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The emergence of spinon quasiparticles, which carry spin but lack charge, is a hallmark of collective quantum phenomena in low-dimensional quantum spin systems. While the existence of spinons has been demonstrated through scattering spectroscopy in ensemble samples, real-space imaging of these quasiparticles within individual spin chains has remained elusive. In this study, we construct individual Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chains using open-shell [2]triangulene molecules as building blocks. Each [2]triangulene unit, owing to its sublattice imbalance, hosts a net spin-$\frac{1}{2}$ in accordance with Lieb's theorem, and these spins are antiferromagnetically coupled within covalent chains with a coupling strength of $J = 45$ meV. Through scanning tunneling microscopy and spectroscopy, we probe the spin states, excitation gaps, and their spatial excitation weights within covalent spin chains of varying lengths with atomic precision. Our investigation reveals that the excitation gap decreases as the chain length increases, extrapolating to zero for long chains, consistent with Haldane's gapless prediction. Moreover, inelastic tunneling spectroscopy reveals an m-shaped energy dispersion characteristic of confined spinon quasiparticles in a one-dimensional quantum box. These findings establish a promising strategy for exploring the unique properties of excitation quasiparticles and their broad implications for quantum information.

Two-dimensional (2D) moir\'e materials have emerged as a highly tunable platform for investigating novel quantum states of matter arising from strong electronic correlations and nontrivial band topology. Recently, topological flat bands formed in 2D semiconducting moir\'e superlattices have attracted great interests. In particular, a series of topological quantum phases, including the long-sought fractional quantum anomalous Hall (FQAH) effect, have recently been experimentally observed in twisted bilayer MoTe2 (tMoTe2). However, the microscopic information of tMoTe2 moir\'e superlattice and its electronic structure is still lacking. Here, we present scanning tunneling microscopy and spectroscopy (STM/STS) studies of the tMoTe2 moir\'e superlattice, with twist angles ranging from about 2.3{\deg} to 2.8{\deg}. We developed a contact-STM mode to apply pressure on tMoTe2 and observed a phase transition from band insulator to metal of tMoTe2 under pressure at the charge neutrality point. STM imaging reveals a pronounced in-plane lattice reconstruction with periodic strain redistribution in the tMoTe2, which serves as gauge fields for generating topological moir\'e bands. Importantly, the electronic states of the low-energy moir\'e flat bands primarily concentrate at the XM and MX regions as revealed by STS imaging. Such spatial distributions are nicely reproduced by our first principal calculations with a large-scale basis, suggesting the low-energy moir\'e flat bands are formed through the hybridization of K valley bands of the top layer and K' valley bands of the bottom layer. Overall, our findings provide compelling real-space evidence of electronic structure under pressure and topological flat bands of tMoTe2, paving the way for further STM/STS investigations of correlated topological states within the topological flat band in gate-tunable tMoTe2 devices.

A two-dimensional quantum spin Hall insulator exhibits one-dimensional gapless spin-filtered edge channels allowing for dissipationless transport of charge and spin. However, the sophisticated fabrication requirement of two-dimensional materials and the low capacity of one-dimensional channels hinder the broadening applications. We introduce a method to manipulate a three-dimensional topological material to host a large number of one-dimensional topological edge channels utilizing surface anisotropy. Taking ZrTe5 as a model system, we realize a highly anisotropic surface due to the synergistic effect of the lattice geometry and Coulomb interaction, and achieve massive one-dimensional topological edge channels -- confirmed by electronic characterization using angle-resolved photoemission spectroscopy, in combination with first-principles calculations. Our work provides a new avenue to engineer the topological properties of three-dimensional materials through nanoscale tunning of surface morphology and opens up a promising prospect for the development of low-power-consumption electronic nano devices based on one-dimensional topological edge channels.

The field of two-dimensional (2D) ferromagnetism has been proliferating over the past few years, with ongoing interests in basic science and potential applications in spintronic technology. However, a high-resolution spectroscopic study of the 2D ferromagnet is still lacking due to the small size and air sensitivity of the exfoliated nanoflakes. Here, we report a thickness-dependent ferromagnetism in epitaxially grown Cr2Te3 thin films and investigate the evolution of the underlying electronic structure by synergistic angle-resolved photoemission spectroscopy, scanning tunneling microscopy, x-ray absorption spectroscopy, and first-principle calculations. A conspicuous ferromagnetic transition from Stoner to Heisenberg-type is directly observed in the atomically thin limit, indicating that dimensionality is a powerful tuning knob to manipulate the novel properties of 2D magnetism. Monolayer Cr2Te3 retains robust ferromagnetism, but with a suppressed Curie temperature, due to the drastic drop in the density of states near the Fermi level. Our results establish atomically thin Cr2Te3 as an excellent platform to explore the dual nature of localized and itinerant ferromagnetism in 2D magnets., Comment: 32 pages, 4 + 10 figures

Topological materials hosting metallic edges characterized by integer quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. However, only integer-dimensional models have been classified, and the interplay of topology and fractals, which may have a non-integer dimension, remained largely unexplored. Quantum fractals have recently been engineered in metamaterials, but up to present no topological states were unveiled in fractals realized in real materials. Here, we show theoretically and experimentally that topological edge and corner modes arise in fractals formed upon depositing thin layers of bismuth on an indium antimonide substrate. Scanning tunneling microscopy reveals the appearance of (nearly) zero-energy modes at the corners of Sierpi\'nski triangles, as well as the formation of outer and inner edge modes at higher energies. Unexpectedly, a robust and sharp depleted mode appears at the outer and inner edges of the samples at negative bias voltages. The experimental findings are corroborated by theoretical calculations in the framework of a continuum muffin-tin and a lattice tight-binding model. The stability of the topological features to the introduction of a Rashba spin-orbit coupling and disorder is discussed. This work opens the perspective to novel electronics in real materials at non-integer dimensions with robust and protected topological states., Comment: Main manuscript 14 pages, supplementary material 34 pages

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction-diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy-Schwarz inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method., Comment: 11 pages, 3 figures(including 8 sub-figures)

This paper studies the robustness of a PDE backstepping delay-compensated boundary controller for a reaction-diffusion partial differential equation (PDE) with respect to a nominal delay subject to stochastic error disturbance. The stabilization problem under consideration involves random perturbations modeled by a finite-state Markov process that further obstruct the actuation path at the controlled boundary of the infinite-dimension plant. This scenario is useful to describe several actuation failure modes in process control. Employing the recently introduced infinite-dimensional representation of the state of an actuator subject to stochastic input delay for ODEs (Ordinary Differential Equations), we convert the stochastic input delay into $r+1$ unidirectional advection PDEs, where $r$ corresponds to the number of jump states. Our stability analysis assumes full-state measurement of the spatially distributed plant's state and relies on a hyperbolic-parabolic PDE cascade representation of the plant plus actuator dynamics. Integrating the plant and the nominal stabilizing boundary control action, all while considering probabilistic delay disturbances, we establish the proof of mean-square exponential stability as well as the well-posedness of the closed-loop system when random phenomena weaken the nominal actuator compensating effect. Our proof is based on the Lyapunov method, the theory of infinitesimal operator for stability, and $C_0$-semigroup theory for well-posedness. Our stability result refers to the $L^2$-norm of the plant state and the $H^2$-norm of the actuator state..., Comment: 16.5 pages, 6 figures

Breaking time reversal symmetry in a topological insulator may lead to quantum anomalous Hall effect and axion insulator phase. MnBi4Te7 is a recently discovered antiferromagnetic topological insulator with TN ~12.5 K, which is constituted of alternatively stacked magnetic layer (MnBi2Te4) and non-magnetic layer (Bi2Te3). By means of scanning tunneling spectroscopy, we clearly observe the electronic state present at a step edge of a magnetic MnBi2Te4 layer but absent at non-magnetic Bi2Te3 layers at 4.5 K. Furthermore, we find that as the temperature rises above TN, the edge state vanishes, while the point defect induced state persists upon temperature increasing. These results confirm the observation of magnetism induced edge states. Our analysis based on an axion insulator theory reveals that the nontrivial topological nature of the observed edge state.

Topological defects in graphene materials introduce exotic properties which are absent in their defect-free counterparts with both fundamental importance and technological implications. Although individual topological defects have been widely studied, collective magnetic behaviors originating from well-organized multiple topological defects remain a great challenge. Here, we studied the collective magnetic properties originating from three pentagon topological defects in truxene-based molecular graphenoids by using scanning tunneling microscopy and non-contact atomic force microscopy. Unpaired $\pi$ electrons are introduced into the aromatic topology of truxene molecular graphenoids one by one by dissociating hydrogen atoms at the pentagon defects via atom manipulation. Scanning tunneling spectroscopy measurements together with density functional theory calculations suggest that the unpaired electrons are ferromagnetically coupled, forming a collective high-spin quartet state of S=3/2. Our work demonstrates that the collective spin ordering can be realized through engineering regular patterned topological defects in molecular graphenoids, providing a new platform for designer one-dimensional ferromagnetic spin chains and two-dimensional ferromagnetic networks., Comment: 25 pages, 12 figures, include Supplementary Information

To realize the braiding operations of Majorana zero mode in the vortex cores of a topological superconductor (TSC), a novel approach is proposed in this letter to replace the common tip (or tip-like) method. Instead of on top of the TSC thin film, arrays of electrically controllable pining centers are built beneath the film, hence detecting can proceed along with braiding. It does not only increase the braiding rate, but also enables the braiding to be performed in an electrically controllable way and to be integrated into large scale. Our work paves the way towards large-scale topological quantum computation., Comment: 5 pages, 2 figures

The magnetic properties of carbon materials are at present the focus of an intense research effort in physics, chemistry and materials science due to their potential applications in spintronics and quantum computations. Although the presence of spins in open-shell nanographenes has been recently confirmed, the ability to control magnetic coupling sign has remained elusive, but the most desirable. Here, we demonstrate an effective approach of engineering magnetic ground states in atomically precise open-shell bipartite/nonbipartite nanographenes using combined scanning probe techniques and mean-field Hubbard model calculations. The magnetic coupling sign between two spins has been controlled via breaking bipartite lattice symmetry of nanographenes. In addition, the exchange-interaction strength between two spins has been widely tuned by finely tailoring their spin density overlap, realizing a large exchange-interaction strength of 42 meV. Our demonstrated method provides ample opportunities for designer above-room-temperature magnetic phases and functionalities in graphene nanomaterials., Comment: 18 pages, 4 figures

Since the early days of Bardeen-Cooper-Schrieffer theory, it has been predicted that a sufficiently large supercurrent can close the energy gap in a superconductor and creates gapless Bogoliubov quasiparticles through the Doppler shift of quasiparticle energy due to the Cooper pair momentum. In this gapless superconducting state, zero-energy quasiparticles reside on a segment of the normal state Fermi surface, while its remaining part is still gapped. The finite density of states of field-induced quasiparticles, known as the Volovik effect, has been observed in tunneling and specific heat measurements on d- and s-wave superconductors. However, the segmented Fermi surface of a finite-momentum state carrying a supercurrent has never been detected directly. Here we use quasiparticle interference (QPI) technique to image field-controlled Fermi surface of Bi$_2$Te$_3$ thin films proximitized by the superconductor NbSe$_2$. By applying a small in-plane magnetic field, a screening supercurrent is induced which leads to finite-momentum pairing on topological surface states of Bi$_2$Te$_3$. Our measurements and analysis reveal the strong impact of finite Cooper pair momentum on the quasiparticle spectrum, and thus pave the way for STM study of pair density wave and FFLO states in unconventional superconductors., Comment: 25 pages, 14 figures

A quantum spin hall insulator(QSHI) is manifested by its conducting edge channels that originate from the nontrivial topology of the insulating bulk states. Monolayer 1T'-WTe2 exhibits this quantized edge conductance in transport measurements, but because of its semimetallic nature, the coherence length is restricted to around 100 nm. To overcome this restriction, we propose a strain engineering technique to tune the electronic structure, where either a compressive strain along a axis or a tensile strain along b axis can drive 1T'-WTe2 into an full gap insulating phase. A combined study of molecular beam epitaxy and in-situ scanning tunneling microscopy/spectroscopy then confirmed such a phase transition. Meanwhile, the topological edge states were found to be very robust in the presence of strain., Comment: 15 pages, 5 figure and 21 pages of supplementary material

Stanene was proposed to be a quantum spin hall insulator containing topological edges states and a time reversal invariant topological superconductor hosting helical Majorana edge mode. Recently, experimental evidences of existence of topological edge states have been found in monolayer stanene films and superconductivity has been observed in few-layer stanene films excluding single layer. An integrated system with both topological edge states and superconductivity are higly pursued as a possible platform to realize topological superconductivity. Few-layer stanene show great potential to meet this requirement and is highly desired in experiment. Here we successfully grow few-layer stanene on bismuth (111) substrate. Both topological edge states and superconducting gaps are observed by in-situ scanning tunneling microscopy/spectroscopy (STM/STS). Our results take a further step towards topological superconductivity by stanene films., Comment: 14 pages,4 figures

Majorana fermions have been intensively studied in recent years for their importance to both fundamental science and potential applications in topological quantum computing1,2. Majorana fermions are predicted to exist in a vortex core of superconducting topological insulators3. However, they are extremely difficult to be distinguished experimentally from other quasiparticle states for the tiny energy difference between Majorana fermions and these states, which is beyond the energy resolution of most available techniques. Here, we overcome the problem by systematically investigating the spatial profile of the Majorana mode and the bound quasiparticle states within a vortex in Bi2Te3/NbSe2. While the zero bias peak in local conductance splits right off the vortex center in conventional superconductors, it splits off at a finite distance ~20nm away from the vortex center in Bi2Te3/NbSe2, primarily due to the Majorana fermion zero mode. While the Majorana mode is destroyed by reducing the distance between vortices, the zero bias peak splits as a conventional superconductor again. This work provides strong evidences of Majorana fermions and also suggests a possible route to manipulating them., Comment: 16 pages, 4 figures

Quasi free-standing monolayer graphene can be produced by intercalating species like oxygen or hydrogen between epitaxial graphene and the substrate crystal. If the graphene is indeed decoupled from the substrate, one would expect the observation of a similar electronic dispersion and many-body effects, irrespective of the substrate and the material used to achieve the decoupling. Here we investigate the electron-phonon coupling in two different types of quasi free-standing monolayer graphene: decoupled from SiC via hydrogen intercalation and decoupled from Ir via oxygen intercalation. Both systems show a similar overall behaviour of the self-energy and a weak renormalization of the bands near the Fermi energy. The electron-phonon coupling is found to be sufficiently weak to make the precise determination of the coupling constant lambda through renormalization difficult. The estimated value of lambda is 0.05(3) for both systems., Comment: 5 pages, 3 figures

Angle-resolved photoelectron spectroscopy is used for a detailed study of the electronic structure of the topological insulator Bi2Se3. Nominally stoichiometric and calcium-doped samples were investigated. The pristine surface shows the topological surface state in the bulk band gap. As time passes, the Dirac point moves to higher binding energies, indicating an increasingly strong downward bending of the bands near the surface. This time-dependent band bending is related to a contamination of the surface and can be accelerated by intentionally exposing the surface to carbon monoxide and other species. For a sufficiently strong band bending, additional states appear at the Fermi level. These are interpreted as quantised conduction band states. For large band bendings, these states are found to undergo a strong Rashba splitting. The formation of quantum well states is also observed for the valence band states. Different interpretations of similar data are also discussed., Comment: 30 pages, 10 figures

One of the salient features of graphene is the very high carrier mobility that implies tremendous potential for use in electronic devices. Unfortunately, transport measurements find the expected high mobility only in freely suspended graphen. When supported on a surface, graphene shows a strongly reduced mobility, and an especially severe reduction for temperatures above 200 K. A temperature-dependent mobility reduction could be explained by scattering of carriers with phonons, but this is expected to be weak for pristine, weakly-doped graphene. The mobility reduction has therefore been ascribed to the interaction with confined ripples or substrate phonons. Here we study the temperature-dependent electronic structure of supported graphene by angle-resolved photoemission spectroscopy, a technique that can reveal the origin of the phenomena observed in transport measurements. We show that the electron-phonon coupling for weakly-doped, supported graphene on a metal surface is indeed extremely weak, reaching the lowest value ever reported for any material. However, the temperature-dependent dynamic interaction with the substrate leads to a complex and dramatic change in the carrier type and density that is relevant for transport. Using ab initio molecular dynamics simulations, we show that these changes in the electronic structure are mainly caused by fluctuations in the graphene-substrate distance., Comment: 14 pages, 2 figures

The electronic structure of Sb(110) is studied by angle-resolved photoemission spectroscopy and first-principle calculations, revealing several electronic surface states in the projected bulk band gaps around the Fermi energy. The dispersion of the states can be interpreted in terms of a strong spin-orbit splitting. The bulk band structure of Sb has the characteristics of a strong topological insulator with a Z2 invariant nu0=1. This puts constraints on the existence of metallic surface states and the expected topology of the surface Fermi contour. However, bulk Sb is a semimetal, not an insulator and these constraints are therefore partly relaxed. This relation of bulk topology and expected surface state dispersion for semimetals is discussed., Comment: 7 pages, 4 figures

The electron dynamics of the topological surface state on Bi2Se3(111) is investigated by temperature-dependent angle-resolved photoemission. The electron-phonon coupling strength is determined in a spectral region for which only intraband scattering involving the topological surface band is possible. The electron-phonon coupling constant is found to be lambda=0.25(5), more than an order of magnitude higher than the corresponding value for intraband scattering in the noble metal surface states. The stability of the topological state with respect to surface irregularities was also tested by introducing a small concentration of surface defects via ion bombardment. It is found that, in contrast to the bulk states, the topological state can no longer be observed in the photoemission spectra and this cannot merely be attributed to surface defect-induced momentum broadening., Comment: 5 pages, 3 figures

Topological insulators are a recently discovered class of materials with fascinating properties: While the inside of the solid is insulating, fundamental symmetry considerations require the surfaces to be metallic. The metallic surface states show an unconventional spin texture, electron dynamics and stability. Recently, surfaces with only a single Dirac cone dispersion have received particular attention. These are predicted to play host to a number of novel physical phenomena such as Majorana fermions, magnetic monopoles and unconventional superconductivity. Such effects will mostly occur when the topological surface state lies in close proximity to a magnetic or electric field, a (superconducting) metal, or if the material is in a confined geometry. Here we show that a band bending near to the surface of the topological insulator Bi$_2$Se$_3$ gives rise to the formation of a two-dimensional electron gas (2DEG). The 2DEG, renowned from semiconductor surfaces and interfaces where it forms the basis of the integer and fractional quantum Hall effects, two-dimensional superconductivity, and a plethora of practical applications, coexists with the topological surface state in Bi$_2$Se$_3$. This leads to the unique situation where a topological and a non-topological, easily tunable and potentially superconducting, metallic state are confined to the same region of space., Comment: 12 pages, 3 figures