Higher-order Phase Research Articles

Prototype design for bidirectional control of stepper motor using features of brain signals and soft computing tools

The brain-computer interface (BCI) plays a significant role in supporting specially-abled people to control devices with the brain signals or electroencephalogram (EEG). The proper functioning of BCI systems requires classification algorithms to distinguish between different tasks based on the features extracted from EEG. Motivated by the requirement of designing a reliable BCI with reduced memory and computational cost, this work proposes an EEG controlled stepper motor drive-based aiding device. The drive system is executed in real-time based on the processing and classification of EEG. The scheme initiates with the extraction of discriminatory features from raw time-domain EEG signals using higher-order statistics (HOS) and phase locking value (PLV).Further, with extracted feature vector, the present study contemplates the operation of backpropagation neural network (BPNN), k-Nearest neighbours (k-NN), and support vector machine (SVM). Signal classification by SVM in conjugation with nonlinear principal component analysis (NLPCA) is implemented to reduce the feature vector dimension. Average classification accuracy of 82.70% and 80.46% is achieved using NLPCA with SVM for PLV and HOS features. The classifier output is utilized to spin the stepper motor clockwise and anticlockwise as needed. The control of stepper motor uses the classifier output and hence the brain signals are implemented on a laboratory-developed digital test bench comprising of TI TMS320F28379D launchpad DSP board and MITSUMI stepper motor. The validation of the proposed scheme for different signals reflects its effectiveness in combining the software and hardware aspects required for realizing an actual BCI system for real-time settings.

Design and Processing Method for Doppler-Tolerant Stepped-Frequency Waveform Using Staggered PRF

Stepped-frequency waveform may be used to synthesize a wideband signal with several narrow-band pulses and achieve a high-resolution range profile without increasing the instantaneous bandwidth. Nevertheless, the conventional stepped-frequency waveform is Doppler sensitive, which greatly limits its application to moving targets. For this reason, this paper proposes a waveform design method using a staggered pulse repetition frequency to improve the Doppler tolerance effectively. First, a generalized echo model of the stepped-frequency waveform is constructed in order to analyze the Doppler sensitivity. Then, waveform design is carried out in the stepped-frequency waveform by using a staggered pulse repetition frequency so as to eliminate the high-order phase component that is caused by the target’s velocity. Further, the waveform design method is extended to the sparse stepped-frequency waveform, and we also propose corresponding methods for high-resolution range profile synthesis and motion compensation. Finally, experiments with electromagnetic data verify the high Doppler tolerance of the proposed waveform.

Higher-Order Dirac Sonic Crystals.

Discovering new topological phases of matter is a major theme in fundamental physics and materials science. Dirac semimetal provides an exceptional platform for exploring topological phase transitions under symmetry breaking. Recent theoretical studies have revealed that a three-dimensional Dirac semimetal can harbor fascinating hinge states, a higher-order topological manifestation not known before. However, its realization in experiment is yet to be achieved. In this Letter, we propose a minimum model to construct a spinless higher-order Dirac semimetal protected by C_{6v} symmetry. By breaking different symmetries, this parent phase transitions into a variety of novel topological phases including higher-order topological insulator, higher-order Weyl semimetal, and higher-order nodal-ring semimetal. Furthermore, for the first time, we experimentally realize this unprecedented higher-order topological phase in a sonic crystal and present an unambiguous observation of the desired hinge states via momentum-space spectroscopy and real-space visualization. Our findings may offer new opportunities to manipulate classical waves such as sound and light.

Higher-order topological insulator phase in a modified Haldane model

We explore the topological properties of a modified Haldane model (MHM) in which the strength of the nearest-neighbor and next-nearest-neighbor hopping terms is made unequal and the threefold rotational symmetry ${\mathcal{C}}_{3}$ is broken by introducing a dimerization term ($|{t}_{1w(2w)}|<{t}_{1s(2s)}$) in the Hamiltonian. Using the parameter $\ensuremath{\eta}={t}_{1w}/{t}_{1s}={t}_{2w}/{t}_{2s}$, we show that this MHM supports a transition from the quantum anomalous Hall insulator to a higher-order topological insulator (HOTI) phase at $\ensuremath{\eta}=\ifmmode\pm\else\textpm\fi{}0.5$. It also hosts a zero-energy corner mode on a nanodisk that can transition to a trivial insulator without gap closing when the inversion symmetry is broken. The gap-closing critical states are found to be magnetic semimetals with a single Dirac node which, unlike the classic Haldane model, can move along the high-symmetry lines in the Brillouin zone. Our MHM offers a rich tapestry of HOTIs and other topological and nontopological phases.

Higher-order topology induced by structural buckling.

Higher-order topological insulator (HOTI) states, such as two-dimension (2D) HOTI featured with topologically protected corner modes at the intersection of two gapped crystalline boundaries, have attracted much recent interest. However, the physical mechanism underlying the formation of HOTI states is not fully understood, which has hindered our fundamental understanding and discovery of HOTI materials. Here we propose a mechanistic approach to induce higher-order topological phases via structural buckling of 2D topological crystalline insulators (TCIs). While in-plane mirror symmetry is broken by structural buckling, which destroys the TCI state, the combination of mirror and rotation symmetry is preserved in the buckled system, which gives rise to the HOTI state. We demonstrate that this approach is generally applicable to various 2D lattices with different symmetries and buckling patterns, opening a horizon of possible materials to realize 2D HOTIs. The HOTIs so generated are also shown to be robust against buckling height fluctuation and in-plane displacement. A concrete example is given for the buckled n}{}beta -Sb monolayer from first-principles calculations. Our finding not only enriches our fundamental understanding of higher-order topology, but also opens a new route to discovering HOTI materials.

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.

Higher-order topological phases in tunable C3 symmetric photonic crystals

We demonstrate that multiple higher-order topological transitions can be triggered via the continuous change of the geometry in kagome photonic crystals composed of three dielectric rods. By tuning a single geometry parameter, the photonic corner and edge states emerge or disappear with higher-order topological transitions. Two distinct higher-order topological insulator phases and a normal insulator phase are revealed. Their topological indices are obtained from symmetry representations. A photonic analog of the fractional corner charge is introduced to distinguish the two higher-order topological insulator phases. Our predictions can be readily realized and verified in configurable dielectric photonic crystals.

Graphyne as a second-order and real Chern topological insulator in two dimensions

Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming three valleys. The bulk bands feature a double band inversion, which is characterized by the real Chern number enabled by the spacetime-inversion symmetry. The real Chern number is explicitly evaluated by both the Wilson loop method and the parity approach. We demonstrate that a nontrivial real Chern number for a 2D system dictates the existence of Dirac-type edge bands and the topological corner states. Furthermore, we find that the topological phase transition in graphyne from the real Chern insulator to a trivial insulator is mediated by a 2D Weyl semimetal phase. The robustness of the corner states against symmetry breaking and possible experimental detection methods are discussed.

Three-Dimensional Higher-Order Topological Insulator Protected by Cubic Symmetry

Recently discovered photonic higher-order topological insulators enable unprecedented flexibility in the robust localization of light in structures of different dimensionality. While the potential of the two-dimensional systems is currently under active investigation, only a few studies explore the physics of the three-dimensional higher-order topological insulators. Here we propose a three-dimensional structure with cubic symmetry exhibiting vanishing bulk polarization but nonzero corner charge and hosting a zero-dimensional corner state mediated by the long-range interactions. We trace the evolution of the corner state with the next-nearest-neighbor coupling strength and prove the topological origin of the corner mode calculating the associated topological invariants. Our results thus reveal the potential of long-range couplings for the formation of three-dimensional higher-order topological phases.

Generic Orbital Design of Higher-Order Topological Quasicrystalline Insulators with Odd Five-Fold Rotation Symmetry.

In addition to crystals, topological phases in quasicrystals and disorder systems have drawn increasing attention lately. Here, we propose a generic double band-inversion mechanism underlying the higher-order topological phase in quasicrystals, that is.,"higher-order topological quasicrystalline insulator" (HOTQI), which exploits local atomic orbital and lattice symmetries. It is generally applicable to both quasicrystals and crystals with either odd-rotational (OR) or even-rotational symmetry (ERS), different from previous HOTI mechanisms whose applicability is limited by symmetry types. The HOTQI is characterized by topological corner states at the nonordinary corners of pentagonal (octagonal) samples of five-fold (eight-fold) quasicrystals, which violate the translational invariance and ordinary crystalline symmetries. The role of quasicrystalline symmetry, the robustness against symmetry breaking, and possible experimental realizations are discussed. Our findings not only provide a concrete example of HOTQIs that is incompatible with classical crystallographic symmetry but also offer useful guidance to the search of higher-order topological materials and metamaterials.

Dynamical characterization of quadrupole topological phases in superconducting circuits

The recent experimental realization of a programmable two-dimensional square superconducting qubit array opens a new avenue for quantum simulation with qubits [Science 372, 948 (2021)]. Here, we present an experimentally feasible method to achieve two-dimensional superconducting qubit lattice with tunable coupling strengths. A configuration with higher-order topological phases is constructed, featuring topologically protected boundary states in lower dimension (corner states). We show that the quadrupole topological phases can be effectively characterized by the dynamics of the single-excitation quantum state. Moreover, we also explore the detection and dynamics of the corner modes in our qubit lattice system. Particularly, we propose an effective scheme to realize quantum information transfer via the corner states. Our work suggests that superconducting circuits systems are a fertile platform to study novel topological quantum phases and may shed light on the ongoing exploration of topologically protected quantum information processing.

Hinge magnons from noncollinear magnetic order in a honeycomb antiferromagnet

We propose that noncollinear magnetic order in magnetic systems can harbor a higher-order topological magnon phase with non-Hermitian topology and hinge magnon modes. We consider a three-dimensional system of interacting local moments on stacked layers of honeycomb lattice. It initially favors a collinear magnetic order along an in-plane direction, which turns into a noncollinear order upon applying an external magnetic field perpendicular to the easy axis. We exploit the non-Hermitian nature of the magnon Hamiltonian to show that this field-induced transition corresponds to the transformation from a topological magnon insulator to a higher-order topological magnon state with a one-dimensional hinge mode. As a concrete example, we discuss the recently discovered monoclinic phase of thin chromium trihalides, which we propose as a promising material candidate of the higher-order topological magnon phase.

Floquet higher-order Weyl and nexus semimetals

This work reports the general design and characterization of two exotic, anomalous nonequilibrium topological phases. In equilibrium systems, the Weyl nodes or the crossing points of nodal lines may become the transition points between higher-order and first-order topological phases defined on two-dimensional slices, thus featuring both hinge Fermi arc and surface Fermi arc. We advance this concept by presenting a strategy to obtain, using time-sequenced normal insulator phases only, Floquet higher-order Weyl semimetals and Floquet higher-order nexus semimetals, where the concerned topological singularities in the three-dimensional Brillouin zone border anomalous two-dimensional higher-order Floquet phases. The fascinating topological phases we obtain are previously unknown and can be experimentally studied using, for example, a three-dimensional lattice of coupled ring resonators.

Generation of quantum states with nonlinear squeezing by Kerr nonlinearity.

In quantum optics, squeezing corresponds to the process in which fluctuations of a quadrature operator are reduced below the shot noise limit. In turn, nonlinear squeezing can be defined as reduction of fluctuations related to nonlinear combination of quadrature operators. Quantum states with nonlinear squeezing are a necessary resource for deterministic implementation of high-order quadrature phase gates that are, in turn, sufficient for advanced quantum information processing. We demonstrate that this class of states can be deterministically prepared by employing a single self-Kerr gate accompanied by suitable Gaussian processing. The required Kerr coupling depends on the energy of the initial system and can be made arbitrarily small. We also employ numerical simulations to analyze the effects of imperfections and to show to which extent can they be neglected.

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.

Using phase dynamics to study partial synchrony: three examples

Partial synchronous states appear between full synchrony and asynchrony and exhibit many interesting properties. Most frequently, these states are studied within the framework of phase approximation. The latter is used ubiquitously to analyze coupled oscillatory systems. Typically, the phase dynamics description is obtained in the weak coupling limit, i.e., in the first-order in the coupling strength. The extension beyond the first-order represents an unsolved problem and is an active area of research. In this paper, three partially synchronous states are investigated and presented in order of increasing complexity. First, the usage of the phase response curve for the description of macroscopic oscillators is analyzed. To achieve this, the response of the mean-field oscillations in a model of all-to-all coupled limit-cycle oscillators to pulse stimulation is measured. The next part treats a two-group Kuramoto model, where the interaction of one attractive and one repulsive group results in an interesting solitary state, situated between full synchrony and self-consistent partial synchrony. In the last part, the phase dynamics of a relatively simple system of three Stuart-Landau oscillators are extended beyond the weak coupling limit. The resulting model contains triplet terms in the high-order phase approximation, though the structural connections are only pairwise. Finally, the scaling of the new terms with the coupling is analyzed.

Simulating higher-order topological insulators in density wave insulators

Since the discovery of the Harper-Hofstadter model, it has been known that condensed matter systems with periodic modulations can be promoted to non-trivial topological states with emergent gauge fields in higher dimensions. In this work, we develop a general procedure to compute the gauge fields in higher dimensions associated to low-dimensional systems with periodic (charge- and spin-) density wave modulations. We construct two-dimensional (2D) models with modulations that can be promoted to higher-order topological phases with $U(1)$ and $SU(2)$ gauge fields in 3D. Corner modes in our 2D models can be pumped by adiabatic sliding of the phase of the modulation, yielding hinge modes in the promoted models. We also examine a 3D Weyl semimetal (WSM) gapped by charge-density wave (CDW) order, possessing quantum anomalous Hall (QAH) surface states. We show that this 3D system is equivalent to a 4D nodal line system gapped by a $U(1)$ gauge field with a nonzero second Chern number. We explain the recently identified interpolation between inversion-symmetry protected phases of the 3D WSM gapped by CDWs using the corresponding 4D theory. Our results can extend the search for (higher-order) topological states in higher dimensions to density wave systems.

The sound of Weyl hinges.

Clever 3D-printed acoustic materials allow probing a new higher-order semimetallic topological phase using audible sound.

Magnetism-mediated transition between crystalline and higher-order topological phases in NpSb

Merging the fields of topology and magnetism expands the scope of fundamental quantum phenomena with novel functionalities for topological spintronics enormously. Here, we theoretically demonstrate that ferromagnetism provides an efficient means to achieve a topological switching between crystalline and higher-order topological insulator phases in two dimensions. Using a tight-binding model and first-principles calculations, we identify layered NpSb as a long-awaited two-dimensional topological crystalline insulator with intrinsic ferromagnetic order with a band gap which is as large as 220 meV. We show that when ${\mathcal{M}}_{z}$ symmetry is preserved for the out of plane magnetization of this material, it exhibits a pair of gapless edge states along all boundaries and carries a nonzero mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=1$. Remarkably, when rotating the magnetization into the plane a higher-order topological insulator phase with a parity-based invariant ${\ensuremath{\nu}}_{2\text{D}}=1$ is achieved, and in-gap topological corner states emerge. Our results pave the way to understanding and engineering topological insulating states in two-dimensional ferromagnets.

Structural-Disorder-Induced Second-Order Topological Insulators in Three Dimensions.

Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge modes protected by crystalline symmetries. Since amorphous solids are ubiquitous, it is important to ask whether such a second-order topological insulator can exist in an amorphous system without any spatial order. Here, we predict the existence of a second-order topological insulating phase in an amorphous system without any crystalline symmetry. Such a topological phase manifests in the winding number of the quadrupole moment, the quantized longitudinal conductance, and the hinge states. Furthermore, in stark contrast to the viewpoint that structural disorder should be detrimental to the higher-order topological phase, we remarkably find that structural disorder can induce a second-order topological insulator from a topologically trivial phase in a regular geometry. We finally demonstrate the existence of a second-order topological phase in amorphous systems with time-reversal symmetry.