Recent acoustic and electrical-circuit experiments have reported the third-order (or octupole) topological insulating phase, while its counterpart in classical magnetic systems is yet to be realized. Here we explore the collective dynamics of magnetic vortices in three-dimensional breathing cuboids, and find that the vortex lattice can support zero-dimensional corner states, one-dimensional hinge states, two-dimensional surface states, and three-dimensional bulk states, when the ratio of alternating intralayer and interlayer bond lengths goes beyond a critical value. We show that only the corner states are stable against external frustrations because of the topological protection. Full micromagnetic simulations verify our theoretical predictions with good agreement.
We study the topological phase in dipolar-coupled two-dimensional breathing square lattice of magnetic vortices. By evaluating the quantized Chern number and $\mathbb{Z}_{4}$ Berry phase, we obtain the phase diagram and identify that the second-order topological corner states appear only when the ratio of alternating bond lengths goes beyond a critical value. Interestingly, we uncover three corner states at different frequencies ranging from sub GHz to tens of GHz by solving the generalized Thiele's equation, which has no counterpart in condensed matter system. We show that the emerging corner…
The edge states of two-dimensional time-reversal topological insulators support a perfect helical conductance on wide ribbons due to the absence of backscattering. Here, we study the changes in the transport properties of topological insulator nanoribbons by introducing a constriction along the ribbon. This setup allows the edge states to hybridize, leading to reflections at the ends of the constriction. We find that the electronic states running along one edge can be reflected back along the opposite edge multiple times, giving rise to Fabry-Pérot resonant vortices within the constriction wit…
We report the observation of the superconducting proximity effect in nanoribbons of a candidate topological insulator (Bi2Se3) which is interfaced with superconducting (tungsten) contacts. We observe a supercurrent and multiple Andreev reflections for channel lengths that are much longer than the inelastic and diffusive thermal lengths deduced from normal state transport. This suggests that the proximity effect couples preferentially to a ballistic surface transport channel, even in the presence of a coexisting diffusive bulk channel. When a magnetic field is applied perpendicular to the plane…
A nonuniform in-plane Zeeman field can induce spontaneous supercurrents of spin-orbit coupled electrons in superconducting two-dimensional systems and thin films. In this work it is shown that current vortices can be created at the ends of a long homogeneously magnetized strip of a ferromagnetic insulator, which is deposited on the surface of a three-dimensional topological insulator. The s-wave superconductivity on its surface is assumed to have an intrinsic origin, or to be induced by the proximity effect. It is shown that vortices with the odd vorticity can localize Majorana zero modes.The …
$\mathbb{Z}_2$ topological insulators for photons and in general bosons cannot be strictly implemented because of the lack of symmetry-protected pseudospins. We show that the required protection can be provided by the real-space topological excitation of an interacting quantum fluid: quantum vortex. We consider a Bose-Einstein Condensate at the $Γ$ point of the Brillouin zone of a quantum valley Hall system based on two staggered honeycomb lattices. We demonstrate the existence of a coupling between the winding number of a vortex and the valley of the bulk Bloch band. This leads to chiral vort…
If superconductivity is induced in the metallic surface states of topological insulators via proximity, Majorana modes will be trapped on the vortex cores. The same effects hold for doped topological insulators which become bulk s-wave superconductors as long as the doping does not exceed a critical values $ μ^{\pm}_c.$ It is this critical chemical potential at which the material forgets it arose from a band-inverted topological insulator; it loses its topological \emph{imprint.} For the most common classes of topological insulators, which can be modeled with a minimal 4-band Dirac model the v…
We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a three-dimensional topological insulator with a proximity induced in-plane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent 0-π crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of…
By analytically solving the Bogoliubov-de Gennes equations we study the fermion bound states at the center of the core of a vortex in a two-dimensional superconductor. We consider three kinds of 2D superconducting models: (a) a standard type II superconductor in the mixed state with low density of vortex lines, (b) a superconductor with strong spin-orbit coupling locking the spin parallel to the momentum and (c) a superconductor with strong spin-orbit coupling locking the spin perpendicular to the momentum. The 2D superconducting states are induced via proximity effect between an $s$-wave supe…
We study the zero-temperature phase diagram and fractional excitation when a thin film of 3D topological insulator has two competing masses: T- symmetric exciton condensation and T- breaking Zeeman effect. Two topologically distinct phases are identified: in one, the quasiparticles can be viewed as in a quantum spin Hall phase, and in the other a quantum anomalous Hall phase. The vortices of the exciton order parameter can carry fractional charge and statistics of electrons in both phases. When the system undergoes the quantum phase transition between these two phases, the charges, statistics …
Departing from the conventional view on superconducting vortices as a parasitic source of dissipation for charge transport, we propose to use mobile vortices as topologically-stable information carriers for spin transport. To this end, we start by constructing a phenomenological theory for the interconversion between spin and vorticity, a topological charge carried by vortices, at the interface between a magnetic insulator and a superconductor, by invoking the interfacial spin Hall effect therein. We then show that a vortex liquid in superconductors can serve as a spin-transport channel betwee…
Recent observation of a metal-insulator phase transition in the $ν=0$ Hall state of graphene has inspired the idea that charge carriers in the metallic state could be fractionally charged vortices. We examine the question of whether vortices in particular gapped states of graphene and subject to external magnetic and pseudo-magnetic fields could have the mid-gap zero mode electron states which would allow them to be charged.
Topological defects, such as domain walls and vortices, have long fascinated physicists. A novel twist is added in quantum systems like the B-phase of superfluid helium He$_3$, where vortices are associated with low energy excitations in the cores. Similarly, cosmic strings may be tied to propagating fermion modes. Can analogous phenomena occur in crystalline solids that host a plethora of topological defects? Here we show that indeed dislocation lines are associated with one dimensional fermionic excitations in a `topological insulator', a novel band insulator believed to be realized in the b…
In this work, we develop an index signature characterising the third order topological phases in 3D systems. This index is an alternating sum of monomial signatures of Higgs triplet values at 3D corners. We extend our method to N-dimensional systems with open boundaries, and demonstrate that the topological invariant can be efficiently generalised to any space dimension including the second order topological insulators. Known results on lower dimensional systems are recovered and an interpretation in the Higgs space parameters is given.
Using Ginzburg-Landau approach we show that the strain of the nematic superconductor can generate a specific (nematic) vorticity. In the case of doped topological insulators that vorticity forms a spin vortex. We find two types of topologically different spin vortices that either enhance (type I) or suppress (type II) superconductivity far from the vortex core. We apply Bogoliubov-de Geunnes equations to study electronic states in the nematic superconductor with spin vortices. We find that in the case of the vortex of type I, zero-energy states are localized near the vortex core. These states …
A three-dimensional topological insulator manifests gapless surface modes, described by two-dimensional Dirac equation. We study magnetic textures, such as domain walls and vortices, in a ferromagnetic thin film deposited on a three-dimensional topological insulator. It is shown that these textures can be electrically charged, ascribed to the proximity effect with the Dirac surface states. We derive a general relation between the electric and the magnetic charges. As a physical consequence, we discuss domain wall motion driven by an applied electric field, which promises magnetic devices with …
In order to stabilize Majorana excitations within vortices of proximity induced topological superconductors, it is mandatory that the Dirac point matches the Fermi level rather exactly, such that the conventionally confined states within the vortex are well separated from the Majorana-type excitation. Here, we show by angle resolved photoelectron spectroscopy that (Bi$_{1-x}$Sb$_{x}$)$_{2}$Te$_{3}$ thin films with $x=0.94$ prepared by molecular beam epitaxy and transferred in ultrahigh vacuum from the molecular beam epitaxy system to the photoemission setup matches this condition. The Dirac po…
The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological 'corner states', as well as three-dimensional (3D) systems having one-dimensional (1D) topological 'hinge states'. Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamond-latt…
Floquet topological insulators are topological phases of matter generated by the application of time-periodic perturbations on otherwise conventional insulators. We demonstrate that spatial variations in the time-periodic potential lead to localized quasi-stationary states in two-dimensional systems. These states include one-dimensional interface modes at the nodes of the external potential, and fractionalized excitations at vortices of the external potential. We also propose a setup by which light can induce currents in these systems. We explain these results by showing a close analogy to px+…
Angle resolved photoemission spectroscopy is used to observe changes in the electronic structure of bulk-doped topological insulator Cu$_x$Bi$_2$Se$_3$ as additional copper atoms are deposited onto the cleaved crystal surface. Carrier density and surface-normal electrical field strength near the crystal surface are estimated to consider the effect of chemical surface gating on atypical superconducting properties associated with topological insulator order, such as the dynamics of theoretically predicted Majorana Fermion vortices.
Current Flow in Topological Insulator Josephson Junctions due to Imperfections
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Weizmann Institute of Science（魏茨曼科学研究所）；Cornell University（康奈尔大学）；Max Planck Institute for Solid State Research（马克斯·普朗克固体研究所）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；HSE University（高等经济大学）；Landau Institute for Theoretical Physics（朗道理论物理研究所）
Gate-tunable large second-order and third-order nonlinear Hall effect in topological insulator BiSbTeSe2
National Laboratory of Solid State Microstructures, Collaborative Innovation Center of Advanced Microstructures, Jiangsu Physical Science Research Center, and School of Physics, Nanjing University（南京大学固体微结构物理国家重点实验室、先进微结构协同创新中心、江苏省物理科学研究中心和物理学院）；Institute of Atom Manufacturing, Nanjing University（南京大学原子制造研究所）；Jiangsu Provincial Key Laboratory of Atomic Level Manufacturing, Nanjing Institute of Atomic Scale Manufacturing（江苏省原子级制造重点实验室、南京原子尺度制造研究所）；School of Physics and Electronic Information, Jiangsu Second Normal University（江苏第二师范学院物理与电子信息学院）
We study the proximity effect between an s-wave superconductor and the surface states of a strong topological insulator. The resulting two dimensional state resembles a spinless p_x+ip_y superconductor, but does not break time reversal symmetry. This state supports Majorana bound states at vortices. We show that linear junctions between superconductors mediated by the topological insulator form a non chiral 1 dimensional wire for Majorana fermions, and that circuits formed from these junctions provide a method for creating, manipulating and fusing Majorana bound states.
Vortices in two-dimensional superconductors with broken time-reversal and spin-rotation symmetry can bind states at zero excitation energy. These socalled Majorana bound states transform a thermal insulator into a thermal metal and may be used to encode topologically protected qubits. We identify an alternative mechanism for the formation of Majorana bound states, akin to the way in which Shockley states are formed on metal surfaces: An atomic-scale electrostatic line defect can have a pair of Majorana bound states at the end points. The Shockley mechanism explains the appearance of a thermal …
At the interface of an s-wave superconductor and a three-dimensional topological insulator, Ma- jorana zero modes and Majorana helical states have been proposed to exist respectively around magnetic vortices and geometrical edges. Here we first show that a single magnetic impurity at such an interface splits each resonance state of a given spin channel outside the superconducting gap, and also induces two new symmetric impurity states inside the gap. Next we find that an increase in the superconducting gap suppresses both the oscillation magnitude and period of the RKKY inter- action between t…
We investigate the quantum phase transition in a one-dimensional chain of ultra-small superconducting grains, considering both the self- and junction capacitances. At zero temperature, the system is transformed into a two-dimensional system of classical vortices, where the junction capacitance introduces anisotropy in the interaction between vortices. This leads to the superconductor-insulator transition of the Berezinskii-Kosterlitz-Thouless type, as the ratios of the Josephson coupling energy to the charging energies are varied. It is found that the junction capacitance plays a role similar …
A highlighting feature of Majorana bound states in two-dimensional topological superconductors is that they gain a phase factor of $π$ upon being orbited by a vortex. This work focuses on the vortex degree of freedom itself and demonstrates that the change in the Majorana state is accompanied by a fermion parity change within the vortex. Such a parity flip is interpreted as a higher dimensional analog of the fermion parity pump mechanism in superconducting wires as well as through general topological arguments. It is demonstrated in terms of level crossings in three different situations - in i…
We theoretically study spin and charge transport induced by a twisted light beam irradiated on a disordered surface of a doped three dimensional topological insulator (TI). We find that various types of spin vortices are imprinted on the surface of the TI depending on the spin and orbital angular momentum of the incident light. The key mechanism for the appearance of the unconventional spin structure is the spin-momentum locking in the surface state of the TI. Besides, the diffusive transport of electrons under an inhomogeneous electric field causes a gradient of the charge density, which then…
The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate th…
We generalize the concept of the spin-momentum locking to magnonic systems and derive the formula to calculate the spin expectation value for one-magnon states of general two-body spin Hamiltonians. We give no-go conditions for magnon spin to be independent of momentum. As examples of the magnon spin-momentum locking, we analyze a one-dimensional antiferromagnet with the Néel order and two-dimensional kagome lattice antiferromagnets with the 120$^\circ$ structure. We find that the magnon spin depends on its momentum even when the Hamiltonian has the $z$-axis spin rotational symmetry, which can…
By coating a three-dimensional topological insulator (TI) with a ferromagnetic film supporting an in-plane magnetic vortex, one breaks the time-reversal symmetry (TRS) without generating a mass gap. It rather yields electronic states bound to the vortex center which have different probabilities associated with each spin mode. In addition, its associate current (around the vortex center) is partially polarized with an energy gap separating the most excited bound state from the scattered ones. Charged zero-modes also appear as fully polarized modes localized near the vortex center. From the magn…
Exploiting the peculiar properties of proximity-induced superconductivity on the surface of a topological insulator, we propose a device which allows the creation of a Majorana fermion inside the core of a pinned Abrikosov vortex. The relevant Bogolyubov-de Gennes equations are studied analytically. We demonstrate that in this system the zero-energy Majorana fermion state is separated by a large energy gap, of the order of the zero-temperature superconducting gap $Δ$, from a band of single-particle non-topological excitations. In other words, the Majorana fermion remains robust against thermal…
We study a robust topological transport carried by vortices in a thin film of an easy-plane ferromagnetic insulator between two metal contacts. A vortex, which is a nonlocal topological spin texture in two-dimensional magnets, exhibits some beneficial features as compared to skyrmions, which are local topological defects. In particular, the total topological charge carried by vorticity is robust against local fluctuations of the spin order-parameter magnitude. We show that an electric current in one of the magnetized metal contacts can pump vortices into the insulating bulk. Diffusion and nonl…
Synchronized rotation of unit cells in a periodic structure provides a novel design perspective for manipulation of band topology. We then design a two-dimensional version of higher-order topological insulators (HOTI), by such rotation in a triangular photonic lattice with $\mathcal{C}_3$ symmetry. This HOTI supports the hallmark zero-dimensional corner states and simultaneously the one-dimensional edge states. We also find that our photonic corner states carry chiral orbital angular momenta locked by valleys, whose wavefunctions are featured by the phase vortex (singularity) positioned at the…
The recent interest in the low-energy states in vortices of semiconductor-superconductor heterostructures are mainly fueled by the prospects of using Majorana zero modes for quantum computation. The knowledge of low-lying states in the vortex core is essential as they pose a limitation on the topological computation with these states. Recently, the low-energy spectra of clean heterostructures, for superconducting-pairing profiles that vary slowly on the scale of the Fermi wavelength of the semiconductor, have been analytically calculated. In this work, we formulate an alternative method based …
Two-dimensional topological superconductivity has attracted great interest due to the emergence of Majorana modes bound to vortices and propagating along edges. However, due to its rare appearance in natural compounds, experimental realizations rely on a delicate artificial engineering involving materials with helical states, magnetic fields and conventional superconductors. Here we introduce an alternative path using a class of three-dimensional antiferromagnet to engineer a two- dimensional topological superconductor. Our proposal exploits the appearance of solitonic states at the interface …
Knots and links are fascinating and intricate topological objects. Their influence spans from DNA and molecular chemistry to vortices in superfluid helium, defects in liquid crystals and cosmic strings in the early universe. Here, we find that knotted structures also exist in a peculiar class of three-dimensional topological insulators---the Hopf insulators. In particular, we demonstrate that the momentum-space spin textures of Hopf insulators are twisted in a nontrivial way, which implies the presence of various knot and link structures. We further illustrate that the knots and nontrivial spi…
Magnetotransport measurements were performed in a ultra-high mobility GaAs/AlGaAs quantum well of density $\sim 3.0 \times 10^{11}$ $cm^{-2}$. The temperature dependence of the magnetoresistance $R_{xx}$ was studied in detail in the vicinity of $ν={9/2}$. In particular, we discovered new minima in $R_{xx}$ at filling factor $ν\simeq 4{1/5}$ and $4{4/5}$, but only at intermediate temperatures $80\lesssim T\lesssim 120$ mK. We interpret these as evidence for a fractional quantum Hall liquid forming in the N=2 Landau level and competing with bubble and Wigner crystal phases favored at lower tempe…
Nonlinear nanophotonics leverages engineered nanostructures to funnel light into small volumes and intensify nonlinear optical processes with spectral and spatial control. Due to its intrinsically large and electrically tunable nonlinear optical response, graphene is an especially promising nanomaterial for nonlinear optoelectronic applications. Here we report on exceptionally strong optical nonlinearities in graphene-insulator-metal heterostructures, demonstrating an enhancement by three orders of magnitude in the third-harmonic signal compared to bare graphene. Furthermore, by increasing the…
We consider an interface between two strong time-reversal invariant topological insulators having surface states with opposite spin chirality, or equivalently, opposite mirror Chern number. We show that such an interface supports gapless modes that are protected by mirror symmetry. The interface states are investigated with a continuum model for the Bi2Se3 class of topological insulators that takes into account terms up to third order in the crystal momentum, which ensures that the model has the correct symmetry. The model parameters are obtained from ab initio calculations. Finally, we consid…
We study the magnetic response of holographic superconductors exhibiting an insulating "normal" phase. These materials can be realized as a CFT compactified on a circle, which is dual to the AdS Soliton geometry. We study the response under i) magnetic fields and ii) a Wilson line on the circle. Magnetic fields lead to formation of vortices and allows one to infer that the superconductor is of type II. The response to a Wilson line is in the form of Aharonov-Bohm-like effects. These are suppressed in the holographic conductor/superconductor transition but, instead, they are unsuppressed for th…
Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurement…
Topological insulators (TIs) have attracted immense interest because they host helical surface states. Protected by time-reversal symmetry, they are robust to non-magnetic disorder. When superconductivity is induced in these helical states, they are predicted to emulate p-wave pairing symmetry, with Majorana states bound to vortices. Majorana bound states possess non-Abelian exchange statistics which can be probed through interferometry. Here, we take a significant step towards Majorana interferometry by observing pronounced Fabry-Perot oscillations in a TI sandwiched between a superconducting…
A 3D fermionic topological insulator has a gapless Dirac surface state protected by time-reversal symmetry and charge conservation symmetry. The surface state can be gapped by introducing ferromagnetism to break time-reversal symmetry, introducing superconductivity to break charge conservation, or entering a topological phase. In this paper, we construct a minimal gapped topological phase that preserves both time-reversal and charge conservation symmetries and supports Ising-type non-Abelian anyons. This phase can be understood heuristically as emerging from a surface s-wave superconducting st…
One dimensional laser-written modulated photonic lattices are known to be particularly suitable for diffraction management purposes. Here, we address the connection between discrete non-diffracting states and topological properties in such devices through the experimental observation and identification of three classes of non-diffracting state. The first one corresponds to topologically protected edge states, recently predicted in Floquet topological insulators, while the second and third are both bulk modes. One of them testifies of a topological transition, although presenting topological fe…
