Session Y32: Topological Insulators: General Theory

Z$_Q$ topological invariants for identification of short rangeentangled states

Since the Berry phase is quantized as $Z_2$ value, i.e., 0 or $\pi$, due to the time-reversal, or lattice-inversion symmetry in any dimension, the quantized Berry phase[1] is useful for characterization of a topological or quantum order in various models including strongly correlated electron systems[2] and spin systems[3]. Recently, we have proved $Z_Q$ quantization of Berry phases for the general lattice symmetry, where $Z_{Q}$ ($Q=d+1$) Berry phases are defined for $d$-dimensional lattices: Polyacetylene, Kagome and Pyrochlore lattice respectively for $d=1,2$ and $3$.[4]. We have also characterized the dimer-plaquette transition of the orthogonal dimer model in two dimension[5]. [1]{Y.Hatsugai, J. Phys. Soc. Jpn, {\bf 75}, 123601 (2006).} [2]{IM, Y. Hatsugai , J. Phys. Soc. Jpn, {\bf 76}, 113601 (2007).} [3]{IM, T.Hirano, Y.Hatsugai, Phys. Rev. B. {\bf 79}, 115107 (2009)}[4]{Y.Hatsugai, IM, Euro. Phys. Lett., {\bf 95}, 20003 (2011) }[5] I.Maruyama, S.Tanaya, M.Arikawa, Y.Hatsugai, J. Phys.:Conf. Ser., {\bf 320}, 01219 (2011)   

Trace index and spectral flow in the entanglement spectrum of topological insulators

We investigate the entanglement spectra of topological insulators, which manifest edge states on a lattice with spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a ``trace index.'' We find that any free-fermion topological insulator that exhibits spectral flow has a nonvanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics.   

Axions strings and Goldstone-Wilczek currents in quantum spin systems and topological insulators

Axion, originally a high-energy physics entity, has in recent years come into focus in the condensed matter community due to its incarnation in the physics of topological insulators. Much of the discussion so far has concentrated on the physics at the interface/junction of these insulators, which can be viewed as axion domain walls. Here we look into a slightly different situation: axion strings, which are vortex configurations of the axion fields. We discuss how these objects can arise in quantum spin systems and topological insulators,along with their electrodynamic and elastic implications. We also point to some novel Goldstone-Wilczek fermionic currents which can be relevant to the systems considered.   

Casimir Interaction Between Topological Insulator Thin Films and Graphene Sheets

The Casimir effect is a peculiar manifestation of quantum vacuum fluctuations of the electromagnetic field, resulting in an attractive force between closely-spaced conductors. Because of the advent of new materials like graphene and topological insulators, it is of fundamental and potentially also of practical interest to investigate the Casimir effect between conductors described by two-dimensional Dirac models. In this work we study the Casimir effect between topological insulator thin films and graphene planes in the presence of a time-reversal breaking perturbation, most practically an external magnetic field, that gives rise to a half-quantized quantum Hall effect. We evaluate the Casimir energies and forces from the reflection properties of the double layer system. We demonstrate the essential role that is normally played by the Dirac system's dissipative longitudinal conductivity which is neglected in topological field theory descriptions of the Casimir effect. We also show that repulsive Casimir forces are possible in the presence of a magnetic field.   

Exact duality between symmetry protected topological order and intrinsic topological order

The discovery of topological insulator(TI) motivates the intensive study of symmetry protected topological(SPT) order. Different from the intrinsic topological order, SPT order is only distinguishable from a trivial disorder phase when certain symmetry is preserved. Indeed, SPT order has a long history in 1D, it has been shown the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems with a time reversal symmetry. Until very recently, it was realized that SPT order also exists in interacting boson/spin systems in higher dimensions. In this talk, I will show in 2D boson/spin systems, there exists an interesting duality map between the intrinsic topological order and the SPT order. The duality map implies the SPT orders are stable and distinguishable against arbitrary perturbations if the symmetry is preserved. I will focus on a simplest exact solvable model with an Ising symmetry and discuss the nature of its symmetry protected low-lying edge excitations. The duality map can of course be generalized into arbitrary symmetry group $G$ in any dimension. In principle, the duality map is also applicable for interacting fermion/electron systems.   

Asymmetric spatial structure of zero modes for birefringent Dirac fermions

We study the zero energy modes that arise in an unusual vortex configuration involving both the kinetic energy and an appropriate mass term in a model which exhibits birefringent Dirac fermions as its low energy excitations. We find the surprising feature that the ratio of the length scales associated with states centered on vortex and anti-vortex topological defects can be arbitrarily varied but that fractionalization of quantum numbers such as charge is unaffected. We discuss this situation from a symmetry point of view and present numerical results for a specific lattice model realization of this scenario.   

Location of Singularities in tight-binding wave function of loop-ordered states with Chern numbers

As a pedagogic exercise, we consider time-reversal violating tight-binding wave-functions for Haldane states in the Graphene model and in a Copper-Oxide model. We locate the singularities in the wave-functions and derive the movement in their location as well as the change in the Berry phase with the choice of Gauge.   

Effects of spin-orbit interaction on a triangular lattice potential patterned two-dimensional electron gas

We have studied theoretically the electronic states of a two-dimensional electron gas (2DEG) in the presence of a triangular lattice of muffin-tin potentials. In particular, spin-orbit interaction due to the in-plane potential gradient is included. At the $K $and $K$' points in the Brilluoin zone, the Dirac cones open up a gap and their respective Berry curvatures are of the same sign which, in turn, depends on the electron spin. This contrasting feature, from that of graphene, is related to the fact that inversion symmetry is preserved in our case. The system thus exhibits Z$_{2}$ physics.   

Smooth gauge for topological insulators

We develop a technique for constructing Bloch functions for ${Z}_2$ quantum spin-Hall insulators that are smooth functions of ${\bf k}$ on the whole Brillouin-zone torus. As the initial step, the occupied subspace of the insulator is decomposed into a direct sum of two ``Chern bands,'' i.e., topologically non-trivial subspaces with opposite Chern numbers. This decomposition remains robust independently of underlying symmetries or specific model features. Starting with the Chern bands obtained in this way, we construct a topologically non-trivial unitary transformation that rotates the occupied subspace into a direct sum of topologically trivial subspaces. The possibility of using such a transformation is validated, and the entire procedure is illustrated, by applying it to the Kane-Mele model.   

Stable nontrivial $Z_2$ topology in ultrathin Bi (111) films: a first-principles study

Recently, there have been intense efforts in searching for new topological insulator (TI) materials. Based on first-principles calculations, we find that all the ultrathin Bi (111) films are characterized by a nontrivial $Z_2$ number independent of the film thickness, without the odd-even oscillation of topological triviality as commonly perceived. The stable nontrivial $Z_2$ topology is retained by the concurrent band gap inversions at multiple time-reversal-invariant k-points and associated with the intermediate inter-bilayer coupling of the multi-bilayer Bi film. Our calculations further indicate that the presence of metallic surface states in thick Bi(111) films can be effectively removed by surface adsorption.   

Breakdown of Landau-Ginzburg-Wilson Scheme by Spontaneous Orbital Currents in Zero-gap Semiconductors

Critical phenomena of phase transitions are classified into a small number of universality classes. For understanding such symmetry-breaking transitions, a Landau-Ginzburg- Wilson (LGW) scheme greatly helps us. Here, we study the critical phenomena of topological Mott insulator (TMI)[1] where spontaneous orbital currents induced by electron correlation bring the system to topological insulator by formally following the spirit of LGW. Then an unconventional criticality surprisingly emerges around the quantum critical point of TMI: It is governed not only by the symmetry as in the LGW framework but also by topology through electron dispersions and spatial dimension. We demonstrate it by studying an extended Hubbard model on several lattices such as honeycomb and pyrochlore lattice[2] within the Hartree-Fock mean-field level. Both fully numerical calculations and analytical calculations using effective band dispersions are performed. \\ $[1]$ S. Raghu {\it et al}., Phys. Rev. Lett. {\bf 100} 156401 (2008) \\ $[2]$ M. Kurita {\it et al}., J. Phys. Soc. Jpn {\bf 80} 044708 (2011)   

Transition in topological phase of Ge-Sb-Te and Ge-Bi-Te heterostructures

We studied the transition of topological phase of ternary chalcogenide compounds using first-principles methods. Our calculations show that they undergo a transition between band insulator and topological insulator phases upon the variation of atomic composition and layer thickness. In order to understand the transition, we developed a model Hamiltonian of 1D Dirac fermionic superlattice. We introduced to the Hamiltonian an effective interaction between Dirac fermions that is mediated by the electrons from the band insulating layers. We obtained the phase diagram of the transition as a function of materials parameters including layer thickness, band gap, and Dirac fermionic mass gap. We also discussed the implication of our ressults to the conducting properties of chalcogenide compounds   

Nontrivial topological effects on the surface of strong and weak topological insulators and superconductors

We investigate states on the surface of strong or weak topological insulators and superconductors that have been gapped by a magnetic material or by a charge density wave. The surface of a strong 3D topological insulator gapped by a magnetic material is well known to possess a half quantum Hall effect. Furthermore, a recent paper (arXiv:1110.3420) showed that the surface of a weak 3D topological insulator gapped by charge density wave has a half quantum spin Hall effect. We generalize these results to all classes of topological insulator and superconductors. We find that if, in d-1 dimension and that symmetry class, there is a $Z$ or $Z_2$ topological invariant, then the resulting surface state may have a nontrivial topological phase.