Session Z32: Topological Insulators: General Theory II

Room temperature quantum spin Hall thin film for buckled honeycomb structures of silicon, germanium, and tin

We have carried out first-principles calculations on buckled honeycomb structures of silicon, germanium and tin. When the spin-orbit coupling is included in the computations, these two dimensional (2D) systems are found to be in a quantum spin Hall phase with nontrivial topological invariant Z$_2$=-1, and a band gap that increases with increasng atomic number. In particular, we predict that buckled honeycomb Sn thin film is a room temperature quantum spin Hall insulator with 0.25 eV band gap.   

Topological order and semions in a strongly correlated quantum spin Hall insulator

We provide a self-consistent mean-field framework to study the effect of strong interactions in a quantum spin Hall insulator on the honeycomb lattice. We identify an exotic phase for large spin-orbit coupling and intermediate Hubbard interaction. This phase is gapped and does not break any symmetry. Instead, we find a four-fold topological degeneracy of the ground state on the torus and fractionalized excitations with semionic mutual braiding statistics. Moreover, we argue that it has gapless edge modes protected by time-reversal symmetry but a trivial $Z_2$ topological invariant. Finally, we discuss the experimental signatures of this exotic phase. Our work highlights the important theme that interesting phases arise in the regime of strong spin-orbit coupling and interactions.   

Spontaneous breakdown of time reversal symmetry in the doped honeycomb lattice with enlarged unit cell

Enlarging the unit cell of simple lattice models is proposed as a simple means to get spontaneous breakdown of time reversal symmetry ($\mathcal{T}$) and to stabilize non trivial topological phases. As a case study we explore the nearest neighbor tight binding model for spinless fermions interacting through short range Coulomb interactions in the honeycomb lattice. Using a variational mean field approach and an enlarged six atom unit cell we obtain a very rich phase diagram as doping and interaction strength are varied. Two broken $\mathcal{T}$ phases, with orbital currents (fluxes) arranged in a Kekul\'e pattern, show up above the Van Hove filling. One of them realizes a topological Fermi liquid with a finite anomalous Hall conductivity. Instabilities towards charge modulated and superconducting phases are also discussed.   

Synchronous and Asynchronous Mott Transitions in Topological Insulator Ribbons

We address how the nature of linearly dispersing edge states of two dimensional (2D) topological insulators evolves with increasing electron-electron correlation engendered by a Hubbard like on-site repulsion $U$. We consider finite ribbons of two systems of topological band insulators with local electronic interactions incorporated. Using an inhomogeneous cluster slave rotor mean-field method developed here, we show that electronic correlations drive the topologically nontrivial phase into a Mott insulating phase via two different routes. In a synchronous transition, the entire ribbon attains a Mott insulating state at one critical $U$. In the second, asynchronous route, Mott localization first occurs on the edge layers at a smaller critical value of electronic interaction which then propagates into the bulk as $U$ is futher increased until all layers of the ribbon become Mott localized.   

Interaction-driven topological and nematic phases on the Lieb lattice

We investigate the interaction-driven instabilities of the band crossing point (BCP), which occurs in the band structure resulting from spinless/spinful fermions moving on the (extended) Lieb lattice. In the non-interacting limit, we show the topological stability of the BCP both from momentum and real space arguments, provided time reversal and $C_4$ point group symmetries are preserved. With short-range repulsive interactions, we find that at zero temperature this BCP is marginally unstable against infinitesimal repulsions and results in topological quantum anomalous/spin Hall, charge nematic, and nematic-spin-nematic phases, separately, depending on the interaction strengths. Possible physical realizations and the existence of a topological nearly flat band are also discussed.   

Geometric effects on surface states in topological insulator Bi$_{2}$Te$_{3 }$ nanowire

Bismuth Telluride (BT) is a 3D topological insulator (TI) with surface states that have energy dispersion linear in momentum and forms a Dirac cone at low energy. In this work we investigate the surface properties of a BT nanowire and demonstrate the existence of TI states. We also show how such states vanish under certain geometric conditions. An atomistic model (sp3d5s* TB) is used to compute the energy dispersion in a BT nanowire. Penetration depth of the surface states is estimated by ratio of Fermi velocity and band-gap. BT possesses a tiny band-gap, which creates small localization of surface states and greater penetration in to the bulk. To offset this large spatial penetration, which is undesirable to avoid a direct coupling between surfaces, we expect that bigger cross-sections of BT nanowires would be needed to obtain stable TI states. Our numerical work validates this prediction. Furthermore, geometry of the nanowire is shown to influence the TI states. Using a combined analytical and numerical approach our results reveal that surface roughness impact electronic structure leading to Rashba type splits along z-direction. Cylindrical and square cross-sections are given as illustrative examples.   

Spinless massless and massive Dirac fermions in a checkerboard lattice magnet

We investigated the theory of the interplay of itinerant electrons and localized magnetic moments on the frustrated checkerboard lattice as function of the super-exchange interaction between the localized moments and the band filling of fermions. We find that at half filling a very robust magnetic ``flux'' phase is lowest in energy. The ordering of the localized spins induces an effective gauge field flux of $\pi$ for the electrons. Consequently, this phase preserves time-reversal symmetry and the low-energy effective theory of the electrons is that of massless Dirac fermions, resembling the situation in graphene except that the spin degree of freedom is absent here. The robustness of this state originates from the geometrical frustration of the checkerboard lattice. In the crossover regime from this flux state and the satured FM state at vanishing super-exchange coupling, these Dirac fermions become massive with opposite sign of the mass at the two degeneracy points. This chiral spin state is then equivalent to a time-reversal breaking anomalous Quantum Hall phase, precisely in the way once envisioned by Haldane in graphene.   

Kondo lattice on the edge of a 2D topological insulator

Much attention has been devoted recently to the experimental and theoretical study of the effect of magnetic impurities on the stability of the gapless boundary modes of topological insulators. When the quantum dynamics of the impurities is considered, those boundary modes constitute novel types of fermionic baths which may affect the nature of possible impurity phases and phase transitions. We study a regular one-dimensional array of quantum magnetic impurities interacting with the helical edge liquid of a two-dimensional time-reversal invariant topological insulator. Exact solutions at the special Toulouse and Luther-Emery points as well as a renormalization group analysis \`{a} la Anderson-Yuval allow us to construct a phase diagram in the space of Kondo coupling, electron-electron interaction strength, and electron density. We point out similarities and differences with the Kondo lattice in a ordinary one-dimensional electron gas.   

Non-Abelian Berry transport, spin coherent states, and Majorana points

We consider the adiabatic evolution of Kramers degenerate pairs of spin states in a half-integer spin molecular magnet as the molecule is slowly rotated, for which it is possible to encounter non-Abelian Berry's phase. To understand the full details of the adiabatic quantum evolution, we invoke Majorana's parametrization of a general spin-$j$ state in terms of the $2j$ Majorana points. As an illustration we consider molecular magnets of the $j=9/2$ Mn4 family and compute the frequency with which the magnetization varies. This frequency is generally different from the frequency of the rotation.   

Exploring Diabolical points and Berry Phases in the Majorana Representation

Diabolical points or level crossings are often observed in spin hamiltonians, where they give rise to a Berry phase. We study their geometric structure using the Majorana representation which associates a spin state to a set of points on the Bloch sphere. Each crossing carries a Chern number which is found to be directly related to the wrapping of Majorana points around it. We apply our method to study model hamiltonians for molecular magnets in an external magnetic field, in which a rich pattern of diabolical points are seen. Our result enables a simple visualization of the Berry phases without the use of perturbation theory.   

Quasiparticle effects in the bulk and surface-state bands of Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$ topological insulators

We investigate the bulk band structures and the surface states of Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$ topological insulators using first-principles many-body perturbation theory based on the \textit{GW} approximation. The quasiparticle self-energy corrections introduce significant changes to the bulk band structures, surprisingly leading to a decrease in the direct band gaps in the band-inversion regime as opposed to the usual situation without band inversion. Parametrized ``scissors operators'' derived from the bulk studies are then used to investigate the electronic structure of slab models which exhibit topologically protected surface states. The introduction of self-energy corrections results in significant shifts of the surface-state Dirac point energies relative to the bulk bands and in enlarged gap openings from the interactions between the surface states across the thin slab, both in agreement with experimental data.   

Search for new topological insulators: ternary Li$_2$AgSb-class semiconductors and related compounds

Topological insulators host a rare quantum phase of electrons which is characterized by a topological invariant number of bulk states of combined spin-orbit and time-reversal symmetry origin. Despite recent progress the available classes of topological insulators are still quite limited for use in device applications and experimental exploration of exotic topological phenomena. For this reason, the search for new materials with greater structural flexibility and tunability in various local order broken symmetry phases is continuing worldwide with great intensity. Here we discuss our effort based on first-principles calculations to show that the adiabatic continuation method can provide a very powerful tool for predicting non-trivial topological phases with the example of ternary intermetallic series, Li$_2M'X$ ($M'$=Cu, Ag, Au, and Cd, $X$=Sb, Bi, and Sn) as well as other compounds with zinc-blende type sublattice. [1-3] Work supported by the Office of Basic Energy Sciences, US DOE.\\ \mbox{[1]} H. Lin, {\it et al.} Nature Materials \textbf{9}, 546 (2010). \\ \mbox{[2]} Y. J. Wang, {\it et al.} New J. Phys. {\bf 13}, 085017 (2011). \\ \mbox{[3]} H. Lin, {\it et al.}, arXiv:1007.5111.