Session J24: Topological Insulators: Theory

Classifying Topological Defects in Insulators and Superconductors

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider a Hamiltonian $\mathcal{H}({\bf k},{\bf r})$ that varies slowly with adiabatic parameters ${\bf r}$ away from the defect. Band theories are grouped into ten classes according to the presence or absence of anti-unitary symmetries, time reversal $\Theta^2=\pm1$ and/or particle-hole $\Xi^2=\pm1$. Both send ${\bf k}\mapsto-{\bf k}$ and ${\bf r}\mapsto{\bf r}$. Stable classification of topological band theories are characterized by a unified set of integral formulae for all the symmetry classes in any dimensions. Examples that fall into this framework include edge and surface states along an interface, 1D chiral, helical and Majorana modes along a line defect, bound charge and Majorana zero mode at a point defect. This approach also applies to time dependent phenomena, such as the Thouless charge pumb, the $Z_2$ spin pumb and the exchange statistics of Majorana bound states in three dimensions.   

Equivalent topological invariants of topological insulators

A time-reversal invariant topological insulator can be generally defined by the effective topological field theory with a quantized theta coefficient, which can only take values of 0 or pi. This theory is generally valid for an arbitrarily interacting system and the quantization of the theta invariant can be directly measured experimentally. Reduced to the case of a non-interacting system, the theta 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 time-reversal invariant momenta. In this paper, we show the complete equivalence between the integral and the discrete invariants of the topological insulator.   

Collective modes of a helical liquid

We study low energy collective modes and transport properties of the ``helical metal" on the surface of a topological insulator. At low energies, electrical transport and spin dynamics at the surface are exactly related by an operator identity equating the electric current to the in-plane components of the spin degrees of freedom. From this relation it follows that an undamped spin wave always accompanies the sound mode in the helical metal -- thus it is possible to `hear' the sound of spins. In the presence of long range Coulomb interactions, the surface plasmon mode is also coupled to the spin wave, giving rise to a hybridized ``spin-plasmon" mode. We make quantitative predictions for the spin-plasmon in ${\rm Bi}_2{\rm Se}_3$, and discuss its detection in a spin-grating experiment.   

Topological Insulators on the Decorated Honeycomb Lattice

We show that the decorated honeycomb lattice supports a number of topological insulating phases with a non-trivial $Z_2$ invariant and time-reversal symmetry protected gapless edge modes. We investigate the stability of these phases with respect to various symmetry breaking perturbations and demonstrate the connection to the recent discovery of an exactly solvable $S=1/2$ chiral spin liquid model [Phys. Rev. Lett. {\bf 99}, 247203 (2007)] with non-Abelian and Abelian excitations on the same lattice. Our work highlights the relationship between topological band insulators and topologically ordered spin systems, and points to promising avenues for enlarging the number of known examples of both.   

Interacting topological insulators on the kagome and decorated honeycomb lattices

We study the phase diagram of an extended Hubbard model on the kagome and decorated honeycomb lattice. At the mean-field level a rich set of conventional and topological phases emerges, including some where spin-orbit coupling is dynamically generated and a $Z_2$ topological band insulator with time-reversal symmetry appears. We discuss the connection between different topological phases, including those involving broken time-reversal symmetry in the strong interaction limit.   

Inducing topological order in 2D using a metallic layer

In our work we consider the possibility of inducing a topological insulator phase in a honeycomb lattice using a metallic gate. We start with a simple nearest-neighbor tight-binding model on a two dimensional honeycomb lattice, without spin-orbit interaction. We then add a metallic layer above the honeycomb sheet and allow density-density interaction without particle tunneling. After integrating out the metal, this induces complex hopping terms with range that depends on the Fermi wavelength of the metal. A strong next-nearest-neighbor hopping, as in the Haldane model, can be achieved by tuning the Fermi wavelength to the honeycomb next-nearest-neighbor distance. This effect might be used to enhance the weak intrinsic spin-orbit coupling in graphene or similar systems to drive the system to a topological insulator phase.   

Vertical Transport in Topological Insulator Thin Films

We present a theory of inter-surface transport in topological insulator thin film. We calculate the transport between two 2-dimensional Dirac fermion surfaces using a phenomenological model with band parameters, obtained from Density Functional calculations. Resonant tunneling between surfaces is absent in the absent of external fields that break inversion symmetry. More generally tunneling is strongest when the Fermi level lies in the conduction band on one surface and in the valence band on the other surface. We discuss manipulation of vertical transport by dual gates and by in-plane magnetic fields and compare with other 2-dimension to 2-dimension tunneling systems.   

Effect of surface curvature on conductivity in 3D topological insulator

The surface spectrum of a three-dimensional (3D) topological insulator consists of massless Dirac fermions. Hence, an electron moving on a curved surface in a 3D topological insulator follows a geodesic trajectory, akin to a photon in curved space. In this work, we study electron scattering due to the surface roughness (defects) modeled as a curved surface. The resulting effect on conductivity is estimated using the Boltzmann Equation. This scattering mechanism leads to a distinguishable signature of the conductivity on the electron density.   

Metallic Line Defects in three-dimensional topological band insulators

Dislocations in topological insulators can host a one-dimensional metallic state that is topologically protected. We discuss experimental consequences for $Bi_{0.9} Sb_{0.1} $ alloys, including an unusual strain-induced conductivity effect. With a view to studying interaction effects, microscopic parameters for the one-dimensional metallic modes are derived, starting from a Liu-Allen tight-binding model. The Luttinger parameter for the one-dimensional metal in $Bi_{0.9} Sb_{0.1} $ is estimated. A different route to a metallic defect line is found in model systems where SU(2) spin rotation symmetry is spontaneously broken, leading to a topological insulator. Line defects of the order parameter are found to be metallic, if a strong topological insulator is realized. We study models exhibiting this phase on the diamond and ideal wurtzite lattices. Prospects for experimental realizations are discussed.   

Wannier representation of $Z_2$ topological insulators

We consider the problem of constructing Wannier functions for $Z_2$ topological insulators. For Chern insulators it is well known that there is a topological obstruction to the construction of Wannier functions, and one may wonder whether this is also true in the $Z_2$ case. We consider a model system for the $Z_2$ problem in 2D. In the $Z_2$-even phase the system is an ordinary insulator, and the usual projection-based scheme can be used to build the Wannier representation. In the $Z_2$-odd phase we do find a topological obstruction, but only if one insists on choosing a gauge that respects the time-reversal symmetry, corresponding to Wannier functions that come in time-reversal pairs. If instead we are willing to violate this gauge condition, a Wannier representation becomes possible. We present a scheme for constructing Wannier functions for the $Z_2$-odd phase, showing explicitly that the Wannier functions do not come in Kramers pairs despite the presence of time-reversal symmetry.   

Calculation of the axion magnetoelectric coupling

Recently it was shown [X.-L.\ Qi {\it et al.}, PRB {\bf 78}, 195424 (2008); A.M.\ Essin {\it et al.}, PRL {\bf 102}, 146805 (2009)] that there exists a purely isotropic (``axionic'') component $\theta$ to the magnetoelectric coupling (MEC). Furthermore, this $\theta$ arises only from the electron orbital motion, and in strong Z$_2$ topological insulators it is unusually large and equals exactly half a quantum ($\theta=\pi$). Experimental observation of this large MEC would require some peculiar breaking of the time-reversal ($T$) symmetry at the surfaces, but $\theta$ might be observed in normal insulators that have $T$ already broken in the bulk. Since there are by now several examples of strong Z$_2$ topological insulators having $\theta=\pi$, we believe there is no strong reason why $\theta$ should necessarily be small in a normal insulator with broken $T$. For this reason, we have used density-functional theory to calculate $\theta$ in various materials. We first consider Cr$_2$O$_3$, a widely studied magnetoelectric material, but we find $\theta$ to be very small there. We attribute this to a weak spin-orbit effect in Cr (and to the fact that even a strong spin-orbit effect by itself does not guarantee a large $\theta$). To calculate $\theta$ we express it in terms of well localized Wannier functions to ensure smoothness of the gauge and also to allow for decomposition of contributions to $\theta$ coming from various electronic bands. The calculation of $\theta$ for BiFeO$_3$ and other materials is currently ongoing.   

Electromagnetic response in a quantum spin Hall insulator with strong correlation

The quantum spin Hall system is a time-reversal invariant band insulator with a non-trivial topological electronic structure caused by the spin-orbit coupling. We investigate such a system with strong electron correlation. The on-site correlation can be expressed by the spin gauge field coupled to the electron spin $(s_x,s_y,s_z)$ in the strong coupling limit. Electromagnetic response of this system can be derived to integrate out Fermions and also spin gauge field. We find that the system shows superconducting response when $s_z$ is conserved, and becomes insulating when $s_z$-conservation is broken by a perturbation like the Rashba term. In our discussions an induced BF-term, which is the topological term with mixing of electromagnetic gauge field and spin gauge field coupled to $s_z$, plays an important role.