Orbital Topology Research Articles

Orbital Hall effect and topology on a two-dimensional triangular lattice: From bulk to edge

Orbital Hall effect and topology on a two-dimensional triangular lattice: From bulk to edge

Bridging the small and large in twisted transition metal dichalcogenide homobilayers: A tight binding model capturing orbital interference and topology across a wide range of twist angles

Many of the important phases observed in twisted transition metal dichalcogenide homobilayers are driven by short-range interactions, which should be captured by a local tight binding description since no Wannier obstruction exists for these systems. Yet, published theoretical descriptions have been mutually inconsistent, with honeycomb lattice tight binding models adopted for some twist angles, triangular lattice models adopted for others, and with tight binding models forsaken in favor of band projected continuum models in many numerical simulations. Here, we derive and study a minimal model containing both honeycomb orbitals and a triangular site that represents the band physics across a wide range of twist angles. The model provides a natural basis to study the interplay of interaction and topology in these heterostructures. It elucidates from generic features of the bilayer the sequence of Chern numbers occurring as twist angle is varied, and the microscopic origin of the magic angle at which flat-band physics occurs. At integer filling, the model successfully captures the Chern ferromagnetic and van Hove-driven antiferromagnetic insulators experimentally observed for small and large angles, respectively, and allows a straightforward calculation of the magnetoelectric properties of the system. Published by the American Physical Society 2024

Floquet engineering of the orbital Hall effect and valleytronics in two-dimensional topological magnets.

We show that circularly polarized light is a versatile way to manipulate both the orbital Hall effect and band topology in two-dimensional ferromagnets. Employing the hexagonal lattice, we proposed that interactions between light and matter allow for the modulation of the valley polarization effect, and then band inversions, accompanied by the band gap closing and reopening processes, can be achieved subsequently at two valleys. Remarkably, the distribution of orbital angular momentum can be controlled by the band inversions, leading to the Floquet engineering of the orbital Hall effect, as well as the topological phase transition from a second-order topological insulator to a Chern insulator with in-plane magnetization, and then to a normal insulator. Furthermore, first-principles calculations validate the feasibility with the 2H-ScI2 monolayer as a candidate material, paving a technological avenue to bridge the orbitronics and nontrivial topology using Floquet engineering.

Visualization of phase-space orbit topological boundary using imaging neutral particle analyzer

A newly-developed Imaging Neutral Particle Analyzer (INPA) in the DIII-D tokamak interrogates phase space occupied by fast ions on multiple different orbit topologies, including passing, stagnation, trapped and potato orbits. Depending on plasma parameters and beam injection geometries, this new INPA system is capable of visualizing distributions of fast ions on the selected orbit topology and its associated orbit topological boundaries. More importantly, the system is able to directly visualize the effective pitch angle scattering ν eff in phase space by measuring fast ions that are scattered across the trapped-passing orbit topological boundaries and from counter-passing orbits to co-passing orbits. It also enables visualization of fast ion confined-loss boundaries and resolves the change of the boundary in phase space, as plasma equilibrium evolves. The key goal of this new INPA system is to directly measure ν eff across phase space induced by drift waves and its interaction with Alfvén eigenmodes, i.e. a key issue towards a future fusion power plant.

Regulating the Spin State Configuration in Bimetallic Phosphorus Trisulfides for Promoting Sulfur Redox Kinetics.

Lithium-sulfur (Li-S) batteries suffer from sluggish kinetics due to the poor conductivity of sulfur cathodes and polysulfide shutting. Current studies on sulfur redox catalysis mainly focus on the adsorption and catalytic conversion of lithium polysulfides but ignore the modulation of the electronic structure of the catalysts which involves spin-related charge transfer and orbital interactions. In this work, bimetallic phosphorus trisulfides embedded in Prussian blue analogue-derived nitrogen-doped hollow carbon nanocubes (FeCoPS3/NCs) were elaborately synthesized as a host to reveal the relationship between the catalytic activity and the spin state configuration for Li-S batteries. Orbital spin splitting in FeCoPS3 drives the electronic structure transition from low-spin to high-spin states, generating more unpaired electrons on the 3d orbit. Specifically, the nondegenerate orbitals involved in the high-spin configuration of FeCoPS3 result in the upshift of energy levels, generating more active electronic states. Such tailored electronic structure increases the charge transfer, influences the d-band center, and further modifies the adsorption energy with lithium polysulfides and the potential reaction pathways. Consequently, the cell with FeCoPS3/NC host exhibits an ultralow capacity decay of 0.037% per cycle over 1000 cycles. This study proposed a general strategy for sculpting geometric configurations to enable spin and orbital topology regulation in Li-S battery catalysts.

Distinct topological configurations of equatorial timelike circular orbit for spherically symmetric (hairy) black holes

Topology is a promising approach toward to the light ring in a generic black hole background, and equatorial timelike circular orbit in a stationary black hole background. In this paper, we consider the distinct topological configurations of the timelike circular orbits in static, spherically symmetric, and asymptotic flat black holes. By making use of the equation of motion of the massive particles, we construct a vector with its zero points exactly relating with the timelike circular orbits. Since each zero point of the vector can be endowed with a winding number, the topology of the timelike circular orbits is well established. Stable and unstable timelike circular orbits respectively have winding number +1 and -1. In particular, for given angular momentum, the topological number of the timelike circular orbits also vanishes whether they are rotating or not. Moreover, we apply the study to the Schwarzschild, scalarized Einstein-Maxwell, and dyonic black holes, which have three distinct topological configurations, representations of the radius and angular momentum relationship, with one or two pairs timelike circular orbits at most. It is shown that although the existence of scalar hair and quasi-topological term leads to richer topological configurations of the timelike circular orbits, they have no influence on the total topological number. These results indicate that the topological approach indeed provides us a novel way to understand the timelike circular orbits. Significantly, different topological configurations can share the same topology number, and hence belong to the same topological class. More information is expected to be disclosed when other different topological configurations are present.

Perspective on spin–orbit torque, topology, and reciprocal and real-space spin textures in magnetic materials and heterostructures

In this Perspective, we present some important aspects of two fundamental concepts of modern spintronics, namely, spin–orbit torque and topology. Although these two fields emerged separately in condensed matter physics, in spintronics they show a deep connection, which requires further theoretical and experimental investigation. The topological features can arise both from momentum space via the wave functions as well as from real space via complex magnetic configurations. These features manifest themselves as unique aspects of different equilibrium and non-equilibrium properties. Physical interactions of such a topological origin can open new possibilities for more efficient mechanisms for manipulating magnetic order with electrical currents, which, in turn, can lead to faster and more efficient spintronics devices.

Structure and stability investigation of oxygen interaction with Fe in bcc-Fe

The precise interaction mechanisms between body-centered cubic Fe and the unavoidable interstitial O atom are critical to understanding oxidation resistance. Even though the Fe–O interaction has been investigated extensively for centuries, the physical evolution processing including the stability, bonding characteristics, and magnetism, remains as one of the most important scientific issues. Here, a four-level (atomic, electronic, orbital, and spin-level) analysis was employed to investigate the precise interactions between Fe and O atoms by multiple complementarity methods (Madelung energy calculation, electronic density analysis, crystal orbital Hamilton population analysis and topology analysis of spin maps). The stability of Fe atoms was enhanced with the O atom at the tetrahedral interstitial site. The bonding strength between Fe and O atoms originates substantially from s orbits. Furthermore, the topology of the minority-spin was changed by the magnetic moment flip of O, and the interaction among Fe atoms were strengthened. A further understanding of the Fe–O mechanism provides a theoretical basis for obtaining excellent physical properties in Fe-based materials.

Topology of equatorial timelike circular orbits around stationary black holes

A topological approach has been successfully used to study the properties of the light ring and the null circular orbit, in a generic black hole background. However, for the equatorial timelike circular orbit, quite different from the light ring case, its radius is closely dependent of the energy and angular momentum of a test particle. This fact seems to restrict the extension of the topological treatment to the timelike circular orbit. In this paper, we confirm that the angular momentum does not affect the asymptotic behavior of the constructed vector with its zero points denoting the equatorial timelike circular orbits. As a result, a well-behaved topology to characterize the equatorial timelike circular orbits can be constructed. Our study shows that the total topological number of the timelike circular orbits vanishes for a generic black hole, which is dependent of the energy of the particle. Significantly, it reveals that if there the timelike circular orbits exist, they always come in pairs for fixed angular momentum. Meanwhile, the stable and unstable timelike circular orbits have positive or negative winding number. Of particular interest is that the marginally stable circular orbit corresponds to the bifurcation point of the zero point of the constructed vector. Moreover, we also examine the case when the particle energy acts as the control parameter. It is shown that there will be topological phase transition when the value of the particle energy is one. Below this value, the timelike circular orbits always come in pairs for fixed energy. Otherwise, we will have one more unstable timelike circular orbit. We further apply the treatment to the Kerr black hole. All the results given in a generic black hole background are exactly reproduced. These strongly indicate that our topological approach can be generalized to the equatorial timelike circular orbits.

Resonant transport of fusion alpha particles in quasisymmetric stellarators

In modern, highly optimized stellarator configurations where prompt fusion alpha particle losses from the plasma core are absent, alpha particles can still be lost due to stochastic orbits which result in delayed losses. One mechanism leading to stochastic orbits are changes in the particle trapping class during drift motion along the contours of the parallel adiabatic invariant J ∥ leading to jumps in J ∥ when crossing class boundaries. Another mechanism, which is of main interest here, is the resonance between particle drift and bounce motion (drift-orbit resonance). The first mechanism affects mainly trapped particles near the trapped-passing boundary in the phase space of quasi-symmetric and quasi-isodynamic devices, and can be minimized there by aligning local magnetic field maxima on a given flux surface. The second mechanism may affect a broader range in the trapped particle domain where contours of J ∥ still remain closed. Drift-orbit resonances modify the topology of orbits leading to island-like structures on Poincaré plots where these islands may overlap thus leading to the stochastic transport. In this report, we study this stochastization mechanism in quasi-symmetric stellarator configurations with help of the Hamiltonian drift-kinetic code NEO-RT as well as orbit classification and direct computation of fusion alpha losses within the symplectic orbit following code SIMPLE. The width and overlap of resonances in phase-space is studied using Hamiltonian perturbation theory. Based on optimized reactor configurations we assess if this approach can be used as a fast metric for fusion alpha losses in stellarator optimization.

Numerical studies on saturated kink and sawtooth induced fast ion transport in JET ITER-like plasmas

This presentation examines the energetic particle transport induced by saturated kink modes and sawtooth crashes in JET deuterium plasmas. It is known that kink mode-resonant transport and phase-space redistribution from sawtooth crashes can drive strong fast ion transport with dependencies on particle pitch and energy. Measurements with JET’s Faraday cup fast ion loss detector array have shown that the internal kink growth phase preceding sawtooth crashes produces substantial fast ion losses. This report will numerically investigate the dominant energetic particle transport mechanism with a detailed examination of the fast ion phase-space dependencies, resonances, orbit topology changes, induced losses, and redistribution associated with the long-lived, resonant, kink mode and non-resonant sawtooth crash. The ORBIT-kick model forms the basis of the transport studies with realistic fast ion distributions produced from TRANSP. A recently created reduced model for sawtooth induced transport is used while the saturated kink modes are modeled with ideal magnetohydrodynamic codes. The simulations were further validated against experiment with a newly developed synthetic Faraday cup fast ion loss detector in addition to scintillator probe and neutron measurements.

Energy-selective confinement of fusion-born alpha particles during internal relaxations in a tokamak plasma

Long-pulse operation of a self-sustained fusion reactor using toroidal magnetic containment requires control over the content of alpha particles produced by D-T fusion reactions. On the one hand, MeV-class alpha particles must stay confined to heat the plasma. On the other hand, decelerated helium ash must be expelled before diluting the fusion fuel. Here, we report results of kinetic-magnetohydrodynamic hybrid simulations of a large tokamak plasma that confirm the existence of a parameter window where such energy-selective confinement can be accomplished by exploiting internal relaxation events known as sawtooth crashes. The physical picture — a synergy between magnetic geometry, optimal crash duration and rapid particle motion — is completed by clarifying the role of magnetic drifts. Besides causing asymmetry between co- and counter-going particle populations, magnetic drifts determine the size of the confinement window by dictating where and how much reconnection occurs in particle orbit topology.

Fast-ion transport and toroidal rotation response to externally applied magnetic perturbations at the ASDEX Upgrade tokamak

This paper studies the effect of 3D magnetic perturbations (MPs) on fast-ion confinement, and its impact on the toroidal rotation velocity profile. Two low collisionality H-mode experiments carried out at the ASDEX Upgrade tokamak have been analysed. The two discharges feature different magnetic field helicity (q 95), and differences in the velocity-space and level of fast-ion losses are observed. A new analysis technique has been developed that sheds light on the dependencies between fast-ion losses and toroidal rotation, providing for the first time correlation patterns resolved in radius and velocity space of the lost fast-ions. The correlation intensifies towards the plasma edge and is strongly dependent on the orbit topology of the lost fast-ions. The ASCOT orbit following code has been used to characterize the fast-ion resonant transport and beam driven torques, using the vacuum approach and including plasma response (PR). The change of the toroidal canonical momentum, which serves as figure of merit for resonant fast-ion transport, has been calculated with ASCOT. The beam geometry and q 95 are found to have a strong impact on the fast-ion transport and losses. The fast-ion transport induced by the MPs affects the beam driven torques. The effect of the changes of the j × B and collisional torques on plasma rotation is analysed using the torques simulated by ASCOT and simple momentum balance calculations. For the low q 95 = 3.8 discharge, which benefits from a resonant amplification, we find excellent agreement with the measured variation of the toroidal velocity. For the high q 95 = 5.5 discharge, the inclusion of the PR improves the comparison with experimental data with respect to the vacuum estimation, but still some differences with experiments are observed. This suggests that other non-resonant effects could play a role for the determination of the toroidal rotation profile.

Orbital embedding and topology of one-dimensional two-band insulators

The topological invariants of band insulators are usually assumed to depend only on the connectivity between orbitals and not on their intracell position (orbital embedding), which is a separate piece of information in the tight-binding description. For example, in two dimensions, the orbital embedding is known to change the Berry curvature but not the Chern number. Here, we consider one-dimensional inversion-symmetric insulators classified by a ${\mathbb{Z}}_{2}$ topological invariant $\ensuremath{\vartheta}=0$ or $\ensuremath{\pi}$, related to the Zak phase, and show that $\ensuremath{\vartheta}$ crucially depends on orbital embedding. We study three two-band models with bond, site, or mixed inversion: the Su-Schrieffer-Heeger model (SSH), the charge density wave model (CDW), and the Shockley model. The SSH (resp. CDW) model is found to have a unique phase with $\ensuremath{\vartheta}=0$ (resp. $\ensuremath{\pi}$). However, the Shockley model features a topological phase transition between $\ensuremath{\vartheta}=0$ and $\ensuremath{\pi}$. The key difference is whether the two orbitals per unit cell are at the same or different positions.

Quantification of the Helicality of Helical Molecular Orbitals.

The frontier molecular orbital (MO) topology of linear carbon molecules, such as polyynes, can be visually identified as helices. However, there is no clear way to quantify the helical curvature of these π-MOs, and it is thus challenging to quantify correlations between the helical curvature and molecular properties. In this paper, we develop a method that enables us to compute the helical curvature of MOs based on their nodal planes. Using this method, we define a robust way of quantifying the helical nature of MOs (helicality) by their deviation from a perfect helix. We explore several limiting cases, including polyynes, metallacumulenes, cyclic allenes, and spiroconjugated systems, where the change in helical curvature is subtle yet clearly highlighted with this method. For example, we show that strain only has a minor effect on the helicality of the frontier orbitals of cycloallenes and that the MOs of spiroconjugated systems are close to perfect helices around the spiro-carbon. Our work provides a well-defined method for assessing orbital helicality beyond visual inspection of MO isosurfaces, thus paving the way for future studies of how the helicality of π-MOs affects molecular properties.

On the topology of geometric and rational orbits for algebraic group actions over valued fields, II

The aim of this paper is twofold. First, we show that if G is a smooth nilpotent group acting on an algebraic variety V defined over an admissible valued field k and then the Zariski closedness of the geometric orbit in is equivalent to the Hausdorff closedness of the rational orbit in V(k). Second, we provide some calculations for the fact that there is a bijection between the set of G(k)-orbits and the kernel of the natural map in flat cohomology. These results are obtained in the framework of studying the rational orbits.

Linear Dynamical Systems of Nilpotent Lie Groups

We study the topology of orbits of dynamical systems defined by finite-dimensional representations of nilpotent Lie groups. Thus, the following dichotomy is established: either the interior of the set of regular points is dense in the representation space, or the complement of the set of regular points is dense, and then the interior of that complement is either empty or dense in the representation space. The regular points are by definition the points whose orbits are locally compact in their relative topology. We thus generalize some results from the recent literature on linear actions of abelian Lie groups. As an application, we determine the generalized \(ax+b\)-groups whose \(C^*\)-algebras are antiliminary, that is, no closed 2-sided ideal is type I.

Orbit topology analyzed from π phase shift of magnetic quantum oscillations in three-dimensional Dirac semimetal

With the emergence of Dirac fermion physics in the field of condensed matter, magnetic quantum oscillations (MQOs) have been used to discern the topology of orbits in Dirac materials. However, many previous researchers have relied on the single-orbit Lifshitz-Kosevich (LK) formula, which overlooks the significant effect of degenerate orbits on MQOs. Since the single-orbit LK formula is valid for massless Dirac semimetals with small cyclotron masses, it is imperative to generalize the method applicable to a wide range of Dirac semimetals, whether massless or massive. This report demonstrates how spin-degenerate orbits affect the phases in MQOs of three-dimensional massive Dirac semimetal, NbSb2 With varying the direction of the magnetic field, an abrupt π phase shift is observed due to the interference between the spin-degenerate orbits. We investigate the effect of cyclotron mass on the π phase shift and verify its close relation to the phase from the Zeeman coupling. We find that the π phase shift occurs when the cyclotron mass is half of the electron mass, indicating the effective spin gyromagnetic ratio as g s = 2. Our approach is not only useful for analyzing MQOs of massless Dirac semimetals with a small cyclotron mass but also can be used for MQOs in massive Dirac materials with degenerate orbits, especially in topological materials with a sufficiently large cyclotron mass. Furthermore, this method provides a useful way to estimate the precise g s value of the material.

Unfolding spatiotemporal dynamics through symmetry reduction based on orbit topology.

Understanding key patterns in a spatially extended system is an essential task of modern physics of complex systems. Just like in low-dimensional nonlinear systems, here we show that orbit topology plays a critical role even for the investigation of spatiotemporal dynamics. First, we design a new scheme to reduce possible continuous symmetries that are prevailing in these systems based on topological consideration. The scheme is successfully demonstrated in the well-known pattern formation systems. Interesting bifurcation routes to chaos are conveniently revealed after symmetry reduction. In particular, we find that near the onset of turbulent dynamics, with an increase of instability, local phase chaos with the same spatial topological index may merge into more complex ones, while those with different indices induce defect chaos necessarily through connections docked with defects. The topological argument is so strong that the scenario presented here should be omnipresent in diverse systems.

Topological bifurcations and reconstruction of travelling waves

This paper is devoted to periodic traveling waves solving Lie–Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which need not be quadratic), provided the wave belongs to the coadjoint orbit of a uniform profile. Equivalently, the corresponding “fluid particle motion” is integrable. Applying this result to the Camassa–Holm equation, we express the drift of particles in terms of parameters labeling periodic peakons and exhibit orbital bifurcations: points in parameter space where the drift velocity varies discontinuously, reflecting a sudden change in the topology of Virasoro orbits.