Boundary Modes Research Articles

Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits

Robust boundary states epitomize how deep physics can give rise to concrete experimental signatures with technological promise. Of late, much attention has focused on two distinct mechanisms for boundary robustness—topological protection, as well as the non-Hermitian skin effect. In this work, we report the experimental realizations of hybrid higher-order skin-topological effect, in which the skin effect selectively acts only on the topological boundary modes, not the bulk modes. Our experiments, which are performed on specially designed non-reciprocal 2D and 3D topolectrical circuit lattices, showcases how non-reciprocal pumping and topological localization dynamically interplays to form various states like 2D skin-topological, 3D skin-topological-topological hybrid states, as well as 2D and 3D higher-order non-Hermitian skin states. Realized through our highly versatile and scalable circuit platform, theses states have no Hermitian nor lower-dimensional analog, and pave the way for applications in topological switching and sensing through the simultaneous non-trivial interplay of skin and topological boundary localizations.

Higher-order topology and corner triplon excitations in two-dimensional quantum spin-dimer models

The concept of free fermion topology has been generalized to $d$-dimensional phases that exhibit $(d-n)$-dimensional boundary modes, such as zero-dimensional (0D) corner excitations. Motivated by recent extensions of these ideas to magnetic systems, we consider 2D quantum paramagnets formed by interacting spin dimers with dispersive triplet excitations. We propose two examples of such dimer models, where the spin-gapped bosonic triplon excitations are shown to host bands with nontrivial higher-order topology. We demonstrate this using real-space Bogoliubov--de Gennes calculations that reveal the existence of mid-bandgap corner triplon modes as a signature of higher-order bulk topology. We provide an understanding of the higher-order topology in these systems via a computation of bulk topological invariants as well as the construction of edge theories, and study their phase transitions as we tune parameters in the model Hamiltonians. We also discuss possible experimental approaches for detecting the emergent corner triplon modes.

Measurement of Corner-Mode Coupling in Acoustic Higher-Order Topological Insulators

Recent developments of band topology have revealed a variety of higher-order topological insulators (HOTIs). These HOTIs are characterized by a variety of different topological invariants, making them different at a fundamental level. However, despite such differences, the fact that they all sustain higher-order topological boundary modes poses a challenge to phenomenologically tell them apart. This work presents experimental measurements of the coupling effects of topological corner modes (TCMs) existing in two different types of two-dimensional acoustic HOTIs. Although both HOTIs have a similar four-site square lattice, the difference in magnetic flux per unit cell dictates that they belong to different types of topologically nontrivial phases—one lattice possesses quantized dipole moments, but the other is characterized by quantized quadrupole moment. A link between the topological invariants and the response line shape of the coupled TCMs is theoretically established and experimentally confirmed. Our results offer a pathway to distinguish HOTIs experimentally.

Evidence of topological boundary modes with topological nodal-point superconductivity

The extension of the topological classification of band insulators to topological semimetals gave way to the topology classes of Dirac, Weyl, and nodal line semimetals with their unique Fermi arc and drum head boundary modes. Similarly, there are several suggestions to employ the classification of topological superconductors for topological nodal superconductors with Majorana boundary modes. Here, we show that the surface 1H termination of the transition metal dichalcogenide compound 4Hb-TaS$_2$, in which 1T-TaS$_2$ and 1H-TaS$_2$ layers are interleaved, has the phenomenology of a topological nodal point superconductor. We find in scanning tunneling spectroscopy a residual density of states within the superconducting gap. An exponentially decaying bound mode is imaged within the superconducting gap along the boundaries of the exposed 1H layer characteristic of a gapless Majorana edge mode. The anisotropic nature of the localization length of the edge mode aims towards topological nodal superconductivity. A zero-bias conductance peak is further imaged within fairly isotropic vortex cores. All our observations are accommodated by a theoretical model of a two-dimensional nodal Weyl-like superconducting state, which ensues from inter-orbital Cooper pairing. The observation of an intrinsic topological nodal superconductivity in a layered material will pave the way for further studies of Majorana edge modes and its applications in quantum information processing.

2n -root weak, Chern, and higher-order topological insulators, and 2n -root topological semimetals

Recently, we have introduced in [A. M. Marques et al., Phys. Rev. B 103, 235425 (2021)] the concept of $2^n$-root topology and applied it to one-dimensional systems. These models require $n$ squaring operations to their Hamiltonians, intercalated with different constant energy downshifts at each level, in order to arrive at a decoupled block corresponding to a known topological insulator (TI) that acts as the source of the topological features of the starting $2^n$-root TI ($\sqrt[2^n]{\text{TI}}$). In the process, $n$ non-topological residual models with degenerate spectra and in-gap impurity states appear, which dilute the topologically protected component of the starting edge states. Here, we generalize this method to several two-dimensional models, by finding the 4-root version of lattices hosting weak and higher-order boundary modes (both topological and non-topological) of a Chern insulator and of a topological semimetal. We further show that a starting model with a non-Hermitian region in parameter space and a complex energy spectrum can nevertheless display a purely real spectrum for all its successive squared versions, allowing for an exact mapping between certain non-Hermitian models and their Hermitian lower root-degree counterparts. A comment is made on the possible realization of these models in artificial lattices.

Nodal Lines and Boundary Modes in Topological Dirac Semimetals with Magnetism

Nodal-line semimetals with magnetic orders have been theoretically predicted and experimentally observed in only few compounds. We theoretically explore the electronic structure in bulk and boundary of such a magnetic nodal-line state by introducing magnetism in topological Dirac semimetal (TDSM). TDSMs, such as $\mathrm{Cd_3 As_2}$ and $\mathrm{Na_3 Bi}$, are characterized by a pair of spin-degenerate Dirac points protected by rotational symmetries of crystals. By introducing local magnetic moments coupled to the electron spins in the lattice model of TDSM, we show that the TDSM can turn into either a Weyl semimetal or a nodal-line semimetal, which is determined by the orbital dependence in the exchange coupling and the direction of magnetization formed by the magnetic moments. In this magnetic nodal-line semimetal state, we find zero modes with drumhead-like band structure at the boundary that are characterized by the topological number of $\mathbb{Z}$. Those zero modes are numerically demonstrated by introducing magnetic domain walls in the lattice model.

Disorder-driven phase transition in the second-order non-Hermitian skin effect

The non-Hermitian skin effect exhibits a collapse of extended bulk modes into an extensive number of localized boundary states in the open boundary condition. Here, we demonstrate a disorder-driven phase transition of a trivial non-Hermitian system to the higher-order non-Hermitian skin effect phase. In contrast to clean systems, the disorder-induced boundary modes form an arc in the complex energy plane, which is a manifestation of the disorder-driven energy-dependent phase transition. At the phase transition, localized corner modes and bulk modes characterized by trivial Hamiltonians coexist within a single band, while they remain separated in the complex energy plane. This behavior is analogous to the mobility edge phenomena in disordered Hermitian systems. Using effective medium theory and numerical diagonalizations, we provide a systematic characterization of the disorder-driven phase transition.

Evidence of topological boundary modes with topological nodal-point superconductivity

Evidence of topological boundary modes with topological nodal-point superconductivity

Gauge-dependent topology in non-reciprocal hopping systems with pseudo-Hermitian symmetry

Energy conservation is not valid in non-Hermitian systems with gain/loss or non-reciprocity, which leads to various extraordinary resonant characteristics. Compared with Hermitian systems, the intersection of non-Hermitian physics and topology generates new phases that have not been observed in condensed-matter systems before. Here, utilizing the designed two-dimensional periodical model with non-reciprocal hopping terms, we show how to obtain both the ellipse-like or hyperbolic-like spectral degeneracy, the topological boundary modes and the bulk-boundary correspondence by the protection of time-reversal symmetry and pseudo-Hermitian symmetry. Notably, the boundary modes and bulk-boundary correspondence can simultaneously appear only for specific selection of the primitive cell, and we explored the analytical solution to verify such gauge-dependent topological behaviors. Our topolectrical circuit simulation provides a flexible approach to confirm the designed properties and clarify the crucial role of pseudo-Hermiticity on the stability of a practical system. In a broader view, our findings can be compared to other platforms such as meta-surface or photonic crystals, for the purpose on the control of resonant frequency and localization properties.

Non-Hermitian flat bands in rhombic microring resonator arrays.

We investigate the flat bands in a quasi-one-dimensional rhombic array composed of evanescently coupled microring resonators (MRRs) with non-Hermitian coupling. By changing the relative position of non-Hermitian coupling in each cell, we construct topologically trivial and nontrivial flat bands, where both the real and imaginary parts of energy bands become flat and coalesce into a single band. We show the nontrivial systems are able to support topological boundary modes isolated from the flat bulk bands although there is no band gap. The elusive topology of flat bands can be geometrically visualized by plotting the trajectories of their eigenvectors on Bloch sphere based on Majorana's stellar representation (MSR). Furthermore, we perform a full wave simulation and show the characteristics of flat bands, associated compact localized modes, and boundary modes are reflected from absorption spectra and field intensity profiles. The study may find potential applications in lasers, narrowband filters, and efficient light harvesting.

Spin-Chern insulator induced by pseudospin-orbit coupling in phononic crystals

The boundary modes of classical waves protected by the topology are of great significance for robust energy transmission and reliable information processing. Many endeavors have been made to realize topological insulators in electromagnetic, mechanical and acoustic systems. Previous studies on acoustic boundary modes mainly focused on the interface states at the domain walls between different topological phases of phononic crystals. In electromagnetic and mechanical systems, although helical boundary states have been observed, the pseudospin in them is conserved, thus these systems can be considered as two independent copies of Chern insulators. When the pseudospin conservation is broken, the topology cannot be described by Chern numbers, and the spin will vary in transport. This general case remains to be explored both in electronic and classical systems. In this work, we constructed a two-dimensional topological phononic crystal induced by pseudospin-orbit coupling, which hosts the gapless helical boundary modes at the edge of a phononic crystal. Since these boundary states originate from non-zero spin-Chern numbers, the system is called the spin-Chern insulator. We have observed these boundary states in experiments, and observed the spin-flipping transport in the H-shaped sample, proving the spin non-conservation of the boundary states. Different from the earlier work, this work demonstrates two important aspects: first, the acoustic topological insulator is induced by the pseudospin-orbit coupling, and possesses the gapless helical edge states on its boundary, thus is the first analogue of topological insulator for acoustic wave; second, the inherent spin conservation is broken, which expands and deepens the knowledge of current topological physics in classical and electronic systems. The helical boundary states may have potential applications in new topological acoustic devices.

Nonlinear second-order photonic topological insulators

Higher-order topological insulators are a novel topological phase beyond the framework of conventional bulk–boundary correspondence1,2. In these peculiar systems, the topologically non-trivial boundary modes are characterized by a co-dimension of at least two3,4. Despite several promising preliminary considerations regarding the impact of nonlinearity in such systems5,6, the flourishing field of experimental higher-order topological insulator research has thus far been confined to the linear evolution of topological states. As such, the observation of the interplay between nonlinearity and the dynamics of higher-order topological phases in conservative systems remains elusive. Here we experimentally demonstrate nonlinear higher-order topological corner states. Our photonic platform enables us to observe nonlinear topological corner states as well as the formation of solitons in such topological structures. Our work paves the way towards the exploration of topological properties of matter in the nonlinear regime, and may herald a new class of compact devices that harnesses the intriguing features of topology in an on-demand fashion.

Quenched topological boundary modes can persist in a trivial system

Topological boundary modes can occur at the spatial interface between a topological and gapped trivial phase and exhibit a wavefunction that exponentially decays in the gap. Here we argue that this intuition fails for a temporal boundary between a prequench topological phase that possess topological boundary eigenstates and a postquench gapped trivial phase that does not possess any eigenstates in its gap. In particular, we find that characteristics of states (e.g., probability density) prepared in a topologically non-trivial system can persist long after it is quenched into a gapped trivial phase with spatial profiles that appear frozen over long times postquench. After this near-stationary window, topological boundary mode profiles decay albeit, slowly in a power-law fashion. This behavior highlights the unusual features of nonequilibrium protocols enabling quenches to extend and control localized states of both topological and non-topological origins.

Loop quantum gravity boundary dynamics and gauge theory

In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3 + 1 space-time dimensions, the boundary theory lives on the 2 + 1-dimensional time-like boundary and is supposed to describe the time evolution of gravitational boundary modes—‘edge modes’—living on the two-dimensional boundary of space, i.e. the space-time corner. We do not analyse the boundary structures of general relativity and their quantization, and focus instead on investigating which boundary theories can be supported by the standard loop quantum gravity formalism and spin network states. Focussing on ‘electric’ excitations—quanta of area—living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background metric on the 2 + 1-dimensional time-like boundary. This leads to a conjecture of a deeper correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a connection, transporting the spinors on the boundary surface and whose SU(2) component would define ‘magnetic’ excitations (tangential Ashtekar–Barbero connection), thereby opening the door to writing the loop quantum gravity boundary dynamics as a 2 + 1-dimensional gauge theory.

Corner states in a second-order mechanical topological insulator

Demonstration of topological boundary modes in elastic systems has attracted a great deal of attention over the past few years due to its unique protection characteristic. Recently, second-order topological insulators have been proposed in manipulating the topologically protected localized states emerging only at corners. Here, we numerically and experimentally study corner states in a two-dimensional phononic crystal, namely a continuous elastic plate with embedded bolts in a hexagonal pattern. We create interfacial corners by adjoining trivial and non-trivial topological configurations. Due to the rich interaction between the bolts and the continuous elastic plate, we find a variety of corner states of and devoid of topological origin. Strikingly, some of the corner states are not only highly-localized but also tunable. Taking advantage of this property, we experimentally demonstrate asymmetric corner localization in a Z-shaped domain wall. This finding could create interest in exploration of tunable corner states for the use of advanced control of wave localization.

Tuning of topological interface modes in an elastic beam array system with inerters

Topological phononic crystals in the mechanical setup became a topic of great interest owing to their applicability in various engineering systems such as waveguides or vibration isolation devices. If such systems are composed of elastic structures, they are usually characterized by a bulk-edge correspondence where the geometrical and material properties can play important role in the existence of stable surface and boundary modes. This work investigates the band transition and topological interface modes in a beam array system, where two sub-lattices of vertically aligned, parallel, and elastically coupled beams are connected at the chain center. To illustrate the existence of interface modes and understand their behavior, the corresponding eigenvalue problem is solved and frequency response function is sought for the system with a finite number of unit cells. Localization of the interface modes is demonstrated based on the steady-state responses of the beam array system to harmonic excitation. The effects of introduced defect masses and inerters on interface states are studied separately. It is revealed that the introduction of a small defect mass in the form of concentrated mass attached to some beam in the system does not affect the interface modes within the observed frequency range. On the other side, inerters produce frequency shifts towards lower values, which even in the case of small values of the inerter parameter causes the interface modes to vanish or even to appear inside another frequency band gap. The obtained results give an insight into the influence of inerter devices and their mass amplification effect on the interface states in complex periodic elastic systems. It also investigates the possibility to tune interface modes without significantly affecting the main band structure properties of the system.

Experimental study of the flow regime effect on the stability of collapsible tubes conveying fluid

Experimental studies of the stability of the collapsible tubes conveying fluid have been previously conducted in the context of cardiovascular mechanics mostly for turbulent flows, although blood flows are laminar under normal conditions. In this paper, the turbulent and laminar regimes with equal flow rates and pressure drops are investigated experimentally to identify the stability boundary and self-exciting oscillation modes of Penrose tubes conveying fluid in the Starling resistor. Four oscillation modes for laminar and for turbulent regimes were observed visually and by measuring the pressure drop and the output pressure. Comparison of amplitudes, frequencies, and boundaries between different oscillation modes for equivalent laminar and turbulent flow regimes is performed.

Amplitude-dependent boundary modes in topological mechanical lattices

Boundary modes localized at the edge or on the interface of topological mechanical lattices are analogous to their electronic counterparts in topological insulators and are robust and immune to structural imperfections. Most studies on the boundary modes in mechanical lattices are based on band theories, focusing on amplitude-independent behaviors at different frequency ranges. However, highly distorted topological mechanical lattices exhibit amplitude-dependent nonlinear effects that are difficult to characterize. In this study, a topological mechanical lattice exhibiting amplitude-dependent boundary modes is explored to uncover the evolution and connection between boundary modes in the linear and nonlinear regimes. Both discrete and continuum models are developed. An iterative method for the discrete model is proposed to numerically capture the boundary mode evolution while, based upon the continuum model, an analytical solution of the decay length scale of the boundary modes is obtained and can reduce to the linear band theory prediction at the small amplitude limit. A design strategy is presented for programmable topological polarization in the lattice that can be used to encode mechanical information. The obtained results provide insight into the modulation of topological protected mechanical response and stimulation of spatially ordered modes.

Statistics-tuned entanglement of the boundary modes in coupled Su-Schrieffer-Heeger chains

We show that mutual statistics between quantum particles can be tuned to generate emergent novel few particle quantum mechanics for the boundary modes of symmetry-protected topological phases of matter. As a concrete setting, we study a system of pseudofermions, defined as quantum particles with tunable algebra, which lie on two distinct Su-Schrieffer-Heeger (SSH) chains. We find that as the mutual statistics of the particles are tuned -- the boundary modes present in the two chains gets non-trivially entangled showing a sudden jump in their mutual entanglement entropy. We further show that, such tuning of statistics engenders a first-order transition between two topologically non-trivial phases which differ in the behavior of inter-chain entanglement. Using a combination of analytical and numerical techniques and effective modeling, we uncover the rich physics that this system hosts. The results are of particular relevance in context of the study of the effective low energy quantum mechanics of topological edge modes in one hand and their recent realization in ultracold atoms on the other. This then provides for controlled manipulation of such low energy modes.

Multi-dimensional wave steering with higher-order topological phononic crystal

The recent discovery and realizations of higher-order topological insulators enrich the fundamental studies on topological phases. Here, we report three-dimensional (3D) wave-steering capabilities enabled by topological boundary states at three different orders in a 3D phononic crystal with nontrivial bulk topology originated from the synergy of mirror symmetry of the unit cell and a non-symmorphic glide symmetry of the lattice. The multitude of topological states brings diverse possibilities of wave manipulations. Through judicious engineering of the boundary modes, we experimentally demonstrate two functionalities at different dimensions: 2D negative refraction of sound wave enabled by a first-order topological surface state with negative dispersion, and a 3D acoustic interferometer leveraging on second-order topological hinge states. Our work showcases that topological modes at different orders promise diverse wave steering applications across different dimensions.