Focus on Topological Insulators

Topological insulators (quantum spin Hall systems) are insulating in the bulk but have gapless edge/surface states, which remain gapless even when nonmagnetic disorder or interaction is present. This robustness stems from the topological nature characterized by the Z ₂ topological number, and this offers us various kinds of new novel properties. We review prominent advances in theories and in experiments on topological insulators since their theoretical proposal in 2005.

We have extended the search for topological insulators to the ternary tetradymite-like compounds M ₂ X ₂ Y ( M=Bi or Sb; X and Y=S, Se or Te), which are variations of the well-known binary compounds Bi ₂Se ₃ and Bi ₂Te ₃. Our first-principles computations suggest that five existing compounds are strong topological insulators with a single Dirac cone on the surface. In particular, stoichiometric Bi ₂Se ₂S, Sb ₂Te ₂Se and Sb ₂Te ₂S are predicted to have an isolated Dirac cone on their naturally cleaved surface. This finding paves the way for the realization of the topological transport regime.

We present first-principles calculations to predict several three-dimensional (3D) topological insulators in quaternary chalcogenide compounds of compositions I ₂–II–IV–VI ₄ and ternary famatinite compounds of compositions I ₃–V–VI ₄. Among the large number of members of these two families, we give examples of naturally occurring compounds that are mainly Cu-based chalcogenides. We show that these materials are candidates for 3D topological insulators or can be tuned to obtain topologically interesting phases by manipulating the atomic number of the various cations and anions. A band inversion can occur at a single point Γ with large inversion strength, in addition to the opening of a bulk bandgap throughout the Brillouin zone. We discuss how the two investigated families of compounds are related to each other by cross-substitution of cations in the underlying tetragonal structure.

We report high-resolution spin-resolved photoemission spectroscopy (spin-ARPES) measurements on the parent compound Sb of the recently discovered three-dimensional topological insulator Bi _{1− x}Sb _x (Hsieh et al 2008 Nature 452 970, Hsieh et al 2009 Science 323 919). By modulating the incident photon energy, we are able to map both the bulk and the (111) surface band structure, from which we directly demonstrate that the surface bands are spin polarized by the spin–orbit interaction and connect the bulk valence and conduction bands in a topologically non-trivial way. A unique asymmetric Dirac surface state gives rise to a k-splitting of its spin-polarized electronic channels. These results complement our previously published works on this class of materials and re-confirm our discovery of topological insulator states in the Bi _{1− x}Sb _x series.

Bi ₂Se ₃, Bi ₂Te ₃ and Sb ₂Te ₃ compounds have recently been predicted to be three-dimensional (3D) strong topological insulators. In this paper, based on ab initio calculations, we study in detail the topological nature and the surface states of this family of compounds. The penetration depth and the spin-resolved Fermi surfaces of the surface states are analyzed. We also present a procedure from which a highly accurate effective Hamiltonian can be constructed based on projected atomic Wannier functions (which keep the symmetries of the systems). Such a Hamiltonian can be used to study the semi-infinite systems or slab-type supercells efficiently. Finally, we discuss the 3D topological phase transition in the Sb ₂(Te _{1− x}Se _x ) ₃ alloy system.

Using k· p theory, we derive an effective four-band model describing the physics of the typical two-dimensional topological insulator (HgTe/CdTe quantum well (QW)) in the presence of an out-of-plane (in the z-direction) inversion breaking potential and an in-plane potential. We find that up to third order in perturbation theory, only the inversion breaking potential generates new elements to the four-band Hamiltonian that are off-diagonal in spin space. When this new effective Hamiltonian is folded into an effective two-band model for the conduction (electron) or valence (heavy hole) bands, two competing terms appear: (i) a Rashba spin–orbit interaction originating from inversion breaking potential in the z-direction and (ii) an in-plane Pauli term as a consequence of the in-plane potential. Spin transport in the conduction band is further analysed within the Landauer–Büttiker formalism. We find that for asymmetrically doped HgTe QWs, the behaviour of the spin-Hall conductance is dominated by the Rashba term.

The spin-polarized surface band structure of the three-dimensional (3D) quantum spin Hall phase of Bi _{1− x}Sb _x ( x=0.12–0.13) was studied by spin- and angle-resolved photoemission spectroscopy (SARPES) using a high-yield spin polarimeter equipped with a high-resolution electron spectrometer. The spin-integrated spectra were also measured and compared to those of Bi _{1− x}Sb _x with x=0.04. Band dispersions of the edge states were fully elucidated between the two time-reversal-invariant points, and , of the (111) surface Brillouin zone. The observed spin-polarized band dispersions at x=0.12–0.13 indicate an odd number of the band crossing at the Fermi energy, giving unambiguous evidence that this system is a 3D strong topological insulator, and determine the 'mirror chirality' to be −1, which excludes the existence of a Dirac point in the middle of the – line. The present research demonstrates that the SARPES measurement with energy resolution ≤50 meV is one of the critical techniques for complementing the topological band theory for spins and spin currents.

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via 'dimensional reduction' by compactifying one or more spatial dimensions (in 'Kaluza–Klein'-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO( N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization ( d=1 spatial dimensions) or to the magnetoelectric polarizability ( d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

The prediction of nontrivial topological phases in Bloch insulators in three dimensions has recently been experimentally verified. Here, I provide a picture for obtaining the Z ₂ invariants for a three-dimensional (3D) topological insulator by deforming suitable 2D planes in momentum space and by using a formula for the 2D Z ₂ invariant based on the Chern number. The physical interpretation of this formula is also clarified through the connection between this formulation of the Z ₂ invariant and the quantization of spin Hall conductance in two dimensions.

Generic non-magnetic disorder effects on topological quantum critical points (TQCP), which intervene between the three-dimensional (3D) topological insulator and an ordinary insulator, are investigated in this work. We first show that, in such 3D TQCP, any backward-scattering process mediated by chemical-potential-type impurity is always cancelled by its time-reversal ( -reversal) counter-process because of the nontrivial Berry phase supported by these two processes in the momentum space. However, this cancellation can be generalized into only those backward-scattering processes that conserve a certain internal degree of freedom, i.e. the parity density, while the 'absolute' stability of the TQCP against any non-magnetic disorders is required by the bulk-edge correspondence. Motivated by this, we further derive the self-consistent Born-phase diagram and the quantum conductivity correction in the presence of generic non-magnetic disorder potentials. The distinctions and similarities between the case with only the chemical-potential-type disorder and that with the generic non-magnetic disorders are summarized.

A time-reversal (TR) invariant topological insulator can be generally defined by the effective topological field theory with a quantized θ coefficient, which can only take values of 0 or π. This theory is generally valid for an arbitrarily interacting system and the quantization of the θ invariant can be directly measured experimentally. Reduced to the case of a non-interacting system, the θ invariant can be expressed as an integral over the entire three-dimensional Brillouin zone. Alternatively, non-interacting insulators can be classified by topological invariants defined over discrete TR invariant momenta. In this paper, we show the complete equivalence between the integral and the discrete invariants of the topological insulator.

Employing first-principles calculations, we studied the electronic structure of ultrathin Bi–Sb films, focusing on the appearance of surface or edge states that are topologically protected. Our calculations show that in ordered structures the Bi–Sb bonds are quite strong, forming well-defined double layers that contain both elements. We find surface states appearing on the (111) surface of a thin film of layerwise ordered Bi–Sb compound, while thin films in (110) orientation are insulating. In the gap of this insulator, edge states can be found in a (110)-oriented ribbon in the A17 (black phosphorus) structure. While these states are strongly spin polarized, their topological properties are found to be trivial. In all structures, we investigate the influence of spin–orbit coupling and analyze spin polarization of the states at the boundaries of the material.

The quantum spin Hall effect shares many similarities (and some important differences) with the quantum Hall effect for electric charge. As with the quantum (electric charge) Hall effect, there exists a correspondence between bulk and boundary physics that allows one to characterize the quantum spin Hall effect in diverse and complementary ways. In this paper, we derive from the network model that encodes the quantum spin Hall effect, namely the so-called network model, a Dirac Hamiltonian in two dimensions. In the clean limit of this Dirac Hamiltonian, we show that the bulk Kane–Mele invariant is nothing but the SU(2) Wilson loop constructed from the SU(2) Berry connection of the occupied Dirac–Bloch single-particle states. In the presence of disorder, the nonlinear sigma model (NLSM) that is derived from this Dirac Hamiltonian describes a metal–insulator transition in the standard two-dimensional symplectic universality class. In particular, we show that the fermion doubling prevents the presence of a topological term in the NLSM that would change the universality class of the ordinary two-dimensional symplectic metal–insulator transition. This analytical result is fully consistent with our previous numerical studies of the bulk critical exponents at the metal–insulator transition encoded by the network model. Finally, we improve the quality and extend the numerical study of boundary multifractality in the topological insulator. We show that the hypothesis of two-dimensional conformal invariance at the metal–insulator transition is verified within the accuracy of our numerical results.

As for a generic parameter-dependent Hamiltonian with time reversal (TR) invariance, a non-Abelian Berry connection with Kramers (KR) degeneracy is introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many-body system with KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without KR degeneracy. Natural dimensions to have nontrivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without KR degeneracy, which describe the monopoles of Dirac and Yang. We have shown that the Yang monopole is literally a quaternionic Dirac monopole.

The generic Berry phases with/without KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry-protected -quantization of these general Berry phases, a sufficient condition of the -quantization is given as the inversion/reflection equivalence.

Topological charges of the SO(3) and SO(5) nonlinear σ-models are discussed in relation to the Chern numbers of the CP ¹ and HP ¹ models as well.

This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have nontrivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, a relation that can be used to investigate the robustness of the edge states against disorder. The paper focuses on the motivations behind creating such tools and on how to use them.