The properties of topological systems are inherently tied to their dimensionality. Higher-dimensional physical systems exhibit topological properties not shared by their lower dimensional counterparts and, in general, offer richer physics. One example is a d-dimensional quantized multipole topological insulator, which supports multipoles of order up to 2^d and a hierarchy of gapped boundary modes with topological 0-D corner modes at the top. While multipole topological insulators have been successfully realized in electromagnetic and mechanical 2D systems with quadrupole polarization, and a 3D octupole topological insulator was recently demonstrated in acoustics, going beyond the three physical dimensions of space is an intriguing and challenging task. In this work, we apply dimensional reduction to map a 4D higher-order topological insulator (HOTI) onto an equivalent aperiodic 1D array sharing the same spectrum, and emulate in this system the properties of a hexadecapole topological insulator. We observe the 1D counterpart of zero-energy states localized at 4D HOTI corners - the hallmark of multipole topological phase. Interestingly, the dimensional reduction guarantees that one of the 4D corner states remains localized to the edge of the 1D array, while all other localize in the bulk and retain their zero-energy eigenvalues. This discovery opens new directions in multi-dimensional topological physics arising in lower-dimensional aperiodic systems, and it unveils highly unusual resonances protected by topological properties inherited from higher dimensions.
We present the exhaustive classification of surface states of topological insulators and superconductors protected by crystallographic magnetic point group symmetry in three spatial dimensions. Recently, Cornfeld and Chapman [Phys. Rev. B {\bf 99}, 075105 (2019)] pointed out that the topological classification of mass terms of the Dirac Hamiltonian with point group symmetry is recast as the extension problem of the Clifford algebra, and we use their results extensively. Comparing two-types of Dirac Hamiltonians with and without the mass-hedgehog potential, we establish the irreducible characte…
The Hopf insulators are characterized by a topological invariant called Hopf index which classifies maps from three-sphere to two-sphere, instead of a Chern number or a Chern parity. In contrast to topological insulator, the Hopf insulator is not protected by any kind of symmetry. By dimensional reduction, we argue that there exists a new type of $\mathbb{Z}_2$ index for 2D Hamiltonian with vanishing Chern number. Specific model Hamiltonian with this nontrivial $\mathbb{Z}_2$ index is constructed. We also numerically calculate the topological protected edge modes of this dimensional reduced Ho…
We study a two-dimensional (2D) tight-binding model of a topological crystalline insulator (TCI) protected by rotation symmetry. The model is built by stacking two Chern insulators with opposite Chern numbers which transform under conjugate representations of the rotation group, e.g. $p_\pm$ orbitals. Despite its apparent similarity to the Kane-Mele model, it does not host stable gapless surface states. Nevertheless the model exhibits topological responses including the appearance of quantized fractional charge bound to rotational defects (disclinations) and the pumping of angular momentum in …
A class of Aubry-André-Harper models of spin-orbit coupled electrons exhibits a topological phase diagram where two regions belonging to the same phase are split up by a multicritical point. The critical lines which meet at this point each defines a topological quantum phase transition with a second-order nonanalyticity of the ground-state energy, accompanied by a linear closing of the spectral gap with respect to the control parameter; except at the multicritical point which supports fourth-order transitions with parabolic gap-closing. Here both types of criticality are characterized through …
A key feature of topological insulators (TI) is symplectic symmetry of the Hamiltonian which changes to unitary when time reversal symmetry is lifted and the topological phase transition occurs. However, such a crossover has never been explicitly observed, by directly probing the symmetry class of the Hamiltonian. In this report, we have probed the symmetry class of topological insulators by measuring the mesoscopic conductance fluctuations in the TI Bi$_{1.6}$Sb$_{0:4}$Te$_2$Se, which shows an exact factor of two reduction on application of a magnetic field due to crossover from symplectic to…
We present a general approach to obtain effective field theories for topological crystalline insulators whose low-energy theories are described by massive Dirac fermions. We show that these phases are characterized by the responses to spatially dependent mass parameters with interfaces. These mass interfaces implement the dimensional reduction procedure such that the state of interest is smoothly deformed into a topological crystal, which serves as a representative state of a phase in the general classification. Effective field theories are obtained by integrating out the massive Dirac fermion…
We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic non-magnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead o…
The effective mass and superfluidity-normal phase transition temperature of magnetoexcitons in topological insulator bilayers are theoretically investigated. The intra-Landau-level Coulomb interaction is treated perturbatively, from which the effective magnetoexciton mass is analytically discussed. The inclusion of inter-Landau-level Coulomb interaction by more exact numerical diagonalization of the Hamiltonian brings out important modifications to magnetoexciton properties, which are specially characterized by prominent reduction in the magnetoexciton effective mass and promotion in the super…
Topological insulators are a novel state of matter that share a common feature: their spectral bands are associated with a nonlocal integer-valued index, commonly manifesting through quantized bulk phenomena and robust boundary effects. In this work, we demonstrate using dimensional reduction that high-order topological insulators are descendants from a chiral semimetal in higher dimensions. Specifically, we analyze the descendants of an ancestor four-dimensional Chern insulator in the limit where it becomes chiral and show their relation to two-dimensional second-order topological insulators.…
Skyrmion nucleation induced by spin-transfer torques at an interface of a topological insulator and a ferromagnetic insulator is investigated. Due to strong spin-orbit coupling on a surface of topological insulators, which enhances the effect of spin torques, efficient manipulation of skyrmions is expected, and therefore, topological insulators could provide the ideal platform to achieve high-performance skyrmionic devices. Using micromagnetic simulations and energetics, we evaluate properties of the skyrmion nucleation on a surface of topological insulators, such as nucleation time, critical …
We investigate the band dispersion and the spin texture of topologically protected surface states in the bulk topological insulators Bi2Se3 and Bi2Te3 by first-principles methods. Strong spin-orbit entanglement in these materials reduces the spin-polarization of the surface states to ~50% in both cases; this reduction is absent in simple models but of important implications to essentially any spintronic application. We propose a way of controlling the magnitude of spin polarization associated with a charge current in thin films of topological insulators by means of an external electric field. …
The chirality associated with broken time reversal symmetry in magnetically doped topological insulators has important implications to the quantum transport phenomena. Here we report the anomalous Hall effect studies in Mn- and Cr-doped Bi$_2$Te$_3$ topological insulators with varied thickness and doping content. By tracing the chirality of the Hall loops, we find that the Mn-type anomalous Hall effect with clockwise chirality is strengthened by the reduction of film thickness, which is opposite to that of the Cr-type anomalous Hall effect with counterclockwise chirality. We provide a phenomen…
The external controllability of the magnetic properties in topological insulators would be important both for fundamental and practical interests. Here we predict the electric-field control of ferromagnetism in a thin film of insulating magnetic topological insulators. The decrease of band inversion by the application of electric fields results in a reduction of magnetic susceptibility, and hence in the modification of magnetism. Remarkably, the electric field could even induce the magnetic quantum phase transition from ferromagnetism to paramagnetism. We further propose a topological transist…
We report on "graphene-like" mechanical exfoliation of single-crystal Bi2Te3 films and thermoelectric characterization of the stacks of such films. Thermal conductivity of the resulting "pseudo-superlattices" was measured by the "hot disk" and "laser flash" techniques. The room-temperature in-plane (cross-plane) thermal conductivity of the stacks decreases by a factor of ~2.4 (3.5) as compared to bulk. The thermal conductivity reduction with preserved electrical properties leads to strong increase in the thermoelectric figure of merit. It is suggested that the film thinning to few-quintuples a…
We measure gate-tuned thermoelectric power of mechanically exfoliated Bi2Se3 thin films in the topological insulator regime. The sign of the thermoelectric power changes across the charge neutrality point as the majority carrier type switches from electron to hole, consistent with the ambipolar electric field effect observed in conductivity and Hall effect measurements. Near charge neutrality point and at low temperatures, the gate dependent thermoelectric power follows the semiclassical Mott relation using the expected surface state density of states, but is larger than expected at high elect…
Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d \ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequenc…
Electrons with a linear energy/momentum dispersion are called massless Dirac electrons and represent the low-energy excitations in exotic materials like Graphene and Topological Insulators (TIs). Dirac electrons are characterized by notable properties like a high mobility, a tunable density and, in TIs, a protection against backscattering through the spin-momentum looking mechanism. All those properties make Graphene and TIs appealling for plasmonics applications. However, Dirac electrons are expected to present also a strong nonlinear optical behavior. This should mirror in phenomena like ele…
The Dirac surface states of topological insulators offer a unique possibility for creating spin polarized charge currents due to the spin-momentum locking. Here we demonstrate that the control over the bulk and surface contribution is crucial to maximize the charge-to-spin conversion efficiency. We observe an enhancement of the spin signal due to surface-dominated spin polarization while freezing out the bulk conductivity in semiconducting Bi1.5Sb0.5Te1.7Se1.3 below 100K. Detailed measurements up to room temperature exhibit a strong reduction of the magnetoresistance signal between 2 and 100K,…
We show that the fundamental time reversal invariant (TRI) insulator exists in 4+1 dimensions, where the effective field theory is described by the 4+1 dimensional Chern-Simons theory and the topological properties of the electronic structure is classified by the second Chern number. These topological properties are the natural generalizations of the time reversal breaking (TRB) quantum Hall insulator in 2+1 dimensions. The TRI quantum spin Hall insulator in 2+1 dimensions and the topological insulator in 3+1 dimension can be obtained as descendants from the fundamental TRI insulator in 4+1 di…
We investigate topological insulating states in both two and three dimensions with the harmonic potential and strong spin-orbit couplings breaking the inversion symmetry. Landau-level like quantizations appear with the full 2D and 3D rotational symmetry and time-reversal symmetry. Inside each band, states are labeled by their angular momenta over which energy dispersions are strongly suppressed by spin-orbit coupling to nearly flat. The radial quantization generates energy gaps between neighboring bands at the order of the harmonic frequency. Helical edge or surface states appear on open bound…
Employing the Foldy-Wouthuysen transformation it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2+1 dimensions is discussed and by means of a dimensional reduction approach the 1+1 dimensional descendant of the 2+1 dimensional Chern-Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derive…
Here, we examine the influence of surface chemical reactivity toward ambient gases on the performance of nanodevices based on two-dimensional materials "beyond graphene" and novel topological phases of matter. While surface oxidation in ambient conditions was observed for silicene and phosphorene with subsequent reduction of the mobility of charge carriers, nanodevices with active channels of indium selenide, bismuth chalcogenides and transition-metal dichalcogenides are stable in air. However, air-exposed indium selenide suffers of p-type doping due to water decomposition on Se vacancies, whe…
When surface states (SSs) form in topological insulators (TIs), they inherit the properties of bulk bands, including the electron-hole (e-h) asymmetry but with much more profound impacts. Here, via combining magneto-infrared spectroscopy with theoretical analysis, we show that e-h asymmetry significantly modifies the SS electronic structures when interplaying with the quantum confinement effect. Compared to the case without e-h asymmetry, the SSs now bear not only a band asymmetry as that in the bulk but also a shift of the Dirac point relative to the bulk bands and a reduction of the hybridiz…
The effect of surface disorder on electronic systems is particularly interesting for topological phases with surface and edge states. Using exact diagonalization, it has been demonstrated that the surface states of a 3D topological insulator survive strong surface disorder, and simply get pushed to a clean part of the bulk. Here we explore a new method which analytically eliminates the clean bulk, and reduces a $D$-dimensional problem to a Hamiltonian-diagonalization problem within the $(D-1)$-dimensional disordered surface. This dramatic reduction in complexity allows the analysis of signific…
