Session K29: Topological Insulators: Theory I

Tunneling Seebeck and Anomalous Nernst effects in three-dimensional topological insulators

We theoretically investigate the longitudinal (Seebeck) and transverse (Nernst) thermopowers generated by thermally-induced tunneling across a magnetic barrier on the surface of a three-dimensional insulator. As a manifestation of Klein tunneling, the tunneling Seebeck coefficient exhibits oscillatory behavior with respect to the barrier thickness. Moreover, in spite of the absence of a source of spin polarization (only the barrier is magnetic), the tunneling anomalous Nernst coefficient is not only finite but can even be much larger than its Seebeck counterpart.   

Amperean pairing mediated by magnetic fluctuations at the surface of a topological insulator

We study the interface between a three-dimensional topological insulator and a ferromagnetic thin film. Due to the Dirac nature of surface states, in-plane magnetization couples to them as a gauge field, leading to emergent electric and magnetic fields. We argue that magnetic fluctuations mediate strongly anisotropic interaction and can be the origin of an unconventional superconductivity with Amperean pairings.   

Edge state contributions to the Loschmidt echo in topological insulators and superconductors

Non-analytic behavior in the time evolution of a quantum system at critical times can be seen in the Loschmidt echo, the overlap between a time evolved state and an initial state. This is referred to as a dynamical phase transition, an analogue of the non-analytic behavior in the free energy across temperature driven phase transitions. In particular it has been demonstrated that in topological systems a quench across the topological phase transition and dynamical phase transitions are intimately related. Here we look at the contribution to the Loschmidt echo of the topologically protected edge states of 1D topological insulators and superconductors. These edge state have already been shown to have interesting contributions to the fidelity and entanglement entropy, which can also be useful for characterizing the topological phases and phase transitions.   

Magnetic ordering and quantum anomalous Hall phase of Cr-doped topological insulators: First principles studies

Realization of transverse electric currents without external magnetic fields, so called the quantum anomalous Hall effect (QAHE), is achieved in Cr-doped topological insulating (Bi,Sb)2Te3 compounds. However, detailed mechanism of QAHE and magnetic ordering in topological insulators (TIs) is still unclear with several models in controversy. We study the origin of QAHE in magnetic impurity-doped TIs using first-principles calculations. We investigate a possibility of the quantum anomalous Hall phase in conventional three-dimensional topological insulators, such as Bi2Se3, Bi2Te3, and Sb2Te3. We find that Sb2Te3 is the most suitable compound for realizing QAHE, because it maintains insulating phase and relatively strong ferromagnetic ordering in a wide range of Cr doping while Bi2Se3 and Bi2Te3 become metallic even by a small amount of Cr doping. Contrary to previous predictions, the kinetic exchange is responsible for the magnetic phase of Cr-doped TIs and it induces spin-polarized valence and conduction bands in Sb2Te3. We also discuss the role of Bi doping in topological surfaces states of Cr-doped Sb2Te3, which leads to QAHE in (Bi,Sb)2Te3.   

Filling-Enforced Quantum Band Insulators in Spin-Orbit Coupled Crystals

While band insulators are usually described in wavevector space in terms of fully filled bands, they are sometimes also described in terms of a complementary Wannier picture in which electrons occupy localized, atom-like orbitals. Under what conditions does the latter picture break down? The presence of irremovable quantum entanglement between different sites can obstruct a localized orbital description, which occurs in systems like Chern and topological insulators. We collectively refer to such states as Quantum Band Insulators (QBIs). Here we report the theoretical discovery of a filling-enforced QBI - that is, a free electron insulator in which the band filling is smaller than the minimum number dictated by the atomic picture. Consequently such insulators have no representation in terms of filling localized orbitals and must be QBIs. This is shown to occur in models of certain cubic crystals with non-symmorphic space groups. Like topological insulators, filling-enforced QBIs require spin-orbit coupling. However, in contrast, they do not typically exhibit protected surface states. Instead their nontrivial nature is revealed by studying the quantum entanglement of their ground state wavefunction.   

Casimir Torque between Topological Insulators: a Physical Implication of the Surface State Hexagonal Warping Effect

We use a variation of the Lifshitz formula to calculate the anisotropic Casimir energy density between two topological insulators in the vacuum. We find that the hexagonal warping effect can induce a Casimir torque between the two topological insulators, $T_c\propto\sin(6\theta)$ with twisted angle $\theta$. The maximal Casimir torque at $\theta=\pi/12$ is estimated to be $\sim 10^{-19} N\cdot{m}/rad$ for Bi$_2$Te$_3$ on the [111] surface when the distance between the two topological insulators is about 20 $nm$ and the surface areas are taken to be $\sim 1 cm^2$.   

Emergent Topological States at Domain Walls in Bismuth

The discovery of topological insulators has brought new perceptions in materials science which allows the understanding of material properties as an inevitable result of symmetry and its breaking. Polyacetylene is one example of topological insulators classified by structural symmetry that exhibits zero modes at a domain wall separating two opposite dimerized phases. The sign reversal of the topological mass across a domain wall is not restricted to 1D systems and is ubiquitous in a wide range of 3D materials. Employing both ab initio and model Hamiltonian calculations we have studied the topological properties of structural domain walls in bismuth, which is the 3D analogue of the domain walls in 1D polyacetylene. The model Hamiltonian can be represented to lowest order by two Pauli matrices yielding a mass gap that closes upon dimerization sign reversal. The calculations demonstrate that zero mode states emerge at the domain wall which exhibit quasi-one dimensional linear dispersions. Our results imply that conducting channels may emerge at structural domain walls such as grain boundaries as a consequence of topological protection, whose properties are determined by global rather than local symmetry.   

Single-ion magnetic anisotropy of transition metal impurities in Bi$_{2}$Se$_{3}$ bulk and thin film topological insulators

The breaking of time reversal symmetry in a topological insulator (TI) by magnetic doping is one of the most studied phenomena among the properties of Dirac materials. The robustness of the topological surface states (TSS) against magnetic impurities is of critical importance for spin-dependent transport in these systems. The interaction between TSS and magnetic impurities can open a gap, provided that the magnetic order is oriented normal to the surface of the TI. Such gap opening is crucial for realizing TI-based spintronic devices and for the observation of different fundamental phenomena, such as the anomalous quantum Hall effect. Using density functional theory as implemented in the WIEN2k ab-initio package, we have investigated the effect of the magnetization orientation on the gap opening at the Dirac point, for substitutional Mn and Fe impurities on the Bi$_{2}$Se$_{3}$ surface, and have calculated the associated single-ion anisotropy (SIA). We also have studied bulk SIA in order to compare the role played by TSS on the surface SIA.   

Strong Correlation effects to Topological Quantum Phase Transitions in Three-Dimensions

Topological Insulating phases of three dimensions are classified in terms of four $\mathbb{Z}_2$ global invariants. In the non-interacting case the Topological Quantum Phase Transition (TQPT), {\it i.e.} the sudden change of such invariants, occurs through the continuous closure of the energy gap as long as the symmetries protecting the Topological phase are preserved. However, the recent progress in engineering or predict Topological Insulating states in heavy-elements compounds, pushed the attention to the effects of large electronic interaction. Here we show that strongly correlated 3-dimensional Topological Insulators are characterized by a substantially different physics with respect to their non-interacting counterpart. Our study shows that the TQPT to the Strong Topological Insulator is dominated by the presence of a Quantum Critical Point, the end of a first-order topological transition. In addition we show that the conventional paradigm of a continuous TQPT breaks down for strong enough correlation, through to a discontinuous transition without closure of the spectral gap.   

Field theory representation of mixed gauge-gravity symmetry-protected topological invariants, group cohomology and beyond

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs, recently observed by Kapustin. We find new examples of mixed gauge-gravity actions for U(1) SPTs in 3+1D and 4+1D via the Stiefel-Whitney class and the gravitational Chern-Simons term. [Work based on Phys. Rev. Lett. 114, 031601 (2015) arXiv:1405.7689]   

Spin texture of topological surface states of side-surfaces in Bi$_2$Se$_3$ from first principles

Topological insulators have recently drawn a lot of attention because of topologically protected surface states with Dirac dispersion and spin-momentum locking induced by time-reversal symmetry and strong spin-orbit coupling, respectively. Recent experiments report interesting transport properties of Bi$_2$Se$_3$ nanowires and nanoribbons with growth directions normal to the (111) surface. However, most of the studies of Bi$_2$Se$_3$ are focused on the (111) surface. Additionally, surfaces other than the (111) surface would facilitate hole doping with adatoms since both Bi and Se atoms are present at these other surfaces. We investigate the spin texture and electronic structure of topologically protected surface states of two representative side-surfaces of Bi$_2$Se$_3$ by using density-functional theory (DFT). In particular, we consider two surfaces normal to the (111) surface, such as (1$\bar{1}$0) and (11$\bar{2}$), where the former has twofold symmetry and the latter has mirror symmetry. We present our calculated spin textures of the surface states of these side surfaces that qualitatively differ from typical Rashba-like features and from the prediction based on the bulk model Hamiltonian by keeping up to quadratic terms in momentum.   

Domain wall of a ferromagnet on a three-dimensional topological insulator

Topological insulators (TIs) show rich phenomena and functions which can never be realized in ordinary insulators. Most of them come from peculiar surface or edge states. Especially, the quantized anomalous Hall effect without an external magnetic field is realized in a two-dimensional ferromagnet on a three-dimensional TI, which supports the dissipationless edge current. Here we demonstrate theoretically that the domain wall of this ferromagnet, which carries edge currents, is charged and can be controlled by an external electric field. The chirality and relative stability of the Neel wall and the Bloch wall depend on the position of the Fermi energy as well as the form of the coupling between the magnetic moment and the orbital of the host TI. These findings will pave a path to utilize the magnets on TI for spintronics applications. \\ R. Wakatsuki, M. Ezawa, and N. Nagaosa, Scientific Reports \textbf{5}, 13638 (2015).   

Topological phases protected by point group symmetry

There has been remarkable progress in the theoretical understanding of symmetry protected topological (SPT) phases. However, most theories focus on internal, or on-site, symmetries, even though spatial symmetries are important in solids. In this talk, we classify bosonic SPT phases protected by crystalline point group symmetry, which we dub point group SPT (pgSPT) phases. Our approach is based on a procedure to reduce a d-dimensional pgSPT phase to lower-dimensional SPT phases protected by internal symmetry. For three-dimensional pgSPT phases, this approach allows us to gain insight into non-trivial properties at symmetry preserving surfaces. In particular, we obtain toy models for the surfaces of certain pgSPT phases at which there is a symmetry preserving $Z_{2}$ topological order with anomalous symmetry fractionalization. We also discuss connections between bosonic pgSPT phases and electronic topological crystalline insulators.   

Visualizing competing trends at topological surfaces

Topological insulators interacting with magnetic impurities are usually described within the framework of gapping the Dirac quasiparticles energy spectrum by time reversal symmetry breaking. However, the overwhelming majority of studies demonstrate the presence of finite density of states near the Dirac node even once the system becomes magnetic. The contradictory observations call for a better understanding of the nature of the mobility and transport in magnetically doped topological insulators. Here, by combining different experimental techniques with theoretical calculations, we map the response of topological states to magnetic impurities at the atomic scale and reveal that, contrary to what generally believed, gapless density of states and magnetic order can coexist.   

Landau Theory of Helical Fermi Liquids

We construct a phenomenological Landau theory for the two-dimensional helical Fermi liquid found on the surface of a three-dimensional time-reversal invariant topological insulator. In the presence of rotation symmetry, interactions between quasiparticles are described by ten independent Landau parameters per angular momentum channel, by contrast with the two (symmetric and antisymmetric) Landau parameters for a conventional spin-degenerate Fermi liquid. We project quasiparticle states onto the Fermi surface and obtain an effectively spinless, projected Landau theory with a single projected Landau parameter per angular momentum channel that captures the spin-momentum locking or nontrivial Berry phase of the Fermi surface. As a result of this nontrivial Berry phase, projection to the Fermi surface can increase or lower the angular momentum of the quasiparticle interactions. We derive equilibrium properties, criteria for Fermi surface instabilities, and collective mode dispersions in terms of the projected Landau parameters. We briefly discuss experimental means of measuring projected Landau parameters.