Session A6: Topological Insulators: Novel States in Topological Insulators

Noncommutative Magneto-Electric Responses of Topological Insulators

Topological magneto-electric response, constructed on a Brillouin torus, defines a $Z_2$ invariant and classifies topological phases. In the presence of disorder or B field, the notion of Brillouin torus is destroyed. This problem can be overcome by using noncommutative geometry. Starting from a generic 3D lattice model, we derive the magneto-electric response on a noncommutative Brillouin torus. Our result is a noncommutative topological formula. We show that its topological stability requires only mobility gap, therefore the robustness extends to strong disorder regime. Our formula doesn't involve gauge or twisted boundary condition, hence can be computational effective.   

Heavy Adatoms on Magnetic Surfaces: A Search for Chern Insulators

The theoretical possibility of a quantum anomalous Hall (QAH) insulator, or Chern insulator, has been known for many years, and several strategies for achieving this topological phase have been proposed. However, no unambiguous experimental realization is yet in hand. In this work, we propose a new QAH search strategy and verify its viability with first-principles calculations. We propose constructing a QAH insulator by depositing a layer of adatoms with large spin-orbit coupling (e.g., Pb, Bi) on the surface of a magnetic insulator. We argue that such systems will typically have surface bands with non-zero Chern numbers, so that if metallic, they will typically have a large anomalous Hall conductivity. Thus, the search for Chern insulators reduces to looking for examples exhibiting a global gap across the entire BZ. Using first-principles calculations, we search through many examples of heavy elements on MnTe, MnSe, and EuS surfaces. Our search reveals several Chern insulators with band gaps of up to 0.14 eV, which may be promising targets for future experimental investigations.   

Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition

A field theory of the Anderson transition in two dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which localization is driven by the proliferation of vortex-like topological defects. The sign of the vortex fugacity determines the $Z_2$ topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator respectively. The critical conductivity and correlation length exponent of these transitions are computed in a $N=1-\epsilon$ expansion in the number of replicas, where for small $\epsilon$ the critical points are perturbatively connected to the Kosterlitz Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.   

Floquet Majorana Fermions for Topological Qubits

We develop an approach to realizing a topological phase transition and non-Abelian statistics with dynamically induced Floquet Majorana Fermions (FMFs). When the periodic driving potential does not break fermion parity conservation, FMFs can encode quantum information. Quasi-energy analysis shows that a stable FMF zero mode and two other satellite modes exist in a wide parameter space with large quasi-energy gaps, which prevents transitions to other Floquet states under adiabatic driving. We also show that in the asymptotic limit FMFs preserve non-Abelian statistics and, thus, behave like their equilibrium counterparts.   

Emergent Supersymmetry in Topological Superconductors

In the absence of interactions, topological superconductors (TSC) host helical Majorana fermion edge states protected by time reversal symmetry. Increasing interactions can lead to spontaneous magnetic order at the boundary. We show that the associated quantum phase transition, if continuous, has emergent space-time supersymmetry at low energies. The magnetic order parameter field is identified as the superpartner of the Majorana fermions. These results are obtained using field theoretical arguments and are verified by numerical DMRG solution of a lattice model that mimics the physics of the phase transition. The emergent supersymmetry, accessed by tuning a single parameter, has striking consequences such as an exact relation between the correlation functions of fermions and those of the order parameter. Similar results are argued to hold for the 2+1 dimensional boundary of a bulk topological superconductor. Generalization to topological insulator surfaces will be mentioned.   

Dynamical manipulation of 2D topological insulator edge states for Majorana fermion braiding

Edge states of 2D topological insulators such as HgTe provide a promising platform for realizing Majorana modes. Networks required for braiding Majoranas along the edge channels can be obtained by adjoining HgTe quantum wells to form corner junctions. Physically cutting quantum wells for this purpose, however, presents technical challenges. Here we propose a more accessible means of forming networks that relies on dynamically manipulating the location of edge states inside of a \emph{single} HgTe sheet. In particular, we show that edge states can effectively be dragged into the system's interior by gating a region near the edge into a metallic regime and then removing the resulting gapless carriers via proximity-induced superconductivity. This method allows one to construct rather general quasi-1D networks along which Majorana modes can be shuttled and exchanged by electrostatic means.   

Topological Excitonic Superfluids in Three Dimensions

We study the equilibrium and non-equilibrium properties of topological dipolar intersurface exciton condensates within time-reversal invariant topological insulators in three spatial dimensions without a magnetic field. We elucidate that, in order to correctly identify the proper pairing symmetry within the condensate order parameter, the full three-dimensional Hamiltonian must be considered. As a corollary, we demonstrate that only particles with similar chirality play a significant role in condensate formation. Furthermore, we find that the intersurface exciton condensation is not suppressed by the interconnection of surfaces in three-dimensional topological insulators as the intersurface polarizability vanishes in the condensed phase. This eliminates the surface current flow leaving only intersurface current flow through the bulk. We conclude by illustrating how the excitonic superfluidity may be identified through an examination of the terminal currents above and below the condensate critical current.   

Quantum interference and quantum oscillation on the surface of mirror symmetric topological insulators

We first study the quasiparticle interference (QPI) of the surface states in crystalline topological insulators which possess mirror symmetry and time-reversal symmetry, by analyzing the Fourier transformed local density of states (FT-LDOS), $\rho(bq,\omega)$ around a single static impurity on the surface. The topological characters of the surface states of these new materials lead to QPI patterns distinct from those of 2D metals and of surface states on 3D time-reversal TI's. We apply the general analysis to the QPI on the $\langle{001}\rangle$-surface of Pb$_{1-x}$Sn$_x$Te and predict all vanishing singularities in $\rho(bq,\omega)$. We also demonstrate that QPI can also be used to probe the Lifshitz transition of the surface states observed in recent ARPES experiment. We next study the Shubnikov de Hass oscillation of these surface states and show that the oscillation has a single period before the Lifshitz transition and two distinct periods after the transition. Adding in-plane magnetic field before the Lifshitz transition leads to splitting of the period into two close periods and a beating thereof.   

Chern and Majorana Modes in QuasiCrystals

The topology of quasicrystals is found to have a novel manifestation in the spatial profile of band edge states as topological invariants transform peaks into doublets of size equals the Chern number. The Chern-dressed peaks form a self-similar pattern encoding topological fingerprints at all length scales. For quasicrystals exhibiting localized states, fluctuations about exponentially localized zero modes describe the onset to topological transition where Majorana modes delocalize. These exotic modes can be captured in their entirety using $U(1)$ symmetry breaking perturbation that supports both the Chern and the Majorana modes. Here topological transition is accompanied by localization as edge-localized modes move to the interior, loosing topological protection.   

Vortex zero modes, chiral anomaly and effective field theory for odd parity topological superconductor in three dimensional Dirac materials

The low energy quasiparticle dispersion of various narrow gap and gapless semiconductors are respectively described by three dimensional massive and massless Dirac fermions. The three dimensional Dirac spinor structure admits an interesting time-reversal invariant, odd parity and Lorentz pseudo-scalar topological superconducting state. In this talk we demonstrate the existence of fermion zero energy states in the vortex core of this odd parity topological superconductor under generic conditions. Guided by the existence of the zero modes and its intimate connection with the chiral anomaly and the index theorem, we derive an effective topological field theory for such a superconducting state. We also discuss the experimental consequences of the zero modes and the topological field theory for the low temperature unconventional superconducting states of copper intercalated bismuth selenide, and indium doped tin telluride.   

Ab initio study of topological insulating property in the heterojunction of Bi (111) bilayer and Bi$_{2}$Te$_{2}$Se

Study of topological insulator (TI) is showing rapid progress in both theory and experiment, particularly on three dimensional materials. Examples of two dimensional TI (quantum spin Hall) materials are, on the other hand, comparatively less common. As such, theoretical predictions of single Bi (111) bilayers as TI draw great attention from experiment. A recent report of ARPES measurements claims verification of the 2D TI property of Bi bilayer, but analysis leaves much room for further clarification. Because Bi (111) bilayers grown on 3D TI substrates such as Bi$_{2}$Te$_{3}$ and Bi$_{2}$Te$_{2}$Se; understanding the details of the interface between the Bi bilayer and 3D TI substrates is essentially required. We study the electronic structures of Bi bilayer-Bi$_{2}$Te$_{2}$Se heterojunction from first-principles calculations. We find a substantial shift of Dirac cone from the TI substrates into Bi bilayer that thus becomes metallic on TI substrates. It is shown that the origin of this change is the inversion-symmetry breaking in Bi bilayer due to TI substrate. By comparing our calculations of Bi bilayer nanoribbons on Bi$_{2}$Te$_{2}$Se with STM/STS measurements, we successfully resolve and locate the edge states of a single Bi bilayer and confirm its 2D TI character.   

Topological indices, defects, and Majorana fermions in chiral superconductors

We study theoretically the role of topological invariants to protect the Majorana fermions in a model of two-dimensional (2D) chiral superconductors which belong to class D of the topological periodic table. A rich phase diagram is revealed. Each phase is characterized by the topological invariants for 2D (Z) and 1D (Z2), which lead to the Majorana fermion at the edge dislocation and the core of the vortex. The topological invariants are determined by the hopping integrals along x and y directions. Interference of the Majorana fermions originating from the different topological invariants is studied. The interaction between zero-energy states at edge dislocations and at vortex cores eliminates the zero-energy states when they coexist at the same position. The stability of the Majorana fermion with respect to the interlayer coupling, i.e., in 3D, is also examined. We found that the zero-energy state survive a finite hopping integral along the z-direction unless the energy gap closes.   

On the possibility of the fractional ac Josephson effect and doubled Shapiro steps in non-topological Josephson junctions

Topological superconductors supporting Majorana Fermions with non-abelian statistics are presently a subject of intense theoretical and experimental effort. It has been proposed that the observation of a half-frequency or a fractional Josephson effect is a more reliable test for topological superconductivity than the search for end zero modes. In fact, the fractional Josephson effect has been observed for the semiconductor nanowire system in the form of doubled Shapiro states. Here we consider the possibility of seeing such a fractional ac Josephson and doubled Shapiro steps from a superconducting nanowire in the non-topological phase. Using a semiclassical treatment we find that both the fractional ac Josephson effect and the doubled Shapiro step can, in principle, occur in the non-topological phase because of non-dynamical Landau Zener processes associated with the Andreev bound state spectrum of the junction. Therefore, while the observation of doubled Shapiro steps can be taken as indicative of a topological phase, it may not be a smoking gun signature for topological superconductivity and Majorana fermions.   

Majorana Flat Bands and Uni-directional Majorana Edge States in Gapless Topological Superconductors

In this work, we show that an in-plane magnetic field can drive a fully gapped $p \pm i p$ topological superconductor into a gapless phase which supports Majorana flat bands (MFBs). Unlike previous examples, the MFBs in the gapless regime are protected from disorder by a chiral symmetry. In addition, novel uni-directional Majorana edge states (MESs) which propagate in the same direction on opposite edges appear when the chiral symmetry is broken by Rashba terms. Unlike the usual chiral or helical edge states, uni-directional MESs appear only in systems with a gapless bulk. We show that the MFBs and the uni-directional MESs induce nearly quantized zero bias conductance in tunneling experiments.