Number Of Topologies Research Articles

Broken Mirror Symmetry of Tracer's Trajectories in Turbulence.

Topological properties of physical systems play a crucial role in our understanding of nature, yet their experimental determination remains elusive. We show that the mean helicity, a dynamical invariant in ideal flows, quantitatively affects trajectories of fluid elements: the linking number of Lagrangian trajectories depends on the mean helicity. Thus, a global topological invariant and a topological number of fluid trajectories become related, and we provide an empirical expression linking them. The relation shows the existence of long-term memory in the trajectories: the links can be made of the trajectory up to a given time, with particles positions in the past. This property also allows experimental measurements of mean helicity.

Bulk-edge correspondence and trapping at a non-Hermitian topological interface.

In Hermitian systems, according to the bulk-edge correspondence, interfacing two topological optical media with different bulk topological numbers implies the existence of edge states, which can trap light at the interface. However, such a general scenario can be violated when dealing with non-Hermitian systems. Here we show that interfacing two semi-infinite Hatano-Nelson chains with different bulk topological numbers can result in the existence of infinitely many edge (interface) states; however, light waves cannot be rather generally trapped at the interface.

A gauge theory for two-band model of Chern insulators and induced topological defects

In this paper a gauge theory is proposed for the two-band model of Chern insulators. Based on the so-called ’t Hooft monopole model, a U(1) Maxwell electromagnetic sub-field is constructed from an SU(2) gauge field, from which arise two types of topological defects, monopoles and merons. We focus on the topological number in the Hall conductance , where C is the Chern number. It is discovered that in the monopole case C is indeterminate, while in the meron case C takes different values, due to a varying on-site energy m. As a typical example, we apply this method to the square lattice and compute the winding numbers (topological charges) of the defects; the C-evaluations we obtain reproduce the results of the usual literature. Furthermore, based on the gauge theory we propose a new model to obtain the high Chern numbers ∣C∣ = 2, 4.

Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: Application to the XXZ model on the square lattice

In this work, we address the challenge of uncovering patterns in variational optimal protocols for taking the system to ground states of many-body Hamiltonians, using variational quantum algorithms. We develop highly optimized classical Monte Carlo (MC) algorithms to find the optimal protocols for transformations between the ground states of the square-lattice XXZ model for finite systems sizes. The MC method obtains optimal bang-bang protocols, as predicted by Pontryagin's minimum principle. We identify the minimum time needed for reaching an acceptable error for different system sizes as a function of the initial and target states and uncover correlations between the total time and the wave-function overlap. We determine a dynamical phase diagram for the optimal protocols, with different phases characterized by a topological number, namely the number of on-pulses. Bifurcation transitions as a function of initial and final states, associated with new jumps in the optimal protocols, demarcate these different phases. The number of pulses correlates with the total evolution time. In addition to identifying the topological characteristic above, i.e., the number of pulses, we introduce a correlation function to characterize bang-bang protocols' quantitative geometric similarities. We find that protocols within one phase are indeed geometrically correlated. Identifying and extrapolating patterns in these protocols may inform efficient large-scale simulations on quantum devices.

Equilateral polygons in a Euclidean space and on deformed spheres via Fadell-Husseini index

We introduce the concept of r-equilateral m-gons. We prove, for odd primes p, the existence of r-equilateral p-gons in Rd if r<d and give conditions for the existence of equilateral p-gons in the image of continuous injective maps f:Sd→Rd+1.We follow the ideas of Soibelman from 1980, in which the topological Borsuk number of R2 is calculated by means of methods of equivariant algebraic topology, and the paper of Blagojević and Ziegler from 2009, where Fadell-Husseini index is used for finding equilateral quadrilaterals with equilateral diagonals in the image of continuous injective maps f:S2→R3.

Non-Hermitian bulk-boundary correspondence and singular behaviors of generalized Brillouin zone

The bulk boundary correspondence, which connects the topological invariant, the continuum band and energies under different boundary conditions, is the core concept in the non-Bloch band theory, in which the generalized Brillouin zone (GBZ), appearing as a closed loop generally, is a fundamental tool to rebuild it. In this work, it can be shown that the recovery of the open boundary energy spectrum by the continuum band remains unchanged even if the GBZ itself shrinks into a point. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of the energy, but not from the topological invariant. Meanwhile, regardless of the fact that the GBZ comes out with the closed loop, the bulk boundary correspondence cannot be well characterized yet because of the ill-definition of the topological number. Here, the results obtained may be useful for improving the existing non-Bloch band theory.

Half-skyrmions with higher topological quantum numbers in homogeneous exciton-polariton condensates.

We investigate the topological excitations of half-quantum vortices (HQVs) with higher topological quantum numbers in a homogeneous spinor exciton-polariton condensate pumped by a laser beam and an additional coherent light carrying orbital angular momentum. The spin texture and integrated topological charge can be controlled through the pump. Among these textures, the polaritonic half-skyrmions (or polaritonic merons) can be created with a suitable excitation condition. Moreover, when the pump polarization is in favor of the vortex component of the HQV, there is an inversion of circular polarization (spin flipping) from the center of the HQV towards the edge. The radial flipping position can be manipulated by the pump polarization or power. Finally, we demonstrate that the HQVs can stably exist from the linear stability analysis.

Topological circuit of a versatile non-Hermitian quantum system

We propose an resistors, inductors and capacitors (RLC) electrical circuit to theoretically analyze and fully simulate a new type of non-Hermitian Su-Schrieffer-Heeger (SSH) model with complex hoppings. We formulate its construction and investigate its properties by taking advantage of the circuit's versatility. Rich physical properties can be identified in this system from the normal modes of oscillation of the RLC circuit, including the highly tunable bulk-edge correspondence between topological winding numbers and edge states and the non-Hermitian skin phenomenon originating from a novel complex energy plane topology. The present study is able to show the wide and appealing topological physics inherent to electric circuits, which is readily generalizable to a plenty of both Hermitian and non-Hermitian nontrivial systems.

Dynamic transition of current-driven single-skyrmion motion in a room-temperature chiral-lattice magnet

Driving and controlling single-skyrmion motion promises skyrmion-based spintronic applications. Recently progress has been made in moving skyrmionic bubbles in thin-film heterostructures and low-temperature chiral skyrmions in the FeGe helimagnet by electric current. Here, we report the motion tracking and control of a single skyrmion at room temperature in the chiral-lattice magnet Co9Zn9Mn2 using nanosecond current pulses. We have directly observed that the skyrmion Hall motion reverses its direction upon the reversal of skyrmion topological number using Lorentz transmission electron microscopy. Systematic measurements of the single-skyrmion trace as a function of electric current reveal a dynamic transition from the static pinned state to the linear flow motion via a creep event, in agreement with the theoretical prediction. We have clarified the role of skyrmion pinning and evaluated the intrinsic skyrmion Hall angle and the skyrmion velocity in the course of the dynamic transition. Our results pave a way to skyrmion applications in spintronic devices.

Electron transport probing the electrically tunable topological phase transition in a Dirac semimetal

Electron transport in a Dirac semimetal (DSM) quantum wire without/with an external electric field is investigated by using the Green-function-based Landauer-B\"uttiker formula. Normal-topological phase transition is certificated by the different characteristics between the DSM quantum wire without and with the electric field, including the orbital decomposition for the energy bands, transversally local charge and spin polarization density distributions, topological Chern number, transport behaviors, and disorder effects. Topologically trivial surface states are found in the former system and their transport can be localized by disorders. However, topologically nontrivial surface states are generated in the latter one and their transport is robust against strong disorders when the electric field strength is larger than a critical value. In addition, switching effect of the electron transport in the DSM quantum wire can be achieved by varying the electric field strength. Further studies indicate that the switching effect shows a strong robustness against disorder, displaying the feasibility of designing a topological field effect transistor based on the considered system.

Particle-like topologies in light

Three-dimensional (3D) topological states resemble truly localised, particle-like objects in physical space. Among the richest such structures are 3D skyrmions and hopfions, that realise integer topological numbers in their configuration via homotopic mappings from real space to the hypersphere (sphere in 4D space) or the 2D sphere. They have received tremendous attention as exotic textures in particle physics, cosmology, superfluids, and many other systems. Here we experimentally create and measure a topological 3D skyrmionic hopfion in fully structured light. By simultaneously tailoring the polarisation and phase profile, our beam establishes the skyrmionic mapping by realising every possible optical state in the propagation volume. The resulting light field’s Stokes parameters and phase are synthesised into a Hopf fibration texture. We perform volumetric full-field reconstruction of the {{{Pi }}}_{{{3}}} mapping, measuring a quantised topological charge, or Skyrme number, of 0.945. Such topological state control opens avenues for 3D optical data encoding and metrology. The Hopf characterisation of the optical hypersphere endows a fresh perspective to topological optics, offering experimentally-accessible photonic analogues to the gamut of particle-like 3D topological textures, from condensed matter to high-energy physics.

Anonymous Stochastic Routing

We propose and analyze a recipient-anonymous stochastic routing model to study a fundamental trade-off between anonymity and routing delay. An agent wants to quickly reach a goal vertex in a network through a sequence of routing actions, whereas an overseeing adversary observes the agent’s entire trajectory and tries to identify the agent’s goal among those vertices traversed. We are interested in understanding the probability that the adversary can correctly identify the agent’s goal (anonymity) as a function of the time it takes the agent to reach it (delay). A key feature of our model is the presence of intrinsic uncertainty in the environment, so that each of the agent’s intended steps is subject to random perturbation and thus may not materialize as planned. Using large-network asymptotics, our main results provide near-optimal characterization of the anonymity–delay trade-off under a number of network topologies. Our main technical contributions are centered on a new class of “noise-harnessing” routing strategies that adaptively combine intrinsic uncertainty from the environment with additional artificial randomization to achieve provably efficient obfuscation.

Topological phases and type-II edge state in two-leg-coupled Su–Schrieffer–Heeger chains

Topological quantum states have attracted great attention both theoretically and experimentally. Here, we show that the momentum-space lattice allows us to couple two Su–Schrieffer–Heeger (SSH) chains with opposite dimerizations and staggered interleg hoppings. The coupled SSH chain is a four-band model which has sublattice symmetry similar to the SSH4. Interestingly, the topological edge states occupy two sublattices at the same time, which can be regarded as a one-dimension analogue of the type-II corner state. The analytical expressions of the edge states are also obtained by solving the eigenequations. Finally, we provide a possible experimental scheme to detect the topological winding number and corresponding edge states.

Distribution Network Reconfiguration for Short-Term Voltage Stability Enhancement: An Efficient Deep Learning Approach

The rapid growth of renewables and dynamic loads has highlighted the short-term voltage stability (STVS) issue in distribution network operation. With the advances in metering, communication and control technologies, deep learning-based distribution network reconfiguration (DNR) becomes available to maintain secure and economic system operation. However, STVS is traditionally evaluated for a specific network topology using time-domain simulations, which cannot be expressed explicitly as a function for optimization, making it intractable to consider STVS in DNR. In this paper, an efficient deep learning-based method is proposed to address this issue. An STVS evaluation network is customized from deep convolution neural networks (CNNs) and trained to learn the relationship between network topology and STVS performance from historical data. To find the optimal topology, a well-trained evaluation network is applied in DNR, where STVS with various network topologies is evaluated without resorting to time-domain simulations. Then, the number of candidate topologies is significantly reduced by a threshold of STVS performance, which enables the direct solution of the DNR for STVS enhancement. The application of the proposed method in large-scale systems is also discussed via integration with heuristic algorithms. Case studies of a modified 69-bus distribution system and a real large-scale distribution system validate the necessity of considering STVS enhancement in DNR and the effectiveness of the proposed solution method.

Focusing properties of power order space-variant phase modulate Bessel-Gaussian vortex beam

The purpose of this paper is to study the focusing properties of power order space-variant phase modulate the Bessel-Gaussian (BG) vortex beam, which is based on vector diffraction theory. Results show that under different power order of space-variant phase, the modulation of topological charge number and phase parameter have different evolution rules. Specifically, when power order n = 1, the position of the focal pattern will be arbitrarily adjusted on the X-axis by modulating the phase parameter C. Under the condition of n = 2, the focal spot will be expanded along y = -x with the increase of topological charge l. Meanwhile, by modulating C, two opposite phenomena of focal spot separation will be obtained. When n = 3, the increase of l will make the focal spot expand along the positive and negative directions of the Y-axis. Besides, increasing the value of C, the original focal spot intensity will converge to the X-axis, and a new intensity peak will be formed on the X-axis. A series of focal properties have been obtained as the power order space-variant phase introduced. These novel properties have potential applications in many fields such as particle trapping, particle movement and optical modulation related fields.

Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems

Quantum anomalous Hall insulator (QAH)/$s$-wave superconductor (SC) hybrid systems are known to be an ideal platform for realizing two-dimensional topological superconductors with chiral Majorana edge modes. In this paper we study QAH/unconventional SC hybrid systems whose pairing symmetry is $p$-wave, $d$-wave, chiral $p$-wave, or chiral $d$-wave. The hybrid systems are a generalization of the QAH/$s$-wave SC hybrid system. In view of symmetries of the QAH and pairings, we introduce three topological numbers to classify topological phases of the hybrid systems. One is the Chern number that characterizes chiral Majorana edge modes and the others are topological numbers associated with crystalline symmetries. We numerically calculate the topological numbers and associated surface states for three characteristic regimes that feature an influence of unconventional SCs on QAHs. Our calculation shows a rich variety of topological phases and unveils the following topological phases that are no counterpart of the $s$-wave case: crystalline symmetry-protected helical Majorana edge modes, a line node phase (crystalline-symmetry-protected Bogoliubov Fermi surface), and multiple chiral Majorana edge modes. The phenomena result from a nontrivial topological interplay between the QAH and unconventional SCs. Finally, we discuss tunnel conductance in a junction between a normal metal and the hybrid systems, and show that the chiral and helical Majorana edge modes are distinguishable in terms of the presence/absence of zero-bias conductance peak.

Nonlinear Topological Effects in Optical Coupled Hexagonal Lattice.

Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson–hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.

Symmetry-Protected Multifold Exceptional Points and Their Topological Characterization.

We investigate the occurrence of n-fold exceptional points (EPs) in non-Hermitian systems, and show that they are stable in n-1 dimensions in the presence of antiunitary symmetries that are local in parameter space, such as, e.g., parity-time (PT) or charge-conjugation parity (CP) symmetries. This implies in particular that threefold and fourfold symmetry-protected EPs are stable, respectively, in two and three dimensions. The stability of such multofold exceptional points (i.e., beyond the usual twofold EPs) is expressed in terms of the homotopy properties of a resultant vector that we introduce. Our framework also allows us to rephrase the previously proposed Z_{2} index of PT and CP symmetric gapped phases beyond the realm of two-band models. We apply this general formalism to a frictional shallow water model that is found to exhibit threefold exceptional points associated with topological numbers ±1. For this model, we also show different non-Hermitian topological transitions associated with these exceptional points, such as their merging and a transition to a regime where propagation is forbidden, but can counterintuitively be recovered when friction is increased furthermore.

Intensity interference characteristics of fractional Laguerre–Gaussian optical vortex

The paper reports the intensity characteristics of fractional vortex beam, and controls the coupling between photon orbital angular momentum and spin angular momentum by coherently manipulating Laguerre–Gaussian vortex beam with linearly polarized plane light and its conjugate beam. By comparing theory with experiment, fractional vortex beams with different topological charges are generated by using pure phase liquid crystal spatial light modulator, and their intensity characteristics and interference characteristics are observed. The results show that the radial notch size of the spot is related to the fractional part of the topological charge value, and the closer the fractional part of the topological charge value is to the half integer, the more obvious the radial notch is. With the increase of integer part of topological charge, the spot radius also increases. In addition, interference with linearly polarized plane light reveals that the number of branches is equal to the number of topological charges. Interfere with its conjugate beam, and judge the topological charge number of vortex beam by the relationship between petal number and topological charge number. The petal number of integer vortex beam is twice the topological charge number, and fractional order vortex beam destroys the ring structure of Laguerre–Gaussian beam.

Topological charge pumping in quasiperiodic systems characterized by the Bott index

We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, regardless of the detail of quasiperiodic models. We apply this formalism to the Fibonacci-Rice-Mele model which we made from Fibonacci lattice, a well-known quasiperiodic system, and Rice-Mele model. We find that these quasiperiodic systems show topological charge pumping with a multi-level behavior due to the fractal nature of the Fibonacci lattice. Such multi-level pumping behaviors can be understood by a real space renormalization group analysis.