can be shifted into the bulk bandgap. Based on this in-gap dislocation states, filtering of acoustic vortex with a selective angular momentum is well achieved.

Observation of fractional spin textures in a Heusler material

Recently a zoology of non-collinear chiral spin textures has been discovered, most of which, such as skyrmions and antiskyrmions, have integer topological charges. Here we report the experimental real-space observation of the formation and stability of fractional antiskyrmions and fractional elliptical skyrmions in a Heusler material. These fractional objects appear, over a wide range of temperature and magnetic field, at the edges of a sample, whose interior is occupied by an array of nano-objects with integer topological charges, in agreement with our simulations. We explore the evolution of these objects in the presence of magnetic fields and show their interconversion to objects with integer topological charges. This means the topological charge can be varied continuously. These fractional spin textures are not just another type of skyrmion, but are essentially a new state of matter that emerges and lives only at the boundary of a magnetic system. The coexistence of both integer and fractionally charged spin textures in the same material makes the Heusler family of compounds unique for the manipulation of the real-space topology of spin textures and thus an exciting platform for spintronic and magnonic applications.

Topological magnon band structure of emergent Landau levels in a skyrmion lattice.

The motion of a spin excitation across topologically nontrivial magnetic order exhibits a deflection that is analogous to the effect of the Lorentz force on an electrically charged particle in an orbital magnetic field. We used polarized inelastic neutron scattering to investigate the propagation of magnons (i.e., bosonic collective spin excitations) in a lattice of skyrmion tubes in manganese silicide. For wave vectors perpendicular to the skyrmion tubes, the magnon spectra are consistent with the formation of finely spaced emergent Landau levels that are characteristic of the fictitious magnetic field used to account for the nontrivial topological winding of the skyrmion lattice. This provides evidence of a topological magnon band structure in reciprocal space, which is borne out of the nontrivial real-space topology of a magnetic order.

Topological dislocation modes in three-dimensional acoustic topological insulators

Dislocations are ubiquitous in three-dimensional solid-state materials. The interplay of such real space topology with the emergent band topology defined in reciprocal space gives rise to gapless helical modes bound to the line defects. This is known as bulk-dislocation correspondence, in contrast to the conventional bulk-boundary correspondence featuring topological states at boundaries. However, to date rare compelling experimental evidences have been presented for this intriguing topological observable in solid-state systems, owing to the huge challenges in creating controllable dislocations and conclusively identifying topological signals. Here, using a three-dimensional acoustic weak topological insulator with precisely controllable dislocations, we report an unambiguous experimental evidence for the long-desired bulk-dislocation correspondence, through directly measuring the gapless dispersion of the one-dimensional topological dislocation modes. Remarkably, as revealed in our further experiments, the pseudospin-locked dislocation modes can be unidirectionally guided in an arbitrarily-shaped dislocation path. The peculiar topological dislocation transport, expected in a variety of classical wave systems, can provide unprecedented control over wave propagations.

Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands

We experimentally realize fractal-like photonic lattices by use of the cw-laser-writing technique, thereby observing distinct compact localized states (CLSs) associated with different flatbands in the same lattice setting. Such triangle-shaped lattices, akin to the first generation Sierpinski lattices, possess a band structure where singular non-degenerate and nonsingular degenerate flatbands coexist. By proper phase modulation of an input excitation beam, we demonstrate not only the simplest CLSs but also their superimposition into other complex mode structures. Our experimental and numerical results are corroborated by theoretical analysis. Furthermore, we show by numerical simulation a dynamical oscillation of the flatband states due to beating of the CLSs that have different eigenenergies. These results may provide inspiration for exploring fundamental phenomena arising from the interplay of fractal structure, flatband singularity, and real-space topology.

Observation of Protected Photonic Edge States Induced by Real-Space Topological Lattice Defects.

Topological defects (TDs) in crystal lattices are elementary lattice imperfections that cannot be removed by local perturbations, due to their real-space topology. In the emerging field of topological photonics, photonic topological edge states arise from the nontrivial topology of the band structure defined in momentum space and are generally protected against defects. Here we show that adding TDs into a valley photonic crystal generates a lattice disclination that acts like a domain wall and hosts photonic topological edge states. Unlike previous topological waveguides, the disclination forms an open arc and functions as a free-form waveguide connecting a pair of TDs of opposite topological charge. This interplay between the real-space topology of lattice defects and momentum-space band topology provides a novel scheme to implement large-scale photonic structures with complex arrangements of robust topological waveguides and resonators.