Topological Boundary Research Articles

Optical bulk-boundary dichotomy in a quantum spin Hall insulator

The bulk-boundary correspondence is a critical concept in topological quantum materials. For instance, a quantum spin Hall insulator features a bulk insulating gap with gapless helical boundary states protected by the underlying Z2 topology. However, the bulk-boundary dichotomy and distinction are rarely explored in optical experiments, which can provide unique information about topological charge carriers beyond transport and electronic spectroscopy techniques. Here, we utilize mid-infrared absorption micro-spectroscopy and pump–probe micro-spectroscopy to elucidate the bulk-boundary optical responses of Bi4Br4, a recently discovered room-temperature quantum spin Hall insulator. Benefiting from the low energy of infrared photons and the high spatial resolution, we unambiguously resolve a strong absorption from the boundary states while the bulk absorption is suppressed by its insulating gap. Moreover, the boundary absorption exhibits strong polarization anisotropy, consistent with the one-dimensional nature of the topological boundary states. Our infrared pump–probe microscopy further measures a substantially increased carrier lifetime for the boundary states, which reaches one nanosecond scale. The nanosecond lifetime is about one to two orders longer than that of most topological materials and can be attributed to the linear dispersion nature of the helical boundary states. Our findings demonstrate the optical bulk-boundary dichotomy in a topological material and provide a proof-of-principal methodology for studying topological optoelectronics.

Universal conductance fluctuations in nanoscale topological insulator devices

Topological materials are promising candidates in fault-tolerant quantum information processing architectures, making it essential to understand the dephasing mechanisms in these materials. Here, we investigate gated, nanoscale mesas fabricated on thin films of cadmium arsenide (Cd3As2), a three-dimensional Dirac semimetal that can be tuned into different topological phases. We observe two independent types of conductance oscillations, one as a function of the applied magnetic field and the other as a function of the gate voltage. Varying the dimensions of the nanostructures allows the discrimination of a variety of scenarios for similar oscillations previously reported in the literature. We conclude that the conductance oscillations are not a signature of topological boundary states per se, but rather are universal conductance fluctuations. These results broadly inform future interpretations of electronic quantum interference in mesoscopic devices made from topological materials.

Spinful Topological Phases in Acoustic Crystals with Projective PT Symmetry.

For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological phases. However, only recently has it been realized theoretically that in the presence of gauge symmetry, the algebraic structure of symmetries can be projectively represented, which possibly enables the switch between spinless and spinful topological phases. Here, we report the experimental demonstration of this idea by realizing spinful topological phases in "spinless" acoustic crystals with projective space-time inversion symmetry. In particular, we realize a one-dimensional topologically gapped phase characterized by a 2Z winding number, which features double-degenerate bands in the entire Brillouin zone and two pairs of degenerate topological boundary modes. Our Letter thus overcomes a fundamental constraint on topological phases by spin classes.

The surface‐reconstruction virtual‐region mesh update method for problems with topology changes

AbstractIn this work, we introduce a novel boundary‐conforming mesh‐update method that is particularly designed for problems with large boundary displacements and topology changes. This method, which we call the surface‐reconstruction virtual‐region mesh update method, integrates the virtual region approach and a surface reconstruction process to handle complex boundary movements. The virtual region approach allows having an activated and deactivated part of the mesh, where elements can freely enter or leave the activated domain. Furthermore, the surface reconstruction ensures boundary conformity of the activated domain. The robustness of the proposed method is shown in two numerical examples: a variation of the benchmark Poiseuille flow, and the flow simulation during a closing diaphragm valve. In particular, the diaphragm valve simulation includes large boundary movement, complex geometry, and closing motion. For this case, both steady and transient simulation results at different closing conditions are presented.

Reconfigurable Nested Photonic Loops Based on Valley Topological Boundary Waveguides

Reconfigurable Nested Photonic Loops Based on Valley Topological Boundary Waveguides

Charge instability of topological Fermi arcs in chiral crystal CoSi

Topological boundary states emerged at the spatial boundary between topological non-trivial and trivial phases, are usually gapless, or commonly referred as metallic states. For example, the surface state of a topological insulator is a gapless Dirac state. These metallic topological boundary states are typically well described by non-interacting fermions. However, the behavior of topological boundary states with significant electron–electron interactions, which could turn the gapless boundary states into gapped ordered states, e.g., density wave states or superconducting states, is of great interest theoretically, but is still lacking evidence experimentally. Here, we report the observation of incommensurable charge density wave (CDW) formed on the topological boundary states driven by the electron–electron interactions on the (001) surface of CoSi. The wavevector of CDW varies as the temperature changes, which coincides with the evolution of topological surface Fermi arcs with temperature. The orientation of the CDW phase is determined by the chirality of the Fermi arcs, which indicates a direct association between CDW and Fermi arcs. Our finding will stimulate the search of more interactions-driven ordered states, such as superconductivity and magnetism, on the boundaries of topological materials.

Coexistence of quantum spin and valley hall effect in two-dimensional dielectric photonic crystals

The location and transmission of light is the core of modern photonic integrated device, and the proposal of topological photonics provides a new way of implementing optical manipulation. Topological photonic structures based on the quantum spin hall effect or quantum valley hall effect have the properties of immunity to defects and suppress backscattering, so they play a key role in designing novel low-loss photonic devices. In this work, we design a two-dimensional dielectric photonic crystal with time-reversal symmetry to achieve the coexistence of the quantum spin hall effect and the quantum valley hall effect in a photonic crystal. The design can be likened to an electronic system in which two pairs of Kramers simplex pairs are constructed to achieve a quadruple simplex pair in a photonic crystal. First, based on the method of shrinking and expanding the silicon pillars arranged in the honeycomb structure, the quadruple degeneracy point at the <i>Γ</i> point of the first Brillouin zone is opened, and the corresponding topologically trivial or non-trivial photonic band gap is formed,thereby realizing quantum spin hall effect. The expanded honeycomb lattice evolves into a Kagome structure, and then positive and negative perturbations are added to the Kagome lattice, breaking the spatial inversion symmetry of the Photonic crystal. When mirror symmetry is broken, different chiral photonic crystals can be created,leading the degeneracy point of the non-equivalent valleys <i>K</i> and <i>K'</i> in the Brillouin zone to be opened and a complete band gap to appear, thus realizing the Quantum valley hall effect. In the common band gap, topologically protected edge states are induced by nontrivial valley Chern number at the interface between two photonic crystals with opposite chirality. The numerical calculations show that unidirectional transport and bending-immune topological boundary states can be realized at the interface composed of topologically trivial (non-trivial) and positively (negatively) perturbed photonic crystals. Finally, a four-channel system based on the coexistence of the two effects is designed, The system is a novel electromagnetic wave router that can be selectively controlled by pseudospin degree of freedom or valley degree of freedom. This system provides a potential method for realizing the optical encoding and robust signal transmission, thereby providing greater flexibility for manipulating electromagnetic waves.

Observation of π/2 Modes in an Acoustic Floquet System.

Topological phases of matter have remained an active area of research in the last few decades. Periodic driving is a powerful tool for enriching such exotic phases, leading to various phenomena with no static analogs. One such phenomenon is the emergence of the elusive π/2 modes, i.e., a type of topological boundary state pinned at a quarter of the driving frequency. The latter may lead to the formation of Floquet parafermions in the presence of interaction, which is known to support more computational power than Majorana particles. In this Letter, we experimentally verify the signature of π/2 modes in an acoustic waveguide array, which is designed to simulate a square-root periodically driven Su-Schrieffer-Heeger model. This is accomplished by confirming the 4T-periodicity (T being the driving period) profile of an initial-boundary excitation, which we also show theoretically to be the smoking gun evidence of π/2 modes. Our findings are expected to motivate further studies of π/2 modes in quantum systems for potential technological applications.

Interstitial‐Electron‐Induced Topological Molecular Crystals

AbstractTopological phases are usually unreachable in molecular solids, which are characterized by weakly dispersed energy bands with a large gap, in contrast to topological materials. In this work, however, it is proposed that nontrivial electronic topology may ubiquitously emerge in a class of molecular crystals that contain interstitial electronic states, the bands of which are prone to be inverted with those of molecular orbitals. Guidelines are provided to hunt for such interstitial‐electron‐induced topological molecular crystals, especially in the topological insulating state. They exhibit a variety of exceptional qualities, as brought about by the intrinsic interplay of molecular crystals, interstitial electrons, and topological nature: 1) They may host cleavable surfaces along multiple orientations, with pronounced topological boundary states free from dangling bonds. 2) Strong response to moderate mechanical perturbations, whereby topological phase transition would occur under relatively low pressure. 3) Inherent high‐efficiency thermoelectricity as jointly contributed by the non‐parabolic band structure (therewith high thermopower), highly mobile interstitial electrons (high electrical conductivity), and soft phonons (small lattice thermal conductivity). 4) Ultralow work function owing to the active interstitial electrons. First‐principles calculations are utilized to demonstrate these properties with the representative candidate K4Ba2[SnBi4]. This work suggests a pathway of realizing topological phases in bulk molecular systems, which may advance the interdisciplinary research between topological and molecular materials.

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Topologically protected dynamics in three-dimensional nonlinear antisymmetric Lotka-Volterra systems

Studies of topological bands and their associated low-dimensional boundary modes have largely focused on linear systems. This work reports robust dynamical features of three-dimensional (3D) nonlinear systems in connection with intriguing topological bands in 3D. Specifically, for a 3D setting of coupled rock-paper-scissors cycles governed by the antisymmetric Lotka-Volterra equation (ALVE) that is inherently nonlinear, we unveil distinct characteristics and robustness of surface-polarized masses and analyze them in connection with the dynamics and topological bands of the linearized Lotka-Volterra equation. Our results show that insights learned from Weyl semimetal phases with type-I and type-II Weyl singularities based on a linearized version of the ALVE are still remarkably useful, even though the system dynamics is far beyond the linear regime. This work indicates the relevance and importance of the concept of topological boundary modes in analyzing high-dimensional nonlinear systems and hopes to stimulate further topological studies where various topological gapless phases of matter may emerge in higher-dimensional nonlinear systems.

Response to the comment on “Do Bloch waves interfere with one another ?”

Here a generalised argument showing the existence of the Bloch superselection rule is presented, in response to the recent comment by Sowinski. In light of the role played by the periodic boundary condition, locality and topology in the system, the claim made in the comment is found untenable.

Topological classification of higher-order topological phases with nested band inversion surfaces

Higher-order topological phases (HOTPs) hold gapped bulk bands and topological boundary states localized in boundaries with codimension higher than one. In this paper, we provide a unified construction and topological characterization of HOTPs for the full Altland-Zirnbauer tenfold symmetry classes, based on a method known as nested band inversion surfaces (BISs). Specifically, HOTPs built on this method are decomposed into a series of subsystems, and higher-order topological boundary states emerges from the interplay of their first-order topology. Our analysis begins with a general discussion of HOTPs in continuous Hamiltonians for each symmetry class, then moves on to several lattice examples illustrating the topological characterization based on the nested-BIS method. Despite the example minimal models possessing several spatial symmetries, our method does not rely on any spatial symmetry, and can be easily extended into arbitrary orders of topology in dimensions. Furthermore, we extend our discussion to systems with asymmetric boundary states induced by two different mechanisms, namely, crossed BISs that break a $\mathcal{C}_4$ rotation symmetry, and non-Clifford operators that break certain chiral-mirror symmetries of the minimal models.

MCI-frcnn: A deep learning method for topological micro-domain boundary detection.

Chromatin structural domains, or topologically associated domains (TADs), are a general organizing principle in chromatin biology. RNA polymerase II (RNAPII) mediates multiple chromatin interactive loops, tethering together as RNAPII-associated chromatin interaction domains (RAIDs) to offer a framework for gene regulation. RAID and TAD alterations have been found to be associated with diseases. They can be further dissected as micro-domains (micro-TADs and micro-RAIDs) by clustering single-molecule chromatin-interactive complexes from next-generation three-dimensional (3D) genome techniques, such as ChIA-Drop. Currently, there are few tools available for micro-domain boundary identification. In this work, we developed the MCI-frcnn deep learning method to train a Faster Region-based Convolutional Neural Network (Faster R-CNN) for micro-domain boundary detection. At the training phase in MCI-frcnn, 50 images of RAIDs from Drosophila RNAPII ChIA-Drop data, containing 261 micro-RAIDs with ground truth boundaries, were trained for 7days. Using this well-trained MCI-frcnn, we detected micro-RAID boundaries for the input new images, with a fast speed (5.26 fps), high recognition accuracy (AUROC = 0.85, mAP = 0.69), and high boundary region quantification (genomic IoU = 76%). We further applied MCI-frcnn to detect human micro-TADs boundaries using human GM12878 SPRITE data and obtained a high region quantification score (mean gIoU = 85%). In all, the MCI-frcnn deep learning method which we developed in this work is a general tool for micro-domain boundary detection.

Celestial Klein spaces

We consider the analytic continuation of ($p+q$)-dimensional Minkowski space (with $p$ and $q$ even) to $(p,q)$ signature, and study the conformal boundary of the resulting ``Klein space.'' Unlike the familiar $(\ensuremath{-}+++\ensuremath{\cdots})$ signature, now the null infinity $\mathcal{I}$ has only one connected component. The spatial and timelike infinities (${i}^{0}$ and ${i}^{\ensuremath{'}}$) are quotients of generalizations of AdS spaces to nonstandard signature. Together, $\mathcal{I}$, ${i}^{0}$, and ${i}^{\ensuremath{'}}$ combine to produce the topological boundary ${S}^{p+q\ensuremath{-}1}$ as an ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fibration over a null segment. The highest weight states (the $L$-primaries) and descendants of $SO(p,q)$ with integral weights give rise to natural scattering states. One can also define $H$-primaries which are highest weight with respect to a signature-mixing version of the Cartan-Weyl generators that leave a point on the celestial ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fixed. These correspond to massless particles that emerge at that point and are Mellin transforms of plane wave states.

Shear-coupling of graphene grain boundaries: Elementary mechanisms, effects of topology, and role of buckling

Engineering grain boundary structure is a promising method for rational control of the microstructure and mechanical properties of two-dimensional materials. In bulk materials, shear stresses can drive grain boundary migration through the dislocations in grain boundaries. However, while shear coupling of grain boundaries has been studied in bulk materials like nanocrystalline copper, its translation to two-dimensional materials where out-of-plane deformation can relieve in-plane shear is not yet established. We investigate how the low flexural rigidity of graphene effects shear coupled grain boundary motion using atomic scale simulations of flat and buckled grain boundaries. We define the coupling shear strain as the strain at which a grain boundary has advanced by one Burgers vector and is at equilibrium and the critical shear strain as the strain at which migration of the first dislocation in the grain boundary becomes thermodynamically favorable. We show that the out-of-plane deformation does not influence the coupling shear strain and is governed only by the grain boundary topology. While the critical shear strain is altered somewhat by the low flexural rigidity due to buckling induced softening, it is also still dominated by the grain boundary topology. Our atomic scale results are synthesized into two models that predict the coupling and critical shears.

Topology-based fluorescence image analysis for automated cell identification and segmentation.

Cell segmentation refers to the body of techniques used to identify cells in images and extract biologically relevant information from them; however, manual segmentation is laborious and subjective. We present Topological Boundary Line Estimation using Recurrence Of Neighbouring Emissions (TOBLERONE), a topological image analysis tool which identifies persistent homological image features as opposed to the geometric analysis commonly employed. We demonstrate that topological data analysis can provide accurate segmentation of arbitrarily-shaped cells, offering a means for automatic and objective data extraction. One cellular feature of particular interest in biology is the plasma membrane, which has been shown to present varying degrees of lipid packing, or membrane order, depending on the function and morphology of the cell type. With the use of environmentally-sensitive dyes, images derived from confocal microscopy can be used to quantify the degree of membrane order. We demonstrate that TOBLERONE is capable of automating this task.

A topological nonlinear parametric amplifier

Topological boundary states are well localized eigenstates at the boundary between two different bulk topologies. As long as bulk topology is preserved, the topological boundary mode will endure. Here, we report topological nonlinear parametric amplification of light in a dimerized coupled waveguide system based on the Su-Schrieffer-Heeger model with a domain wall. The good linear transmission properties of the topological waveguide arising from the strong localization of light to the topological boundary is demonstrated through successful high-speed transmission of 30 Gb/s non-return-to-zero and 56 Gb/s pulse amplitude 4-level data. The strong localization of a co-propagating pump and probe to the boundary waveguide is harnessed for efficient, low power optical parametric amplification and wavelength conversion. A nonlinear tuning mechanism is shown to induce chiral symmetry breaking in the topological waveguide, demonstrating a pathway in which Kerr nonlinearities may be applied to tune the topological boundary mode and control the transition to bulk states.

A simple elastic phononic crystal plate with adjustable topological valley transmission paths

The study of topological insulator devices has now been extended from electromagnetic and acoustic waves to the field of elastic waves. Topology-protected wave transmission has stimulated great interest due to its strong robustness. In this paper, we design an elastic phononic crystal plate (EPCP) with tunable topological valley transmission. The inversion of the topological energy band is attained just by adjusting the relative height of the bolts. Moreover, by calculating the topological invariants, the topological property of this EPCP is further demonstrated. The topological boundary transport immunity to defect, sharp bend, and disturbance is verified using the finite element method and experiments. By simply regulating the relative heights of the bolts in the unit cell, topological valley transmission for different shaped paths is achieved, which is important for the practical application of tunable topology elastic wave transmission devices.

Non-Hermitian Topological Phenomena: A Review

The past decades have witnessed an explosion of interest in topological materials, and a lot of mathematical concepts have been introduced in condensed matter physics. Among them, the bulk-boundary correspondence is the central topic in topological physics, which has inspired researchers to focus on boundary physics. Recently, the concepts of topological phases have been extended to non-Hermitian Hamiltonians, whose eigenvalues can be complex. Besides the topology, non-Hermiticity can also cause a boundary phenomenon called the non-Hermitian skin effect, which is an extreme sensitivity of the spectrum to the boundary condition. In this article, we review developments in non-Hermitian topological physics by focusing mainly on the boundary problem. As well as the competition between non-Hermitian and topological boundary phenomena, we discuss the topological nature inherent in non-Hermiticity itself.