Topological Character Research Articles

Some topological aspects of fluid dynamics

Abstract

Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids

Spin-orbital liquids are quantum disordered states in systems with entangled spin and orbital degrees of freedom. We study exactly solvable spin-orbital models in two dimensions with selected Heisenberg-, Kitaev-, and $\mathrm{\ensuremath{\Gamma}}$-type interactions, as well as external magnetic fields. These models realize a variety of spin-orbital-liquid phases featuring dispersing Majorana fermions with Fermi surfaces, nodal Dirac or quadratic band touching points, or full gaps. In particular, we show that Zeeman magnetic fields can stabilize nontrivial flux patterns and induce metamagnetic transitions between states with different topological character. Solvable nearest-neighbor biquadratic spin-orbital perturbations can be tuned to stabilize zero-energy flat bands. We discuss in detail the examples of $\mathrm{SO}(2)$- and $\mathrm{SO}(3)$-symmetric spin-orbital models on the square and honeycomb lattices, and use group-theoretical arguments to generalize to $\mathrm{SO}(\ensuremath{\nu})$-symmetric models with arbitrary integer $\ensuremath{\nu}>1$. These results extend the list of exactly solvable models with spin-orbital-liquid ground states and highlight the intriguing general features of such exotic phases. Our models are thus excellent starting points for more realistic modelling of candidate materials.

⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills

A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.

Critical thickness for the emergence of Weyl features in Co3Sn2S2 thin films

Magnetic Weyl semimetals are quantum phases of matter arising from the interplay of linearly dispersive bands, spin-orbit coupling, and time reversal symmetry breaking. This can be realised, for example, in Co3Sn2S2, based on a cobalt kagome lattice and characterised by intriguing phenomena such as large anomalous Hall effect, Nernst effect, and water oxidation. Here, we attempt to determine the robustness of the twofold necessary conditions for the emergence of the magnetic Weyl semimetal phase in Co3Sn2S2 ultrathin films. Except for two-dimensional layered materials, a reduction of thickness generally makes it difficult to develop topological character and ferromagnetic long-range order. In Co3Sn2S2 films, while ferromagnetic ordering appears robustly even in average thicknesses of one or two unit cells with island-like polycrystalline domains, the anomalous Hall conductivity appears only above a critical thickness of approximately 10 nm. The emergence of surface conduction and large anomalous Hall effect implies the distinct contribution of Weyl nodes and their Berry curvature. These findings reveal an exotic feature of Weyl physics in thin-film based superstructures as well as a potential for future applications in electronic devices.

Landau Levels as a Probe for Band Topology in Graphene Moiré Superlattices.

We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magnetization in momentum space that is governed by the rotational C_{2} and time-reversal T symmetries. Our results can be readily extended to other twisted graphene multilayers and h-BN/graphene heterostructures thus establishing the Hofstadter butterfly spectra as a powerful tool for detecting the nontrivial valley band topology.

Square-root topological semimetals

We propose topological semimetals generated by the square-root operation for tight-binding models in two and three dimensions, which we call square-root topological semimetals. The square-root topological semimetals host topological band touching at finite energies, whose topological protection is inherited from the squared Hamiltonian. Such a topological character is also reflected in emergence of boundary modes with finite energies. Specifically, focusing on topological properties of squared Hamiltonian in class AIII, we reveal that a decorated honeycomb (decorated diamond) model hosts finite-energy Dirac cones (nodal lines). We also propose a realization of a square-root topological semimetal in a spring-mass model, where robustness of finite-energy Dirac points against the change of tension is elucidated.

Correlation-driven topological phases in magic-angle twisted bilayer graphene.

Magic-angle twisted bilayer graphene (MATBG) exhibits a range of correlated phenomena that originate from strong electron-electron interactions. These interactions make the Fermi surface highly susceptible to reconstruction when ±1, ±2 and ±3 electrons occupy each moiré unit cell, and lead to the formation of various correlated phases1-4. Although some phases have been shown to have a non-zero Chern number5,6, the local microscopic properties and topological character of many other phases have not yet been determined. Here we introduce a set of techniques that use scanning tunnelling microscopy to map the topological phases that emerge in MATBG in a finite magnetic field. By following the evolution of the local density of states at the Fermi level with electrostatic doping and magnetic field, we create a local Landau fan diagram that enables us to assign Chern numbers directly to all observed phases. We uncover the existence of six topological phases that arise from integer fillings in finite fields and that originate from a cascade of symmetry-breaking transitions driven by correlations7,8. These topological phases can form only for a small range of twist angles around the magic angle, which further differentiates them from the Landau levels observed near charge neutrality. Moreover, we observe that even the charge-neutrality Landau spectrum taken at low fields is considerably modified by interactions, exhibits prominent electron-hole asymmetry, and features an unexpectedly large splitting between zero Landau levels (about 3 to 5 millielectronvolts). Our results show how strong electronic interactions affect the MATBG band structure and lead to correlation-enabled topological phases.

Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics.

Direct measurement of a bulk topological observable in topological phase of matter has been a long-standing issue. Recently, detection of bulk topology through quench dynamics has attracted growing interests. Here, we propose that topological characters of a quantum quadrupole insulator can be read out by quench dynamics. Specifically, we introduce a quantity, a quadrupole moment weighted by the eigenvalues of the chiral operator, which takes zero for the trivial phase and finite for the quadrupolar topological phase. By utilizing an efficient numerical method to track the unitary time evolution, we elucidate that the quantity we propose indeed serves as an indicator of topological character for both noninteracting and interacting cases. The robustness against disorders is also demonstrated.

A Novel Hybrid (PID + MRAC) Adaptive Controller for an Air Levitation System

The air levitation system belongs to a class of systems with fast dynamics and low damping. Such characteristics make the plant intrinsically unstable and respond in a non-linear form. Thus, it is prohibitive to use classic control techniques, such as the PID (Proportional-Integral-Derivative) controller, to track the position of the sphere. The control system must be able to compensate the non-linearities, high oscillation and reject disturbances. Thus, this research proposes to create a new approach for the hybrid controller (PID + MRAC) present in the literature. The topological character of the proposed MRAC (Model Reference Adaptive Controller) consists of three parts: a feedforward controller, a derivative portion and an ordinary feedback. The feedforward portion has the purpose of rejecting undesirable disturbances. The derivative portion increases the stability of the system and the ordinary feedback makes the error null in steady state. Due to the convergence time of the adjustment parameters, MRAC performs poorly during reference changes and in the rejection of disturbances. Thus, it is common practice to use the MRAC with the PID controller. In its methodological aspect, the control law was created from Lyapunov's theory, with the purpose of ensuring asymptotic stability for the system. As a result, the proposed controller (Hybrid-MRAC or H-MRAC) showed better results than a literature reference (A-PID), in terms of mean absolute (MAE), mean square (MSE) and root mean square (RMSE) errors. In MAE simulations it was 51,25% lower on average, MSE was 51,65% and RMSE 31,40%. In the experiments, the MAE was on average 19,72% lower, the MSE 42,92% and the RMSE 18,58%.

Hybridized quadrupolar excitations in the spin-anisotropic frustrated magnet FeI2

Magnetic order is usually associated with well-defined magnon excitations. Exotic magnetic fluctuations with fractional, topological or multipolar character, have been proposed for radically different forms of magnetic matter such as spin-liquids. As a result, considerable efforts have searched for, and uncovered, low-spin materials with suppressed dipolar order at low temperatures. Here, we report neutron-scattering experiments and quantitative theoretical modeling of an exceptional spin-1 system -- the uniaxial triangular magnet FeI2 -- where a bright and dispersive band of mixed dipolar-quadrupolar fluctuations emerges just above a dipolar ordered ground-state. This excitation arises from anisotropic exchange interactions that hybridize overlapping modes carrying fundamentally different quantum numbers. Remarkably, a generalization of spin-wave theory to local SU(3) degrees of freedom accounts for all details of the low-energy dynamical response of FeI2 without going beyond quadratic order. Our work highlights that quantum excitations without classical counterparts can be realized even in presence of fully developed magnetic order.

Topological character of three-dimensional nexus triple point degeneracies

Recently a generic class of three-dimensional band structures was identified that host two-fold line degeneracies meeting at three-fold or triple point degeneracies, which resist the usual topological characterization of isolated point degeneracies as in Dirac/Weyl semimetals. For these so-called ``Nexus" fermions which lie beyond Dirac/Weyl fermions, we lay out several concepts to characterize the wavefunction geometry and spell out its topology. Our approach is based on an understanding of the analyticity properties of Nexus wavefunctions building on a two-dimensional analogue studied recently by us. We use this to write down a homological classification of various Nexus triple point degeneracies in three dimensions.

Community detection based on first passage probabilities

Community detection is of fundamental significance for understanding topology characters and spreading dynamics on complex networks. While random walk is widely used in previous algorithms, there still exist two major defects: (i) the maximal length of random walk in some methods is too large to distinguish different communities; (ii) the useful community information at all other step lengths are missed if using a pre-assigned maximal length. Here we propose a novel community detection algorithm named as First Passage Probability Method (FPPM), equipped with a new similarity measure that incorporates the complete structural information within the maximal step length. The diameter of the network is chosen as an appropriate boundary of random walks, which is adaptive to different networks. Numerical simulations show that FPPM performs best compared to several classic algorithms on both synthetic benchmarks and real-world networks, especially those with weak community structures.

Strongly correlated Chern insulators in magic-angle twisted bilayer graphene.

Interactions between electrons and the topology of their energy bands can create unusual quantum phases of matter. Most topological electronic phases appear in systems with weak electron-electron interactions. The instances in which topological phases emerge only as a result of strong interactions are rare and mostly limited to those realized in intense magnetic fields1. The discovery of flat electronic bands with topological character in magic-angle twisted bilayer graphene (MATBG) has created a unique opportunity to search for strongly correlated topological phases2-9. Here we introduce a local spectroscopic technique using a scanning tunnelling microscope to detect a sequence of topological insulators in MATBG with Chern numbers C=±1, ±2 and ±3, which form near filling factors of±3, ±2 and ±1 electrons per moiré unit cell, respectively, and are stabilized by modest magnetic fields. One of the phases detected here (C=+1) was previously observed when the sublattice symmetry of MATBG was intentionally broken by a hexagonal boron nitride substrate, with interactions having a secondary role9. We demonstrate that strong electron-electron interactions alone can produce not only the previously observed phase, but also other unexpected Chern insulating phases in MATBG. The full sequence of phases that we observe can be understood by postulating that strong correlations favour breaking time-reversal symmetry to form Chern insulators that are stabilized by weak magnetic fields. Our findings illustrate that many-body correlations can create topological phases in moiré systems beyond those anticipated from weakly interacting models.

Knotted trajectories of neutral and charged particles in Gaussian light beams

Making use of the equivalence between paraxial wave equation and two-dimensional Schr\"odinger equation, Gaussian beams of monochromatic light, possessing knotted nodal structures are obtained in an analytical way. These beams belong to the wide class of paraxial beams called the Hypergeometric-Gaussian beams [E. Karimi, G. Zito, B. Piccirillo, L. Marrucci and E. Santamato, Opt. Lett. {\bf 32}, 3053(2007)]. Four topologies are dealt with: the unknot, the Hopf link, the Borromean rings and the trefoil. It is shown in the numerical way that neutral polarizable particles placed in such light fields, upon precise tuning of the initial conditions, can be forced to follow the identical knotted trajectories. A similar outcome is also valid for charged particles that are subject to a ponderomotive potential. This effect can serve to precisely steer particles along chosen complicated pathways exhibiting non-trivial topological character, guide them around obstacles and seems to be helpful in engineering more complex nanoparticles.

Strain-induced excitonic instability in twisted bilayer graphene

The low-energy bands of twisted bilayer graphene form Dirac cones with approximate electron-hole symmetry at small rotation angles. These crossings are protected by the emergent symmetries of moir\'e patterns, conferring a topological character to the bands. Strain accumulated between layers (heterostrain) shifts the Dirac points both in energy and momentum. The overlap of conduction and valence bands favors an excitonic instability of the Fermi surface close to the neutrality point. The spontaneous condensation of electron-hole pairs breaks time reversal symmetry and the separate conservation of charge within each valley sector. The order parameter describes interlayer circulating currents in a Kekul\'e-like orbital magnetization density wave. Vortices in this order parameter carry fermion numbers owing to the underlying topology of the bands. This mechanism may explain the occurrence of insulating states at neutrality in the most homogenous samples, where uniform strain fields contribute both to stabilizing the relative orientation between layers and to the formation of an excitonic gap.

Boundary-Obstructed Topological High- Tc Superconductivity in Iron Pnictides

Non-trivial topology and unconventional pairing are two central guiding principles in the contemporary search for and analysis of superconducting materials and heterostructure compounds. Previously, a topological superconductor has been predominantly conceived to result from a topologically non-trivial band subject to intrinsic or external superconducting proximity effect. Here, we propose a new class of topological superconductors which are uniquely induced by unconventional pairing. They exhibit a boundary-obstructed higher-order topological character and, depending on their dimensionality, feature unprecedently robust Majorana bound states or hinge modes protected by chiral symmetry. We predict the 112-family of iron pnictides, such as Ca$_{1-x}$La$_x$FeAs$_2$, to be a highly suited material candidate for our proposal, which can be tested by edge spectroscopy. Because of the boundary-obstruction, the topologically non-trivial feature of the 112 pnictides does not reveal itself for a bulk-only torus band analysis without boundaries, and as such had evaded previous investigations. Our proposal not only opens a new arena for highly stable Majorana modes in high-temperature superconductors, but also provides the smoking gun evidence for extended s-wave order in the iron pnictides.

Multiple Dirac nodes and symmetry protected Dirac nodal line in orthorhombic α -RhSi

Owing to their chiral cubic structure, exotic multifold topological excitations have been predicted and recently observed in transition metal silicides like $\beta$-RhSi. Herein, we report that the topological character of RhSi is also observed in its orthorhombic $\alpha$-phase which displays multiple types of Dirac nodes very close to the Fermi level ($\varepsilon_F$) with the near absence of topologically trivial carriers. We discuss the symmetry analysis, band connectivity along high-symmetry lines using group representations, the band structure, and the nature of the Dirac points and nodal lines occurring near $\varepsilon_F$. The de Haas-van Alphen effect (dHvA) indicates a Fermi surface in agreement with the calculations. We find an elliptically-shaped nodal line very close to $\varepsilon_F$ around and near the $S$-point on the $k_y-k_z$ plane that results from the intersection of two upside-down Dirac cones. The two Dirac points of the participating Kramers degenerate bands are only 5 meV apart, hence an accessible magnetic field might induce a crossing between the spin-up partner of the upper-Dirac cone and the spin-down partner of the lower Dirac cone, possibly explaining the anomalies observed in the magnetic torque.

Butterfly-Like Anisotropic Magnetoresistance and Angle-Dependent Berry Phase in a Type-II Weyl Semimetal WP2Supported by the National Natural Science Foundation of China (Grant Nos. 11974324, 11804326, U1832151, and 11674296), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDC07010000), the National Key Research and Development Program of China (Grant No. 2017YFA0403600), the Anhui Initiative in Quantum Information Technologies

The Weyl semimetal has emerged as a new topologically nontrivial phase of matter, hosting low-energy excitations of massless Weyl fermions. Here, we present a comprehensive study of a type-II Weyl semimetal WP2. Transport studies show a butterfly-like magnetoresistance at low temperature, reflecting the anisotropy of the electron Fermi surfaces. This four-lobed feature gradually evolves into a two-lobed variant with an increase in temperature, mainly due to the reduced relative contribution of electron Fermi surfaces compared to hole Fermi surfaces for magnetoresistance. Moreover, an angle-dependent Berry phase is also discovered, based on quantum oscillations, which is ascribed to the effective manipulation of extremal Fermi orbits by the magnetic field to feel nearby topological singularities in the momentum space. The revealed topological character and anisotropic Fermi surfaces of the WP2 substantially enrich the physical properties of Weyl semimetals, and show great promises in terms of potential topological electronic and Fermitronic device applications.

Highly responsive broadband photodetection in topological insulator - Carbon nanotubes based heterostructure

Topological insulators represent a new electronic phase which comes from the topological character of the bulk wave functions in some special materials. These exotic phases can be attributed to the strong spin-orbit interaction and can be explained with the band theory of solids. Recent studies show that topological insulators (TIs) such as Bi2Te3 or Bi2Se3 have promising candidature for optoelectronic industry and some of these interesting properties have been already demonstrated such as strong light absorption, photocurrent sensitivity to the polarization of light, layer thickness and size dependent tuning in the band gap. Theoretically, it has been proposed that TIs based broad spectral photodetector has potential in analogy with that of a graphene based photodetector. Combining TIs with carbon-based nanomaterials provides novel and promising technique for photodetection, and can show broad spectral properties. Along this line, we have grown photodetector based on hybrid of TIs with carbon nanotubes. Efficient light absorption and electron hole pair generation has been observed under the illumination of visible (532 nm) and NIR (1064 nm) light. High photoresponsivity and photoconductive gain of 79.4 AW-1 and 185, respectively, along with very good detector response time were observed in our device under visible laser (532 nm) illumination.

Testing nonlinearity in topological organization of functional brain networks.

Aiming to provide an argumentation on the underlying nonlinearity of the overall functional brain network via surrogate data method and graph theory. Taking the functional magnetic resonance imaging data as original data set and then shuffled the time series of each region of interest to generate surrogate data sets, corresponding original network and its 400 surrogates were obtained via computing connectivity matrixes. The results show that both the global correlation level and corresponding small-world topological characters exhibited obvious differences between the original network and its surrogates. And the following statistical testing results demonstrate their significant distinction, and this topological difference has been proved to be caused by the intrinsic nonlinear dynamics. Accordingly, the nonlinearity of the original functional network and its superior dynamical complexity have been confirmed. The results of this study could provide a novel angle into exploring the underlying mechanism of the neural brain system and offer an essential evidence in explaining complex brain activities.