Session P31: Topological Insulators: Disorder

Disordered topological conductor

A topological conductor, like a topological insulator is a system in which the bands are characterized by non-trivial topological invariants such as the Chern numbers. However, unlike a topological insulator, in this system the Fermi energy does not lie in an energy gap but instead intersects at least one of the bulk bands. Although not an insulator the topological conductor supports chiral edge modes. In this work we consider a disordered topological conductor and analyze its properties. In particular we find that moderate disorder reduces the edge conductivity from its quantum value and stronger disorder increases it before the whole system is localized and the conductivity drops to zero. This effect is seen numerically on a lattice system and analytically in a disorder averaged continuum model.   

Dislocations and their braiding in topological insulators

We demonstrate the fundamental importance of crystal lattice dislocations in two-dimensional topological insulators. These defects characterize the topological state through the appearance of electronic localized midgap states. The states turn out to be robust even for the class of materials where they are not protected. At the same time, these localized electronic states have interesting quantum properties. We show that adiabatic braiding of dislocations, which can be achieved using lattice shear induced dislocation glide, brings out the quantum statistics of the electronic bound states.   

Study of the robustness of two dimensional topological insulators

Two dimensional topological insulators present gapless spin filtered edge states which are topologically protected against backscattering. As long as disorder does not mix the states of opposite edges or with bulk ones, these states contribute to the two terminal conductance as a single quantum channel regardless of the amount of non-magnetic disorder present in the sample. We address this problem studying the effect of different types of disorder: constrictions and Anderson disorder, for two different materials that have been predicted to present the quantum spin Hall insulator phase, graphene and a bilayer of Bi(111). We also study the effect of the zigzag edge reconstruction of graphene over the robust behavior of the edge states. We describe their electronic structure using an orthogonal tight-binding model in the Slater-Koster approximation including the intra-atomic spin-orbit interaction. The conductance is computed using the Landauer formula making use of the ALACANT transport package.   

Investigating Impurities on 3D Topological Insulators with Multiple Scattering Theory

Recently a new class of materials, three-dimensional topological insulators (3DTI), have generated significant theoretical and experimental interest due to surface states (SS) exhibiting linear dispersion at a Dirac pont that are protected by time-reversal symmetry (TRS). Using magnetism to break TRS is of particular interest, especially via the deposition of magnetic impurities (MI) on the 3DTI surface. Experimental studies are in the early stages and consensus on the effect of MI of 3DTI SS has yet to be reached. Multiple scattering theory (MST) has proven useful in investigating surfaces, impurities and disordered systems and provides an ideal framework for first-principles, computational study. This presentation will report on progress in adapting MST to 3DTI systems with impurities.   

LDOS Multifractal Hunter's Guide to Dirty Topological Insulators

We compute the multifractal spectra associated to local density of states (LDOS) fluctuations due to weak quenched disorder, for a single Dirac fermion in two spatial dimensions. Our results are relevant to the surfaces of $Z_2$ topological insulators such as Bi$_2$Se$_3$ and Bi$_2$Te$_3$, where LDOS modulations can be directly probed via scanning tunneling microscopy. We find a qualitative difference in spectra obtained for magnetic versus non-magnetic disorder. Randomly polarized magnetic impurities induce quadratic multifractality at first order in the impurity density; by contrast, no operator exhibits multifractal scaling at this order for a non-magnetic impurity profile. For the time-reversal invariant case, we compute the first non-trivial multifractal correction, which appears at two loops (impurity density squared). We discuss spectral enhancement approaching the Dirac point due to renormalization, and we survey known results for the opposite limit of strong disorder.   

Effects of Strong Disorder in a 3-Dimensional Topological Insulators: Phase Diagram and Mapping of the Z2 Invariant

We study the effect of strong disorder in a 3-dimensional topological insulators with time-reversal symmetry and broken inversion symmetry. Using level statistics analysis, we demonstrate first the persistence of delocalized bulk states even at large disorder. The delocalized spectrum displays the levitation and pair annihilation effect, indicating that the delocalized states continue to carry the Z2 invariant after the onset of disorder. The Z2 invariant is computed via twisted boundary conditions using a novel and efficient numerical algorithm. We demonstrate that the Z2 invariant remains well defined and quantized even after the spectral gap closes and becomes filled with dense localized states. In fact, our results indicate that the Z2 remains quantized until the mobility gap closes or until the Fermi level touches the mobility edge. Based on such data, we compute the phase diagram as function of disorder strength and position of the Fermi level.   

Strong potential impurities on the surface of a three-dimensional topological insulator

Topological insulators (TIs) are said to be stable against non-magnetic impurity scattering due to suppressed backscattering in the Dirac surface states. We solve a lattice model of a three-dimensional TI in the presence of strong potential impurities on the surface and find that both the Dirac point and low-energy states are significantly modified: low-energy impurity resonances are formed that produce a peak in the density of states near the Dirac point, which is destroyed and split into two nodes that move off-center. The impurity-induced states penetrate up to 10 layers into the bulk of the TI. These findings demonstrate the importance of bulk states for the stability of TIs and how they can destroy the topological protection of the surface. Extensions to sub-surface and extended defects, as well as direct comparisons to recent experimental results are also made.   

Disorder induced quantized conductance with fractional value and universal conductance fluctuation in three-dimensional topological insulators

We report a theoretical investigation on the conductance and its fluctuation of three-dimensional topological insulators (3D TI) in \textit{Bi}2\textit{Se}3 and \textit{Sb}2\textit{Te}3 in the presence of disorders. Extensive numerical simulations are carried out. We find that in the diffusive regime the conductance is quantized with fractional value. Importantly, the conductance fluctuation is also quantized with a universal value. For 3D TI connected by two terminals, three independent conductances $G$\textit{zz}, $G$\textit{xx} and $G$\textit{zx} are identified where z is the normal direction of quintuple layer of 3D TI. The quantized conductance are found to be $\left\langle {G_{zz} } \right\rangle $ = 1, $\left\langle {G_{xx} } \right\rangle $ = 4/3 and $\left\langle {G_{zx} } \right\rangle $ = 6/5 with corresponding quantized conductance fluctuation 0.54, 0.47, and 0.50. The quantization of average conductance and its fluctuation can be understood by theory of mode mixing. The experimental realization that can observe the quantization of average conductance is discussed.   

Conducting state of GeTe by defect-induced topological insulating order

Topological insulating order protected by time-reversal symmetry is robust under structural disorder. Interestingly, recent studies on phase change materials like GeSbTe showed that their topological insulating order is sensitive to atomic stacking sequences. It was also shown that their structural phase transition is correlated with topological insulating order. GeTe, a well-known phase change material, is trivial insulator in its equilibrium structure. In this study, we discuss how atomic defects such as Ge tetrahedral defect observed in amorphous GeTe can change its topological insulating order based on first-principles calculations and model Hamiltonian. We also investigated the critical density of such tetrahedral defects to induce topological insulating order in GeTe. Our study will help explore hidden orders in GeTe.   

Mapping the conductivity tensor of disordered topological insulators in the presence of magnetic fields

Transport measurements on topological insulators revealed extremely interesting effects and generated data that contain extremely valuable information. It is now possible to control the carriers' concentration using finely tuned gate voltages and to do the transport measurements under controlled applied magnetic fields. As such, accurate maps of the conductivity tensor are now available, as function of the Fermi level and the magnetic field strength. To extract useful information, we need a quantitative theory of charge transport for aperiodic systems in presence of magnetic fields. Such theory has been developed in the past using C*-Algebras and Non-Commutative calculus, the result being closed and exact formulas for the conductivity tensor. In this talk we explain how to manipulate the algebras and how to implement the non-commutative calculus on a computer, in order to compute the conductivity tensor of topological insulating materials in the presence of disorder and magnetic fields. Quantitative simulations of the transport experiments on 2D (and possibly 3D) topological insulators will be presented. Since the methodology can treat disorder and magnetic fields in the same time, it enable us to reproduce, for example, the quantization and the plateaus of the Hall conductance.   

Magnetic field induced Quasi Helical Liquid state in a disordered 1D electron system with strong spin-orbit interaction

We study the crossover from a Luttinger liquid to a quasi-helical liquid state in a one-dimensional system of interacting electrons with strong spin-orbit interaction in the presence of a transverse magnetic field, which leads to a gap for one-half of the conducting modes. In particular, we study the effect of gap opening by electron localization in the presence of non-magnetic disorder. We show that the localization length has a nonuniform behavior as a function of the magnetic field. With increasing field, the localization length grows from its zero-field Luttinger-liquid value to a maximum, after which it crosses over to again smaller values corresponding to the localization length of spinless fermions.   

Delocalized States and Topological Edge States of Quantum Walks in Random Environment

The quantum walk (QWs) describe quantum mechanical time evolution of particles, which is identified as random walks when systems are brought to classical limit. QWs can be applied for efficient algorithms of quantum computation and have been realized in experiments. Remarkably, QWs realized by many experiments possess chiral symmetry. Thereby, QWs in a one dimensional (1D) space possibly have non-trivial topological phases and show edge states near boundaries of the system. In this work, we consider QWs interacting with spatial and temporal disorders and study how the edge states of QWs are influenced. Even by introducing the weak spatial disorder to the QWs, the edge states are robust. However, in the strong disorder limit, the energy gap vanishes and the edge states disappear. We found that critical states due to the Anderson transition in the 1D chiral class alternatively appear at energy $\omega=0$ in this case. Significantly, these critical states also appear at $\omega=\pm \pi/2$ for any strength of static disorder, since the extra sublattice symmetry of the time-evolution operator $U$ makes $\omega=\pm \pi/2$ singular. Consequently, for the QWs with relatively weak spatial disorder, the edge states, critical states, and Anderson localized sates are coexist.   

The strong side of weak topological insulators

Three-dimensional topological insulators are classified into ``strong'' (STI) and ``weak'' (WTI) according to the nature of their surface states. While the surface states of the STI are topologically protected, in the WTI they are believed to be very fragile to disorder. In this work we show that the WTI surface states are actually protected from any random perturbation which does not break time-reversal symmetry, and does not close the bulk energy gap. Consequently, the conductivity of metallic surfaces in the clean system will remain finite even in the presence of strong disorder of this type. In the weak disorder limit the surfaces are perfect metals, and strong surface disorder only acts to push them inwards. We find that WTI's differ from STI's primarily in their anisotropy, and that the anisotropy is not a sign of their weakness but rather of their richness.