Momentum Space Topology Research Articles

Polarization textures in crystal supercells with topological bands

Two-dimensional materials are a highly tunable platform for studying the momentum space topology of the electronic wavefunctions and real space topology in terms of skyrmions, merons, and vortices of an order parameter. Such textures for electronic polarization can exist in moiré heterostructures. A quantum-mechanical definition of local polarization textures in insulating supercells was recently proposed. Here, we propose a definition for local polarization that is also valid for systems with topologically nontrivial bands. We introduce semilocal hybrid polarizations, which are valid even when the Wannier functions in a system cannot be made exponentially localized in all dimensions. We use this definition to explicitly show that nontrivial real-space polarization textures can exist in topologically nontrivial systems with nonzero Chern number under (1) an external superlattice potential, and (2) under a stacking-induced moiré potential. In the latter, we find that while the magnitude of the local polarization decreases discontinuously across a topological phase transition from trivial to topologically nontrivial, the polarization does not completely vanish. Our findings suggest that band topology and real-space polar topology may coexist in real materials. Published by the American Physical Society 2024

Neutrino zeromodes on electroweak strings in light of topological insulators

We examine neutrino zeromode solutions on the electroweak Z-string and their effect on the stability of the string in the standard model and its extensions. We propose using topological invariants constructed from the momentum (and real) space topology of Green’s functions, often used for investigating edge modes in condensed matter physics. We analyze the standard model and then examine type-I and type-II extensions of the neutrino sector as well as their hybrid. Based on this analysis, we also comment on proposals in the literature to stabilize the Z-string.

Emergence of Inductance and Capacitance from Topological Electromagnetism

Topological electromagnetism owing to nontrivial momentum-space topology of electrons in insulators gives rise to diverse anomalous magnetoelectric responses. While conventional inductors and capacitors are based on classical electromagnetism described by Maxwell’s equations, here we show that topological electromagnetism in combination with spin dynamics in magnets also generates an inductance or a capacitance. We build a generic framework to extract the complex impedance on the basis of topological field theory, and demonstrate the emergence of an inductance or a capacitance in several heterostructure setups. In comparison with the previously-studied emergent inductances in metallic magnets, insulators highly suppress the power loss, because of the absence of Joule heating. We show that the inductance from topological electromagnetism is achieved at low current and high frequency, and is also advantageous in its power efficiency, as characterized by the high quality factor (Q-factor).

Spin-charge separation and quantum spin Hall effect of beta-bismuthene

Multiple works suggest the possibility of classification of quantum spin Hall effect with magnetic flux tubes, which cause separation of spin and charge degrees of freedom and pumping of spin or Kramers-pair. However, the proof of principle demonstration of spin-charge separation is yet to be accomplished for realistic, ab initio band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on beta-bismuthene, and demonstrate spin-charge separation, and quantized pumping of spin for three insulating states, that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of bands supporting even integer invariants, which cannot be addressed with symmetry-based indicators. Our work sets a new standard for the computational diagnosis of two-dimensional, quantum spin-Hall materials by going beyond the mathbb {Z}_{2} paradigm and providing an avenue for precise determination of the bulk invariant through computation of quantized, real-space response.

Emergent geometry, torsion and anomalies in non-relativistic topological matter

I review and discuss aspects of the interplay of emergent geometry and anomalies in topological semimetals and insulators, focusing on effects of torsion. This correspondence identifies torsional topological responses in terms of anomalies and anomaly related hydrodynamic phenomena involving gauge fields and geometry. I discuss how torsional emergent geometry arises from elastic deformations in crystalline materials and how this background couples to thee low-energy continuum models inherited from lattice models, utilizing the semiclassical expansion. Via the coupling of momentum space topology and emergent vielbein geometry, non-relativistic topological matter can realise new geometrical responses of mixed gauge-gravitational character. The topological low-energy torsional responses depend momentum space geometry, lattice momenta and the regularization and UV completion, provided by the non-relativistic physics and symmetries of topological materials.

Continuously tunable topological defects and topological edge states in dielectric photonic crystals

Topological defects in solid-state materials are crystallographic imperfections that local perturbations cannot remove. Owing to their nontrivial real-space topology, topological defects such as dislocations and disclinations could trap anomalous states associated with nontrivial momentum-space topology. The real-space topology of dislocations and disclinations can be characterized by the Burgers vector $\mathbf{B}$, which is usually a fixed fraction and integer of the lattice constant in solid-state materials. Here we show that in a dielectric photonic crystal---an artificial crystalline structure---it is possible to tune $\mathbf{B}$ continuously as a function of the dielectric constant of dislocations. Through this unprecedented tunability of $\mathbf{B}$, we achieve proper controls of topological interfacial states, i.e., reversal of their helicities. Based on this fact, we propose a topological optical switch controlled by the dielectric constant of the tunable dislocation. Our results shed light on the interplay of real and reciprocal space topologies and offer a scheme to implement scalable and tunable robust topological waveguides in dielectric photonic crystals.

Index Theorems, Generalized Hall Currents, and Topology for Gapless Defect Fermions.

We show how the index of the fermion operator from the Euclidean action can be used to uncover the existence of gapless modes living on defects (such as edges and vortices) in topological insulators and superconductors. The 1-loop Feynman diagram that computes the index reveals an analog of the quantum Hall current flowing on and off the defect-even in systems without conserved currents or chiral anomalies-and makes explicit the interplay between topology in momentum and coordinate space. We provide several explicit examples.

Observation of a Flat and Extended Surface State in a Topological Semimetal.

A flat band structure in momentum space is considered key for the realization of novel phenomena. A topological flat band, also known as a drumhead state, is an ideal platform to drive new exotic topological quantum phases. Using angle-resolved photoemission spectroscopy experiments, we reveal the emergence of a highly localized surface state in a topological semimetal BaAl and provide its full energy and momentum space topology. We find that the observed surface state is localized in momentum, inside a square-shaped bulk Dirac nodal loop, and in energy, leading to a flat band and a peak in the density of state. These results imply this class of materials as an experimental realization of drumhead surface states and provide an important reference for future studies of the fundamental physics of correlated quantum effects in topological materials.

Active quasi-BIC optical vortex generators for ultrafast switching

The Pancharatnam–Berry phase induced by the winding topology of polarization around a vortex singularity at bound states in the continuum (BIC) provides a unique approach to optical vortex (OV) generation. The BIC-based OV generators have the potential to outperform their counterparts that rely on spatial variations in terms of design feasibility, fabrication complexity, and robustness. However, given the fact that this class of OV generators originates from the topological property of the photonic bands, their responses are generally fixed and cannot be dynamically altered, which limits their applications to photonic systems. Here, we numerically demonstrate that a silicon photonic crystal slab can be used to realize optically switchable OV generation by simultaneously exploiting the vortex topology in momentum space in conjunction with silicon’s nonlinear dynamics. Picosecond switching of OV beams at near-infrared wavelengths are observed. The demonstrated nontrivial topological nature of the active generators can significantly expand the application of BIC toward ultrafast vortex beam generation, high-capacity optical communication, and mode-division multiplexing.

Intrinsic Torques Emerging from Anomalous Velocity in Magnetic Textures.

Momentum-space topology of electrons under strong spin-orbit coupling contributes to the electrically induced torques exerting on magnetic textures insensitively to disorder or thermal fluctuation. We present a direct connection between band topology and the torques by classifying the whole torques phenomenologically. As well as the intrinsic anomalous Hall effect, the torques also emerge intrinsically from the anomalous velocity of electrons regardless of a nonequilibrium transport current. We especially point out the intrinsic contribution arising exclusively in magnetic textures, which we call the "topological Hall torque (THT)." The THT emerges in bulk crystals without any interface or surface structures. We numerically demonstrate the enhancement of the THT in comparison with the conventional spin-transfer torque in the bulk metallic ferromagnet, which accounts for the giant current-induced torque measured in ferromagnetic SrRuO_{3}.

Topological Control of 2D Perovskite Emission in the Strong Coupling Regime.

Momentum space topology can be exploited to manipulate radiation in real space. Here we demonstrate topological control of 2D perovskite emission in the strong coupling regime via polaritonic bound states in the continuum (BICs). Topological polarization singularities (polarization vortices and circularly polarized eigenstates) are observed at room temperature by measuring the Stokes parameters of photoluminescence in momentum space. Particularly, in symmetry-broken structures, a very large degree of circular polarization (DCP) of ∼0.835 is achieved in the perovskite emission, which is the largest in perovskite materials to our knowledge. In the strong coupling regime, lower polariton modes shift to the low-loss spectral region, resulting in strong emission enhancement and large DCP. Our reciprocity analysis reveals that DCP is limited by material absorption at the emission wavelength. Polaritonic BICs based on 2D perovskite materials combine unique topological features with exceptional material properties and may become a promising platform for active nanophotonic devices.

Topological Magnets: Functions Based on Berry Phase and Multipoles

Macroscopic responses of magnets are often governed by magnetization and, thus, have been restricted to ferromagnets. However, such responses are strikingly large in the newly developed topological magnets, breaking the conventional scaling with magnetization. Taking the recently discovered antiferromagnetic (AF) Weyl semimetals as a prime example, we highlight the two central ingredients driving the significant macroscopic responses: the Berry curvature enhanced because of nontrivial band topology in momentum space, and the cluster magnetic multipoles in real space. The combination of large Berry curvature and multipoles enables large macroscopic responses such as the anomalous Hall and Nernst effects, the magneto-optical effect, and the novel magnetic spin Hall effect in antiferromagnets with negligible net magnetization, but also allows us to manipulate these effects by electrical means. Furthermore, nodal-point and nodal-line semimetallic states in ferromagnets may provide the strongly enhanced Berry curvature near the Fermi energy, leading to large responses beyond the conventional magnetization scaling. These significant properties and functions of the topological magnets lay the foundation for future technological development such as spintronics and thermoelectric technology.

Biasing topological charge injection in topological matter

We explore the interplay between topologies in the momentum and real spaces to formulate a thermodynamic description of nonequilibrium injection of topological charges under external bias. We show that the edge modes engendered by the momentum-space topology can play a functional role of connecting the external reservoirs to the bulk transport of topological charges in the real space. We illustrate our general results with two examples: the spin-torque injection of skyrmions in an electrically-biased integer quantum Hall system, and the vortex injection in a topological $p\,+\,i\,p$ superconductor coupled to heat reservoirs. Based on the universal fractional entropy of the Majorana zero modes bound to the vortices, their controllable injection proposed in this work could provide a route for creating and manipulating Majorana fermions.

Unquantized anomalies in topological semimetals

Topological semimetals are a new class of metallic materials, which exist at band fillings that ordinarily correspond to insulators or compensated accidental semimetals with zero Luttinger volume. Their metallicity is a result of nontrivial topology in momentum space and crystal symmetry, wherein topological charges may be assigned to point band-touching nodes, preventing gap opening, unless protecting crystal symmetries are violated. These topological charges, however, are defined from noninteracting band eigenstates, which raises the possibility that the physics of topological semimetals may be modified qualitatively by electron-electron interactions. Here we ask the following question: what makes the topological semimetals nontrivial beyond band theory? Alternatively, can strong electron-electron interactions open a gap in topological semimetals without breaking the protecting symmetries or introducing topological order? We demonstrate that the answer is generally no, and what prevents it is their topological response, or quantum anomalies. While this is familiar in the case of magnetic Weyl semimetals, where the topological response takes the form of an anomalous Hall effect, analogous responses in other types of topological semimetals are more subtle and involve crystal symmetry as well as electromagnetic gauge fields. Physically these responses are detectable as fractional symmetry charges induced on certain gauge defects. We discuss the cases of type-I Dirac semimetals and time-reversal invariant Weyl semimetals in detail. For type-I Dirac semimetals, we also show that the anomaly vanishes, in a nontrivial manner, if the momenta of the Dirac nodes satisfy certain exceptional conditions.

Thermal Hall effect of chiral spin fluctuations

Using a two-dimensional square lattice Heisenberg model with a Rashba-type Dzyaloshinskii-Moriya interaction, we demonstrate that chiral spin fluctuations can give rise to a thermal Hall effect in the absence of any static spin texture or momentum space topology. It is shown by means of Monte Carlo and stochastic spin dynamics simulations that the thermal Hall response is finite at elevated temperature outside of the linear spin wave regime and consistent with the presence of thermal fluctuation-induced nontrivial topology. Our result suggests that the high-fluctuation phases outside of the conventional regime of magnonics may yet be a promising area of exploration for spin-based electronics.

Vortex states in an acoustic Weyl crystal with a topological lattice defect

Crystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a one-dimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a one-dimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in three-dimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space.

Transverse spin polarization of a Rashba-Zeeman-coupled Fermi superfluid

We apply the Bogoliubov-de Gennes theory to a superfluid Fermi gas that is confined to a disc, and analyze its transverse spin polarization that is produced by the simultaneous presence of an out-of-plane Zeeman field and an in-plane Rashba spin-orbit coupling. This is a finite-size effect whose physical origin is directly associated with the equilibrium mass-current density. Our numerical findings for the ground state suggest a bulk-boundary correspondence between characteristic features of the edge-bound transverse polarization and the changes in the momentum-space topology of the bulk.

Fragility of surface states in topological superfluid 3He

Superfluid 3He, with unconventional spin-triplet p-wave pairing, provides a model system for topological superconductors, which have attracted significant interest through potential applications in topologically protected quantum computing. In topological insulators and quantum Hall systems, the surface/edge states, arising from bulk-surface correspondence and the momentum space topology of the band structure, are robust. Here we demonstrate that in topological superfluids and superconductors the surface Andreev bound states, which depend on the momentum space topology of the emergent order parameter, are fragile with respect to the details of surface scattering. We confine superfluid 3He within a cavity of height D comparable to the Cooper pair diameter ξ0. We precisely determine the superfluid transition temperature Tc and the suppression of the superfluid energy gap, for different scattering conditions tuned in situ, and compare to the predictions of quasiclassical theory. We discover that surface magnetic scattering leads to unexpectedly large suppression of Tc, corresponding to an increased density of low energy bound states.

Topology in momentum space becomes real

Using topological singular points, the topological charge of photonic crystals in momentum space is successfully transferred to optical vortex beams in real space.

Layer construction of topological crystalline insulator LaSbTe

Topological crystalline insulator (TCI) is one of the symmetry-protected topological states. Any TCI can be deformed into a simple product state of several decoupled two-dimensional (2D) topologically nontrivial layers in its lattice respecting its crystalline symmetries called the layer construction (LC) limit. In this work, based on first-principles calculations we have revealed that both tetragonal LaSbTe (t-LaSbTe) and orthorhombic LaSbTe (o-LaSbTe) can be interpreted as stacking of 2D topological insulators in each lattice space. The structural phase transition from t-LaSbTe to o-LaSbTe due to soft phonon modes demonstrates how the real space change can lead to the modification of topological states. Their symmetry-based indicators and topological invariants have been analyzed based on LC. We propose that LaSbTe is an ideal example demonstrating the LC paradigm, which bridges the crystal structures in real space to the band topology in momentum space.