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In topological band theory, phonon boundary modes consequence of a topologically non-trivial band structure feature desirable properties for atomically-precise technologies, such as robustness against defects, waveguiding, and one-way transport. These topological phonon boundary modes remain to be studied both theoretically and experimentally in synthetic materials, such as polymers and supramolecular assemblies at the atomistic level under thermal fluctuations. Here we show by means of molecular simulations, that surface-confined Su-Schrieffer-Heeger (SSH) phonon analogue models express robust topological phonon boundary modes at heavy boundaries and under thermal fluctuations. The resulting bulk-heavy boundary correspondence enables patterning of boundary modes in polymer chains and weakly-interacting supramolecular lattices. Moreover, we show that upon excitation of a single molecule, propagation along heavy-boundary modes differs from free boundary modes. Our work is an entry to topological vibrations in supramolecular systems, and may find applications in the patterning of phonon circuits and realization of Hall effect phonon analogues at the molecular scale. Topological phonon boundary modes (TBM) hold promises for advanced phononic applications. Here, the authors introduce the bulk-heavy boundary correspondence to pattern TBM in polymers and supramolecular assemblies under thermal fluctuations. [ABSTRACT FROM AUTHOR]

Topologically ordered phases of matter elude Landau's symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. Recently, there have been proposals to extend the concept of time crystals to topological order. Here the authors observe a prethermal topologically ordered time crystal on a superconducting quantum processor, where discrete time-translation symmetry breaking manifests for nonlocal rather than local observables. [ABSTRACT FROM AUTHOR]

Følner density is a very natural notion of density which is defined on any semigroup satisfying the Strong Følner Condition (SFC). (These include all commutative semigroups and all left cancellative left amenable semigroups.) Piecewise syndetic and thick are notions of largeness arising from topological dynamics. It has been known that if S satisfies SFC and is either left cancellative or satisfies a weak right cancellation requirement, then every thick subset has density 1. We show here that in any semigroup S satisfying SFC a subset of S is thick if and only if it has density 1. As a consequence, every piecewise syndetic set has positive density. [ABSTRACT FROM AUTHOR]

Current-driven dynamics of topological spin textures, such as skyrmions and antiskyrmions, have garnered considerable attention in condensed matter physics and spintronics. As compared with skyrmions, the current-driven dynamics of their antiparticles – antiskyrmions − remain less explored due to the increased complexity of antiskyrmions. Here, we design and employ fabricated microdevices of a prototypical antiskyrmion host, (Fe0.63Ni0.3Pd0.07)3P, to allow in situ current application with Lorentz transmission electron microscopy observations. The experimental results and related micromagnetic simulations demonstrate current-driven antiskyrmion dynamics confined within stripe domains. Under nanosecond-long current pulses, antiskyrmions exhibit directional motion along the stripe regardless of the current direction, while the antiskyrmion velocity is linearly proportional to the current density. Significantly, the antiskyrmion mobility could be enhanced when the current flow is perpendicular to the stripe direction. Our findings provide novel and reliable insights on dynamical antiskyrmions and their potential implications on spintronics. Antiskyrmions, like skyrmions, are a form of topological spin texture, with a topological charge of opposite sign to the equivalent skyrmion with the same polarity. While antiskyrmions have been less explored, they offer some potential advantages for applications, and here, Guang et al demonstrate antiskyrmion motion within stripe domains. [ABSTRACT FROM AUTHOR]

The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs. [ABSTRACT FROM AUTHOR]

Programmable quantum simulators may one day outperform classical computers at certain tasks. But at present, the range of viable applications with noisy intermediate-scale quantum (NISQ) devices remains limited by gate errors and the number of high-quality qubits. Here, we develop an approach that places digital NISQ hardware as a versatile platform for simulating multi-dimensional condensed matter systems. Our method encodes a high-dimensional lattice in terms of many-body interactions on a reduced-dimension model, thereby taking full advantage of the exponentially large Hilbert space of the host quantum system. With circuit optimization and error mitigation techniques, we measured on IBM superconducting quantum processors the topological state dynamics and protected mid-gap spectra of higher-order topological lattices, in up to four dimensions, with high accuracy. Our projected resource requirements scale favorably with system size and lattice dimensionality compared to classical computation, suggesting a possible route to useful quantum advantage in the longer term. Previous quantum simulations of higher-order topological phases have utilized synthetic dimensions. Here the authors simulate high-order topological phases on lattices with spatial dimension up to four on a superconducting quantum computer using a mapping to a many-body Hamiltonian in a reduced dimension. [ABSTRACT FROM AUTHOR]

This article is about the periodic points characterization of automorphisms of some solenoids, whose duals are subgroups of algebraic number fields. Here, we use the theory of adeles for describing a solenoid and the periodic points of its automorphisms. This is in line with the earlier characterizations which considered other classes of solenoids. [ABSTRACT FROM AUTHOR]

Advancements in photonics technologies have significantly enhanced their capability to facilitate experiments involving quantum light, even at room temperature. Nevertheless, fully integrating photonic chips that include quantum light sources, effective manipulation and transport of light minimizing losses, and appropriate detection systems remains an ongoing challenge. Topological photonic systems have emerged as promising platforms to protect quantum light properties during propagation, beyond merely preserving light intensity. In this work, we delve into the dynamics of non-classical light traversing a Su-Schrieffer-Heeger photonic lattice with topological domain walls. Our focus centers on how topology influences the quantum properties of light as it moves across the array. By precisely adjusting the spacing between waveguides, we achieve dynamic repositioning and interaction of domain walls, facilitating effective beam-splitting operations. Our findings demonstrate high-fidelity transport of non-classical light across the lattice, replicating known results that are now safeguarded by the topology of the system. This protection is especially beneficial for quantum communication protocols with continuous variable states. Our study enhances the understanding of light dynamics in topological photonic systems and paves the way for high-fidelity, topology-protected quantum communication. [ABSTRACT FROM AUTHOR]

The abundant topological magnetic textures in three-dimensional systems provide opportunities to investigate the fundamental spin dynamics and realize spintronic applications. The dynamics of such magnetic textures have however rarely been studied, especially for those driven by spin waves, which allow applications with ultralow energy consumption and ease of implementation even in insulating systems. Here, we report our micromagnetic simulations on the spin-wave-driven dynamics of a skyrmion tube (SkT) and chiral bobber (ChB) in a thick magnetic film. We predict tornado-like dynamics in both SkT and ChB, where the topological centers present a lateral rotation with the rotation centers forming a distorted profile in the thickness direction. While the velocity of SkT scales with the driving power, the ChB motion presents a threshold in the driving field, which is found to depend linearly on its penetration length. This distinct behavior could be useful to differentiate ChB from SkT and estimate its penetration length experimentally. The magnetic dynamics driven by spin waves can be useful for low-consumption devices, which is however still largely unexplored in three-dimensional system. Here, the authors find, using micromagnetic simulations, spin-wave-driven tornado-like dynamics of skyrmion tube and chiral bobber in thick magnetic films. [ABSTRACT FROM AUTHOR]

The purpose of this study is to analyze the topological structure dynamics of the complex network of stocks before and after the outbreak of the COVID-19, so as to provide a basis for preventing financial risks. We calculate Pearson correlation coefficient between enterprises according to logarithmic rate of return and trading volume ratio of enterprises' stocks, and then constructed a complex network of stock market price and volume before and after the outbreak of the COVID-19. First, through thresholding and heat map imaging of the correlation matrix, the change characteristics of the correlation between various industries in 2019 and 2020 are studied. Second, the node degree, average weighted degree, graph density, clustering coefficient, and average clustering coefficient are used to study the topological structure change of the complex network of stock correlation. Third, the principle of node betweenness centrality is used to analyze the characteristics of a complex network after removing the core nodes. The research shows that, first, under the influence of the COVID-19 pandemic, the correlation among industries has the characteristics of industrial clusters, that is, the correlation in a industry is strengthened. In addition to banking, the correlation between industries has weakened, and the correlation between the banking industry and other industries has strengthened. Second, the node difference in betweenness centrality of core nodes in 2020 is higher than that in 2019, indicating that the network stability in 2019 is higher than that in 2020. These two points indicate that under the influence of the COVID-19 epidemic, the complex network topology of China's entire stock market has changed, and companies need to undertake countermeasures in the face of the crisis to effectively prevent and control systemic risks. [ABSTRACT FROM AUTHOR]

The metasurface platform with time-varying characteristics has emerged as a promising avenue for exploring exotic physics associated with Floquet materials and for designing photonic devices like linear frequency converters. However, the limited availability of materials with ultrafast responses hinders their applications in the terahertz range. Here we present a time-varying metasurface comprising an array of superconductor-metal hybrid meta-molecules. Each meta-molecule consists of two meta-atoms that are "bonded" together by double superconducting microbridges. Through experimental investigations, we demonstrate high-efficiency linear terahertz frequency conversion by rapidly breaking the bond using a coherent ultrashort terahertz pump pulse. The frequency and relative phase of the converted wave exhibit strong dependence on the pump-probe delay, indicating phase controllable wave conversion. The dynamics of the meta-molecules during the frequency conversion process are comprehensively understood using a time-varying coupled mode model. This research not only opens up new possibilities for developing innovative terahertz sources but also provides opportunities for exploring topological dynamics and Floquet physics within metasurfaces. The time-varying metasurfaces show promise for exploring exotic physics and photonic applications. The authors introduce a time-varying metasurface with superconductor-metal hybrid meta-molecules, demonstrating phase-controllable frequency conversion with high efficiency. [ABSTRACT FROM AUTHOR]

Features of the dynamics of topological defects in an electroconvective structure formed in a π/2-twisted nematic liquid crystal have been studied. The electroconvective structure (Williams domains) is a system of rolls similar to the system of convective cells in thermal convection. Hydrodynamic flows in rolls of the twisted nematic liquid crystal are helicoidal because the axial velocity component existing in addition to the tangential component has opposite directions in the neighboring rolls. This feature is responsible for the formation of stable localized extended objects—linear defects—oriented normally to Williams domains. The continuity condition for a helicoidal flow of an anisotropic liquid in twisted nematic liquid crystals prevents the decay of a linear defect into single dislocations. The length of the linear defect and the number of dislocations in it are controlled by an ac voltage applied to a liquid crystal cell. In contrast to the case of planar orientation, where linear defects decay into single dislocations with increasing applied voltage, zigzag oscillations appear in this case, but the structure of domains remains stationary. The boundaries between zig and zag regions in the core of the linear defect are dislocations with the topological charges S = +1 and –1. An "elementary" decay of a dislocation with the topological charge S = +1 (kink) into the dislocation with S = –1 (antikink) and two dislocations the topological charges S = +1 has been found for the first time in the linear defect of a certain length. The decay of the topological defect is possibly due to the appearance of the local instability of the orientational twist mode of the director n in the defect core caused by the critical growth of hydrodynamic fluctuations with increasing applied voltage. It has been shown that the detected decay of the topological soliton is qualitatively well described by the perturbed sine-Gordon eq-uation. [ABSTRACT FROM AUTHOR]

The recent observations of exotic quantum phenomena in AV3Sb5 (A = K, Rb, Cs) kagome superconductors have attracted significant attention in materials physics. Here, we propose an innovative two-frequencies laser model for ultrafast control of transient structural distortions. Using first-principles density functional theory in conjunction with the perturbative regime of nonlinear phononics, we investigate the nonharmonic potential energy, the crystal lattice dynamics and the topological properties of CsV3Sb5. We find that driving two infrared-active phonons of different frequencies promotes the desired Raman phonon vibrations, in which the displacement of Sb atoms is closely related to superconductivity. We demonstrate that the dimensional crossover and the topological nontrivial to trivial state transition of superconductor CsV3Sb5 can be triggered by ultrafast optical control. This work can be applied to other layered quantum materials and provide guidance for experiments related to photoinduced topology and superconductivity. [ABSTRACT FROM AUTHOR]

The recent observations of exotic quantum phenomena in AV3Sb5 (A = K, Rb, Cs) kagome superconductors have attracted significant attention in materials physics. Here, we propose an innovative two-frequencies laser model for ultrafast control of transient structural distortions. Using first-principles density functional theory in conjunction with the perturbative regime of nonlinear phononics, we investigate the nonharmonic potential energy, the crystal lattice dynamics and the topological properties of CsV3Sb5. We find that driving two infrared-active phonons of different frequencies promotes the desired Raman phonon vibrations, in which the displacement of Sb atoms is closely related to superconductivity. We demonstrate that the dimensional crossover and the topological nontrivial to trivial state transition of superconductor CsV3Sb5 can be triggered by ultrafast optical control. This work can be applied to other layered quantum materials and provide guidance for experiments related to photoinduced topology and superconductivity. [ABSTRACT FROM AUTHOR]

In this paper we study the static structure and the dynamics of topological defects associated with isotropic droplets in nematic environment. Investigations were made in confined geometry of optical cells when the droplet size was of the order of or larger than the gap of the cell. We observed the coexistence of point boojums and Saturn ring or modified Saturn ring defects. We found transformation of the Saturn ring defect to two localized broad defects at increasing the droplet size. At droplet coalescence antipodes of point and localized broad defects were born and the dynamics of their annihilation with existing defects was investigated. We found strong difference in the process of annihilation of point and localized broad defects. [ABSTRACT FROM AUTHOR]

The realization of Majorana zero modes in condensed matter have been attracting enormous interests from fundamental science such as topological quantum computation. Recently iron based superconductors were identified as a high-temperature platform for realizing topological superconductivity and Majorana modes. As unconventional superconductors, one of the most important characteristics of them is that they are in the vicinity of magnetic states due to the strong Hund's coupling in iron atoms. Here we propose that the line defects with missing Te/Se anions in Fe(Se, Te) superconductors provide the realization of intrinsic antiferromagnetic (AFM) chains with Rashba spin-orbit coupling. Against conventional wisdom, Majorana zero modes (MZMs) can be robustly generated at these AFM chain ends. These results can consistently explain the recent experimental observation of zero-energy end states in line defects of monolayer Fe(Te, Se)/SrTiO3 by scanning tunneling microscopy (STM) measurements. Our research not only demonstrates an unprecedented interplay among native line defect, emergent magnetism and topological superconductivity but also explores a high-temperature platform for Majorana fermions. [ABSTRACT FROM AUTHOR]

Laser-driven coherent phonons can act as modulated strain fields and modify the adiabatic ground state topology of quantum materials. Here we use time-dependent first-principles and effective model calculations to simulate the effect of the coherent phonon induced by strong terahertz electric field on electronic carriers in the topological insulator ZrTe5. We show that a coherent A1g Raman mode modulation can effectively pump carriers across the band gap, even though the phonon energy is about an order of magnitude smaller than the equilibrium band gap. We reveal the microscopic mechanism of this effect which occurs via Landau–Zener-Stückelberg tunneling of Bloch electrons in a narrow region in the Brillouin zone center where the transient energy gap closes when the system switches from strong to weak topological insulator. The quantum dynamics simulation results are in excellent agreement with recent pump-probe experiments in ZrTe5 at low temperature. Coherent phonons can modify the wavefunction topology of quantum materials, yet the implications for electron dynamics remain to be addressed. The authors use time-dependent approaches to simulate the effect of coherent phonons, induced by strong terahertz laser field, on the electronic carrier dynamics in the topological insulator Zirconium Pentatelluride. [ABSTRACT FROM AUTHOR]

We give a new categorical approach to the Halmos–von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos–von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka–Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group. [ABSTRACT FROM AUTHOR]

The Besicovitch pseudometric is a shift-invariant pseudometric over the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. It corresponds to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudometric, the Feldman–Katok pseudometric, where the Hamming distance is replaced by the Levenshtein distance. Like in the Besicovitch space, cellular automata are Lipschitz in this space, but here also all substitutions are Lipschitz. In both spaces, we discuss equicontinuity of these systems, give a number of examples, and generalize our results to the class of dill maps, that embeds both cellular automata and substitutions. [ABSTRACT FROM AUTHOR]

We explore the notion of discrete spectrum and its various characterizations for ergodic measure-preserving actions of an amenable group on a compact metric space. We introduce a notion of 'weak-tameness', which is a measure theoretic version of a notion of 'tameness' introduced by E. Glasner, based on the work of A. Köhler, and characterize such topological dynamical systems as systems for which every invariant measure has a discrete spectrum. Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of 'witness of irregularity' which is based on 'up-crossings'. Then we establish that 'strong Veech systems' are tame. In particular, for any countable amenable group T, the flow on the orbit closure of the translates of a 'Veech function' f ∈ K (T) is tame. Thus Sarnak's Möbius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi–Ramachandra theorem of 1976 on the Mertens function in short intervals. We further improve Motohashi–Ramachandra's bound to 1/2 under the Chowla conjecture. [ABSTRACT FROM AUTHOR]

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016). [ABSTRACT FROM AUTHOR]

A Möbius strip, fascinating for its unique topological property of being a one-side nonorientable surface, has inspired mathematicians, physicists, engineers, and artists for many centuries. In a coherent system, coherent waves on the nonorientable surfaces reveal rich topological dynamics due to the interplay of coherence and topology. Here we experimentally observe topological resonances in a Möbius ring resonator formed in a twisted optical fiber loop. The twisted polarization-maintaining fiber ring encourages the hybridization of two polarization states, giving rise to the crucial Berry phase. This geometrical phase leads to the frequency shifts of fiber resonant modes with a non-trivial fractional mode number. Moreover, the resonant modes are topological, only resonating with certain polarized modes with circular chirality. These topological features introduce geometrical factors into coherent wave resonances, paving the way for topological information processing for quantum information, and coherent wave dynamics. The Möbius strip exhibits unique topological resonances, which are fundamental for topology-based signal processing. The authors realise a Möbius ring resonator via polarization-maintaining fibers, observing topological resonances emerging from the coupling between the two polarization modes along the fast and slow axes of the ring. [ABSTRACT FROM AUTHOR]

We study ultrafast population dynamics in the topological surface state of Sb 2 Te 3 in two-dimensional momentum space with time- and angle-resolved two-photon photoemission spectroscopy. Linearly polarized mid-infrared pump pulses are used to permit a direct optical excitation across the Dirac point. We show that this resonant excitation is strongly enhanced within the Dirac cone along three of the six Γ ¯ – M ¯ directions and results in a macroscopic photocurrent when the plane of incidence is aligned along a Γ ¯ – K ¯ direction. Our experimental approach makes it possible to disentangle the decay of transiently excited population and photocurent by elastic and inelastic electron scattering within the full Dirac cone in unprecedented detail. This is utilized to show that doping of Sb 2 Te 3 by vanadium atoms strongly enhances inelastic electron scattering to lower energies, but only scarcely affects elastic scattering around the Dirac cone. [ABSTRACT FROM AUTHOR]

The popularity of football among fans to analyze the game has been immense with the advent of internet. The concept of making a dream team in football has become a new fashion for the football lovers. The paper focuses in helping achieving this prediction of a football dream team. The aim of this research is to assess the dynamics of a complex topological structure when prompted with random entities whose attributes are known to us. Using graph theory and vectorial distances, the dream team is evaluated on the basis of individual abilities and interplayer synergy. Instead of focusing on discrete events in a match, this framework proposes an idea in which a dream team is quantified on the basis of their positional attributes. Each player is rated in accordance to the position he is playing, which eventually helps in finding the overall team rating. The second part of this research uses graph theory to evaluate structural and topological properties of interpersonal interactions of teammates. Teammates are treated as nodes of a graph, where each edge exemplifies the strength of their interpersonal interaction. The strength of the bond depends on on-field interactions via ball passing, ball receiving and communication which depend on experience of playing together, Nationality and Club. The methodology adopted in this paper can be a formidable basis for similarly situated larger setups involving much larger intricacies. Using this framework, we can see the behavior of a hypothetical topological structure whose node attributes are known to us, thus projecting its performance as a team and individual entities. [ABSTRACT FROM AUTHOR]

The interplay between system dynamics and topological structure plays a crucial role in nonlinear system and modern network science. In this paper, we intend to investigate the effects of both heterogeneity and asymmetric coupling on the collective behaviors of Kuramoto oscillators, and reveal the processes how the synchronization transits to multi-cluster or disorder states. We find that the complete synchronization and anti-phase synchronization states in a star motif change into the remote synchronization state, owing to the disparity in the values of oscillators' natural frequencies, and further become into multi-cluster due to the existence of multiple hubs in complete bipartite network, and finally achieve disorder state in complex networks. These findings are helpful to understand the forming processes of remote synchronization and general multi-cluster in coupled systems. [ABSTRACT FROM AUTHOR]

We consider the following singularly perturbed Kirchhoff-type equations - ε 2 M ε 2 - N ∫ R N | ∇ u | 2 d x Δ u + V (x) u = | u | p - 2 u in R N , u ∈ H 1 (R N) , N ≥ 1 , where M ∈ C ([ 0 , ∞)) and V ∈ C (R N) are given functions. Under very mild assumptions on M, we prove the existence of single-peak or multi-peak solution u ε for above problem, concentrating around topologically stable critical points of V, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. (Arch Ration Mech Anal 213(3):931–979, 2014) [ABSTRACT FROM AUTHOR]

In this paper, we investigate normalized solutions for the Chern–Simons–Schrödinger system with a trapping potential V (x) = ω | x | 2 and a exponential critical growth f(u). The solutions correspond to critical points of the underlying energy functional subject to the L 2 -norm constraint, namely, ∫ R 2 | u | 2 d x = c for c > 0 given. Under some suitable assumptions on f, we show that the system has at least two normalized solutions u c , u ^ c ∈ H 1 (R 2) , depending on the trapping frequency ω and the mass c, where u c is a ground state with positive energy and orbitally stable, while u ^ c is a high-energy solution with positive energy. In addition, the asymptotic behavior of the solution u c as c → 0 is described. [ABSTRACT FROM AUTHOR]

Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur J Appl Math 31(4), 682–708, 2020) that there exist switched server systems with 3 buffers (tanks), a server, filling rates ρ 1 = ρ 2 = ρ 3 = 1 3 and parameters d 1 , d 2 , d 3 > 0 whose ω -limit set is a fractal set. In this article, we give an explicit large subset of parameters for which the corresponding switched server systems have no fractal ω -limit set. More precisely, the Poincaré map of each system has a finite ω -limit set. The approach we use is to study the topological dynamics of a family of piecewise λ -affine contractions that includes the Poincaré maps of the switched server systems as a particular case. [ABSTRACT FROM AUTHOR]

The anomalous Hall effect (AHE), typically observed in ferromagnetic (FM) metals with broken time-reversal symmetry, depends on electronic and magnetic properties. In Co3Sn2-xInxS2, a giant AHE has been attributed to Berry curvature associated with the FM Weyl semimetal phase, yet recent studies report complicated magnetism. We use neutron scattering to determine the spin dynamics and structures as a function of x and provide a microscopic understanding of the AHE and magnetism interplay. Spin gap and stiffness indicate a contribution from Weyl fermions consistent with the AHE. The magnetic structure evolves from c-axis ferromagnetism at x = 0 to a canted antiferromagnetic (AFM) structure with reduced c-axis moment and in-plane AFM order at x = 0.12 and further reduced c-axis FM moment at x = 0.3 . Since noncollinear spins can induce non-zero Berry curvature in real space acting as a fictitious magnetic field, our results revealed another AHE contribution, establishing the impact of magnetism on transport. [ABSTRACT FROM AUTHOR]

The persistent dynamics in systems out of equilibrium, particularly those characterized by annihilation and creation of topological defects, is known to involve complicated spatiotemporal processes and is deemed difficult to control. Here the complex dynamics of defects in active smectic layers exposed to strong confinements is explored, through self-propulsion of active particles and a variety of confining geometries with different topology, ranging from circular, flower-shaped epicycloid, to hypocycloid cavities, channels, and rings. We identify a wealth of dynamical behaviors during the evolution of complex spatiotemporal defect patterns as induced by the confining shape and topology, particularly a perpetual creation-annihilation dynamical state at intermediate activity with large fluctuations of topological defects and a controllable transition from oscillatory to damped time correlation of defect number density via mechanisms governed by boundary cusps. Our results are obtained by using an active phase field crystal approach. Possible experimental realizations are also discussed. It is intriguing but challenging to control the complex dynamics of topological defects that emerge in nonequilibrium spatiotemporal systems. Through an effective modelling this work explores a viable way to controllably vary the properties of persistent defect dynamics in self-propelled active smectics via the competition of activity and boundary confinements with different topology and geometry. [ABSTRACT FROM AUTHOR]

Elucidating collective dynamics in crystalline systems is a common scientific question in multiple fields. In this work, by combination of high-precision numerical approach and analytical normal mode analysis, we systematically investigate the dynamical response of two-dimensional Lennard–Jones crystal as a purely classical mechanical system under random disturbance of varying strength, and reveal rich microscopic dynamics. Specifically, we observe highly symmetric velocity field composed of sharply divided coherent and disordered regions, and identify the order–disorder dynamical transition of the velocity field. Under stronger disturbance, we reveal the vacancy-driven shattering of the crystal. This featured disruption mode is fundamentally different from the dislocation-unbinding scenario in two-dimensional melting. We also examine the healing dynamics associated with vacancies of varying size. The results in this work advance our understanding about the formation of collective dynamics and crystal disruption, and may have implications in elucidating relevant non-equilibrium behaviors in a host of crystalline systems. Microscopic dynamics and underlying topological defects in the disruption of the Lennard–Jones lattice under random disturbance. [ABSTRACT FROM AUTHOR]

Real-world network data consisting of social interactions can be incomplete due to deliberately erased or unsuccessful data collection, which cause the misleading of social interaction analysis for many various time-aware applications. Naturally, the link prediction task has drawn much research interest to predict the missing edges in the incomplete social network. However, existing studies of link prediction cannot effectively capture the entangling topological and temporal dynamics already residing in the social network, thus cannot effectively reasoning the missing interactions in dynamic networks. In this paper, we propose the NEAWalk, a novel model to infer the missing social interaction based on topological-temporal features of patterns in the social group. NEAWalk samples the query-relevant walks containing both the historical and evolving information by focusing on the temporal constraint and designs a dual-view anonymization procedure for extracting both topological and temporal features from the collected walks to conduct the inference. Two-track experiments on several well-known network datasets demonstrate that the NEAWalk stably achieves superior performance against several state-of-the-art baseline methods. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of systems with the Coulomb friction expressed as Lagrange's equations of the second kind with solutions in the sense of Filippov. The friction coefficients are assumed to have dependence on generalized states, velocities, and time which can arise for various reasons such as temperature change, surface roughness, and other characteristics of the rubbing bodies. Combining the direct method of Lyapunov functions with semidefinite derivatives and the method of limit equations stemming from the works of Sell (1967) and Artstein (1977, 1978) on the topological dynamics of nonautonomous systems, we construct the limit differential inclusions by the methods of set-valued analysis and the theory of discontinuous systems. [ABSTRACT FROM AUTHOR]

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees. [ABSTRACT FROM AUTHOR]

In this paper, we study aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. In topological dynamics, it is a well-known fact that every continuous potential on a compact metric space has a maximizing measure. Unfortunately, for non-compact spaces this fact is no longer true. For the geodesic flow we provide a criterion that ensures the existence of a maximizing measure for uniformly continuous potentials. We prove that the only obstruction to the existence of a maximizing measure is the full escape of mass phenomenon. To the best of our knowledge, this is the first general result on the existence of maximizing measures for non-compact topological spaces which does not require the potential to be coercive. We study zero temperature limits of equilibrium measures for a suitable family of potentials. We prove some convergence and divergence results for the limiting behavior of such measures. Among some consequences, we obtain that the geodesic flow has the intermediate entropy property and that equilibrium states are dense in the space of invariant probability measures. [ABSTRACT FROM AUTHOR]

With the emerging interdisciplinary study between transportation and network topological dynamics, exploring vulnerability bottlenecks (which consider knowledge differences between infrastructure disruptions and route service disruptions) of large-scale bus transit networks has become prevailing. This study aims to distinguish and simulate two types of disruptions by considering different concerns of different stakeholders on vulnerability. An integrated vulnerability evaluation indicator is designed to realize the comparability of attacks at various scales (which include single target, route, and area scales) and the diversity of measuring perspectives. Then, a case study is conducted based on the designed attack strategy set, from which the most efficient deliberate attack to trigger network vulnerability is quantitatively explored. The results indicate that the areas of the large-scale bus transit network accessing and interacting with urban rail transit stations are vulnerability bottlenecks, and the destructiveness of a failure edge is driven by the scale of sudden failures it constitutes. This work deepens the understanding of two types of disruptions, improves the comparability of various vulnerability studies, and provides insights for formulating operational management policies. [ABSTRACT FROM AUTHOR]

The chaotic evolution resulting from the interplay between topology and nonlinearity in photonic systems generally forbids the sustainability of optical currents. Here, we systematically explore the nonlinear evolution dynamics in topological photonic lattices within the framework of optical thermodynamics. By considering an archetypical two-dimensional Haldane photonic lattice, we discover several prethermal states beyond the topological phase transition point and a stable global equilibrium response, associated with a specific optical temperature and chemical potential. Along these lines, we provide a consistent thermodynamic methodology for both controlling and maximizing the unidirectional power flow in the topological edge states. This can be achieved by either employing cross-phase interactions between two subsystems or by exploiting self-heating effects in disordered or Floquet topological lattices. Our results indicate that photonic topological systems can in fact support robust photon transport processes even under the extreme complexity introduced by nonlinearity, an important feature for contemporary topological applications in photonics. The nonlinear evolution dynamics in topological photonic lattices is systematically investigated within the framework of optical thermodynamics. This approach allows for the precise prediction of topological currents even under the extreme complexity introduced by nonlinearity. [ABSTRACT FROM AUTHOR]

Vanadium diselenide (VSe2) has recently been highlighted as an efficient 2D electrode owing to its extra-high conductivity, thickness controllability, and van der Waals contact. However, as the electrode, applications of VSe2 to various materials are still lacking. Here, by employing ultrafast time-resolved spectroscopy, we study VSe2-thickness-dependent interfacial effects in heterostructures with topological insulator Bi2Se3 that is severely affected by contact with conventional 3D electrodes. Our results show unaltered Dirac surface state of Bi2Se3 against forming junctions with VSe2, efficient ultrafast hot electron transfer from VSe2 to Bi2Se3 across the interface, shortened metastable carrier lifetimes in Bi2Se3 due to dipole interactions enabling efficient current flow, and the electronic level shift (~tens meV) of bulk states of Bi2Se3 by interfacial interactions, which is ~10 times lower compared to conventional electrodes, implying weak Fermi level pinning. Our observations confirm VSe2 as an ideal electrode for efficient Bi2Se3-based-applications with full utilization of topological insulator characteristics. Topological insulators could be ideal materials for use in electronic devices but complications arising at the interface with metallic electrodes degrades performance values. Here, the authors propose VSe2 as an electrode material investigating the charge dynamics and interface quality using ultrafast transient reflectance measurements and demonstrating the preservation of Dirac surface states of the topological insulator. [ABSTRACT FROM AUTHOR]

Vanadium diselenide (VSe2) has recently been highlighted as an efficient 2D electrode owing to its extra-high conductivity, thickness controllability, and van der Waals contact. However, as the electrode, applications of VSe2 to various materials are still lacking. Here, by employing ultrafast time-resolved spectroscopy, we study VSe2-thickness-dependent interfacial effects in heterostructures with topological insulator Bi2Se3 that is severely affected by contact with conventional 3D electrodes. Our results show unaltered Dirac surface state of Bi2Se3 against forming junctions with VSe2, efficient ultrafast hot electron transfer from VSe2 to Bi2Se3 across the interface, shortened metastable carrier lifetimes in Bi2Se3 due to dipole interactions enabling efficient current flow, and the electronic level shift (~tens meV) of bulk states of Bi2Se3 by interfacial interactions, which is ~10 times lower compared to conventional electrodes, implying weak Fermi level pinning. Our observations confirm VSe2 as an ideal electrode for efficient Bi2Se3-based-applications with full utilization of topological insulator characteristics. Topological insulators could be ideal materials for use in electronic devices but complications arising at the interface with metallic electrodes degrades performance values. Here, the authors propose VSe2 as an electrode material investigating the charge dynamics and interface quality using ultrafast transient reflectance measurements and demonstrating the preservation of Dirac surface states of the topological insulator. [ABSTRACT FROM AUTHOR]

Extending of nanostructures into the third dimension has become a major research avenue in condensed-matter physics, because of geometry- and topology-induced phenomena. In this regard, superconductor 3D nanoarchitectures feature magnetic field inhomogeneity, non-trivial topology of Meissner currents and complex dynamics of topological defects. Here, we investigate theoretically topological transitions in the dynamics of vortices and slips of the phase of the order parameter in open superconductor nanotubes under a modulated transport current. Relying upon the time-dependent Ginzburg–Landau equation, we reveal two distinct voltage regimes when (i) a dominant part of the tube is in either the normal or superconducting state and (ii) a complex interplay between vortices, phase-slip regions and screening currents determines a rich FFT voltage spectrum. Our findings unveil novel dynamical states in superconductor open nanotubes, such as paraxial and azimuthal phase-slip regions, their branching and coexistence with vortices, and allow for control of these states by superimposed dc and ac current stimuli. [ABSTRACT FROM AUTHOR]

We investigate the quantum dynamics of the antiferromagnetic transverse field Ising model on the triangular lattice through large-scale quantum Monte Carlo simulations and stochastic analytic continuation. This model effectively describes a series of triangular rare-earth compounds, for example, TmMgGaO4. At weak transverse field, we capture the excitations related to topological quantum strings, which exhibit continuum features described by XY chain along the strings and those in accord with 'Luttinger string liquid' in the perpendicular direction. The continuum features can be well understood from the perspective of topological strings. Furthermore, we identify the contribution of strings from the excitation spectrum. Our study provides characteristic features for the experimental search for string-related excitations and proposes a theoretical method to pinpoint topological excitations in the experimental spectra. [ABSTRACT FROM AUTHOR]

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations ∣ M and ∣ ~ . The set of its minimal elements proves to be very rich, and the ∣ ~ -hierarchy is used to get a better intuition of this richness. We find the place of the set of ∣ ~ -maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets of N , and compare it to other such notions, important for infinite combinatorics and topological dynamics. [ABSTRACT FROM AUTHOR]

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years. [ABSTRACT FROM AUTHOR]

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics. [ABSTRACT FROM AUTHOR]

Programmable quantum processors are suitable platforms for simulating quantum systems, of which topological phases are of particular interest. We simulate the quench dynamics of a one-dimensional system on IBM Q devices. The topological properties of the dynamics are described by the dynamical topological invariants, the dynamical winding number, and the time-dependent Berry phase, which are simulated with the quantum circuit model. The results show that despite the noise present in the current quantum computers, the dynamical topological invariants are robust. Moreover, to investigate the influence of open quantum system, we analytically solve the master equation in Lindblad form and show that the dynamical winding number and the change in Berry phase are not affected by the dissipation. This study sheds light on the robustness of topological phases on the noisy intermediate-scale quantum computers. [ABSTRACT FROM AUTHOR]

Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. We give a precise description of the set of sequences generated by these graphs. This provides a geometric explanation of many former abstract results in this domain. Our results are illustrated by many examples. [ABSTRACT FROM AUTHOR]

Floquet-Bloch lattices are systems in which wave packets are subjet to periodic modulations both in time and space, showing rich dynamics. While this type of lattice is difficult to implement in solid-state physics, optical systems have provided excellent platforms to probe their physics: among other effects, they have revealed genuine phenomena such as the anomalous Floquet topological insulator and the funnelling of light into localised interface modes. Despite the crucial importance of the band dispersion in the photon dynamics and the topological properties of the lattice, the direct experimental measurement of the Floquet-Bloch bands has remained elusive. Here we report the direct measurement of the Floquet-Bloch bands of a photonic lattice with a single shot method. We use a system of two coupled fibre rings that implements a time-multiplexed Floquet-Bloch lattice. By Fourier transforming the impulse response of the lattice we obtain the band structure together with an accurate characterization of the lattice eigenmodes, i. e. the amplitudes and the phases of the Floquet-Bloch eigenvectors over the entire Brillouin zone. Our results open promising perspectives for the observation of topological effects in the linear and nonlinear regime in Floquet systems. Floquet-Bloch lattices are systems that are relatively difficult to implement in solid-state physics. Here, using a photonic lattice as an alternative, the authors experimentally observe Floquet-Bloch bands using a single-shot method which allows them to fully characterise the lattice eigenmode structure. [ABSTRACT FROM AUTHOR]

Topologically protected surface states present rich physics and promising spintronic, optoelectronic, and photonic applications that require a proper understanding of their ultrafast carrier dynamics. Here, we investigate these dynamics in topological insulators (TIs) of the bismuth and antimony chalcogenide family, where we isolate the response of Dirac fermions at the surface from the response of bulk carriers by combining photoexcitation with below-bandgap terahertz (THz) photons and TI samples with varying Fermi level, including one sample with the Fermi level located within the bandgap. We identify distinctly faster relaxation of charge carriers in the topologically protected Dirac surface states (few hundred femtoseconds), compared to bulk carriers (few picoseconds). In agreement with such fast cooling dynamics, we observe THz harmonic generation without any saturation effects for increasing incident fields, unlike graphene which exhibits strong saturation. This opens up promising avenues for increased THz nonlinear conversion efficiencies, and high-bandwidth optoelectronic and spintronic information and communication applications. [ABSTRACT FROM AUTHOR]

The local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of systems with Coulomb friction represented as Lagrange's equations of the second kind is investigated. Lyapunov's direct method is used in combination with the method of limiting equations, which goes back to the works by G.R. Sell (1967) and Z. Artstein (1977, 1978) on topological dynamics of nonautonomous systems. The results generalize LaSalle's principle of invariance. [ABSTRACT FROM AUTHOR]