Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge modes protected by crystalline symmetries. Since amorphous solids are ubiquitous, it is important to ask whether such a second-order topological insulator can exist in an amorphous system without any spatial order. Here we predict the existence of a secondorder topological insulating phase in an amorphous system without any crystalline symmetry. Such a topological phase manifests in the winding number of the quadrupole moment, the quantized longitudinal conductance and the hinge states. Furthermore, in stark contrast to the viewpoint that structural disorder should be detrimental to the higher-order topological phase, we remarkably find that structural disorder can induce a second-order topological insulator from a topologically trivial phase in a regular geometry. We finally demonstrate the existence of a second-order topological phase in amorphous systems with time-reversal symmetry.
A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either …
We investigate the effects of disorder in Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are induced by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a mobility gap at the resonant quasi-energy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator. Interestingly, the effects of disorder are not necessarily adverse: we show that in the same…
Effects of disorder on two-dimensional Z2 topological insulator are studied numerically by the transfer matrix method. Based on the scaling analysis, the phase diagram is derived for a model of HgTe quantum well as a function of disorder strength and magnitude of the energy gap. In the presence of sz non-conserving spin-orbit coupling, a finite metallic region is found that partitions the two topologically distinct insulating phases. As disorder increases, a narrow-gap topologically trivial insulator undergoes a series of transitions; first to metal, second to topological insulator, third to m…
We investigate the possibility of realizing a disorder-induced topological Floquet spectrum in two-dimensional periodically-driven systems. Such a state would be a dynamical realization of the topological Anderson insulator. We establish that a disorder-induced trivial-to-topological transition indeed occurs, and characterize it by computing the disorder averaged Bott index, suitably defined for the time-dependent system. The presence of edge states in the topological state is confirmed by exact numerical time-evolution of wavepackets on the edge of the system. We consider the optimal driving …
A higher-order topological insulator is a new concept of topological states of matter, which is characterized by the emergent boundary states whose dimensionality is lower by more than two compared with that of the bulk, and draws a considerable interest. Yet, its robustness against disorders is still unclear. Here we investigate a phase diagram of higher-order topological insulator phases in a breathing kagome model in the presence of disorders, by using a state-of-the-art machine learning technique. We find that the corner states survive against the finite strength of disorder potential as l…
Different disorders lead to various localization and topological phenomena in condensed matter and artificial systems. Here we study the topological and localization properties in one-dimensional Su-Schrieffer-Heeger model with spatially correlated random binary disorders. It is found that random binary disorders can induce the topological Anderson insulating phase from the trivial insulator in various parameter regions. The topological Anderson insulators are characterized by the disorder-averaged winding number and localized bulk states revealed by the inverse participation ratio in both rea…
We study the nature of the superfluid--insulator quantum phase transition in a one-dimensional system of lattice bosons with off-diagonal disorder in the limit of large integer filling factor. Monte Carlo simulations of two strongly disordered models show that the universality class of the transition in question is the same as that of the superfluid--Mott-insulator transition in a pure system. This result can be explained by disorder self-averaging in the superfluid phase and applicability of the standard quantum hydrodynamic action. We also formulate the necessary conditions which should be s…
The role of disorder in the field of three-dimensional time reversal invariant topological insulators has become an active field of research recently. However, the computation of Z2 invariants for large, disordered systems still poses a considerable challenge. In this paper we apply and extend a recently proposed method based on the scattering matrix approach, which allows the study of large systems at reasonable computational effort with few-channel leads. By computing the Z2 invariant directly for the disordered topological Anderson insulator, we unambiguously identify the topological nature…
We studied effects of disorder in a three dimensional layered Chern insulator. By calculating the localization length and density of states numerically, we found two distict types of metallic phases between Anderson insulator and Chern insulator; one is diffusive metallic (DM) phase and the other is renormalized Weyl semimetal (WSM) phase. We show that longitudinal conductivity at the zero energy state remains finite in the renormalizd WSM phase as well as in the DM phase, while goes to zero at a semimetal-metal quantum phase transition point between these two. Based on the Einstein relation c…
We are honored to dedicate this article to Emmanuel Rashba on the occasion of his 95 birthday. In the ideal disorder-free situation, a two-dimensional band gap insulator has an activation energy for conductivity equal to half the band gap, $Δ$. But transport experiments usually exhibit a much smaller activation energy at low temperature, and the relation between this activation energy and $Δ$ is unclear. Here we consider the temperature-dependent conductivity of a two-dimensional narrow gap semiconductor on a substrate containing Coulomb impurities, mostly focusing on the case when amplitude o…
We report the theoretical investigation of the disorder effects on the bulk states of inverted InAs/GaSb quantum wells. As disorder sources we consider the interface roughness and donors/acceptors supplied by intentional doping. We use a $\bm{k}\cdot\bm{p}$ approach combined with a numerical diagonalization of the disordered Hamiltonian to get a full insight of the disordered eigenenergies and eigenfunctions of the electronic system. While interface roughness slightly pertubs the carrier motion, we show that dopants strongly bind and localize the bulk states of the structure. Moreover, both ty…
Employing scaling analysis of the localization length, we deduce the critical exponent of the metal-topological insulator (TI) transitions induced by disorder. The obtained exponent nu~2.7 shows no conspicuous deviation from the value established for metal-ordinary insulator transitions in systems of the symplectic class. We investigate the topological phase diagram upon carrier doping to reveal the nature of the so-called topological Anderson insulator (TAI) region. The critical exponent of the metal-TAI transition is also first estimated, shown to be undistinguishable from the above value wi…
The Friedel oscillations resulting from coupling a quantum dot to one edge of a disordered one-dimensional wire in the Mott insulator regime, are calculated numerically using the DMRG method. By investigating the influence of the disorder on the Friedel oscillations decay we find that the effect of disorder is reduced by increasing the interaction strength. This behavior is opposite to the recently reported influence of disorder in the Anderson insulator regime, where disorder led to a stronger decay of the Friedel oscillations.
In magnetically doped thin-film topological insulators, aligning the magnetic moments generates a quantum anomalous Hall phase supporting a single chiral edge state. We show that as the system de-magnetizes, disorder from randomly oriented magnetic moments can produce a `quantum anomalous parity Hall' phase with \emph{helical} edge modes protected by a unitary reflection symmetry. We further show that introducing superconductivity, combined with selective breaking of reflection symmetry by a gate, allows for creation and manipulation of Majorana zero modes via purely electrical means and at ze…
We investigate the presence of percolating states in disordered two-dimensional topological insulators. In particular, we uncover a close connection between these states and the so-called topological Anderson insulator (TAI), which is a topologically non-trivial phase induced by the presence of disorder. The decay of this phase could previously be connected to a delocalization of bulk states with increasing disorder strength. In this work we identify this delocalization to be the result of a percolation transition of states that circumnavigate the hills of the bulk disorder potential.
Recently, a new family of symmetry-protected higher-order topological insulators has been proposed and was shown to host lower-dimensional boundary states. However, with the existence of the strong disorder in the bulk, the crystal symmetry is broken, and the associated corner states are disappeared. It is well known that the emergence of robust edge states and quantized transport can be induced by adding sufficient disorders into a topologically trivial insulator, that is the so-called topological Anderson insulator. The question is whether disorders can also cause the higher-order topologica…
We propose a new method to numerically compute the $\mathbb{Z}_2$ indices for disordered topological insulators in Kitaev's periodic table. All of the $\mathbb{Z}_2$ indices are known to be derived from the index formulae which are expressed in terms of a pair of projections introduced by Avron, Seiler, and Simon. For a given pair of projections, the corresponding index is determined by the spectrum of the difference between the two projections. This difference exhibits remarkable and useful properties, as it is compact and has a supersymmetric structure in the spectrum. These properties make …
The effect of surface disorder on electronic systems is particularly interesting for topological phases with surface and edge states. Using exact diagonalization, it has been demonstrated that the surface states of a 3D topological insulator survive strong surface disorder, and simply get pushed to a clean part of the bulk. Here we explore a new method which analytically eliminates the clean bulk, and reduces a $D$-dimensional problem to a Hamiltonian-diagonalization problem within the $(D-1)$-dimensional disordered surface. This dramatic reduction in complexity allows the analysis of signific…
In two dimensions chaotic level-statistics is expected for massless Dirac fermions in the presence of disorder. For weakly disordered graphene flakes with zigzag edges the obtained level-spacing distribution in the Dirac region is neither chaotic (Wigner) nor localized (Poisson) but similar to that at the critical point of the Anderson metal-insulator transition. The quantum transport in finite graphene can occur via critical edge states as in topological insulators, for strong disorder the Dirac region vanishes and graphene behaves as ordinary Anderson insulator.
In disordered two dimensional Chern insulators, a single bulk extended mode is predicted to exist per band, up to a critical disorder strength; all the other bulk modes are localized. This behavior contrasts strongly with topologically trivial two-dimensional phases, whose modes all become localized in the presence of disorder. Using a tight-binding model of a realistic photonic Chern insulator, we show that delocalized bulk eigenstates can be observed in an experimentally realistic setting. This requires the selective use of resonator losses to suppress topological edge states, and acquiring …
The disorder effects on higher-order topological phases in periodic systems have attracted much attention. However, in aperiodic systems, such as quasicrystalline systems, the interplay between disorder and higher-order topology is still unclear. In this paper, we investigate the effects of disorder on two types of second-order topological insulators, including a quasicrystalline quadrupole insulator and a modified quantum spin Hall insulator, in a two-dimensional Amman-Beenker tiling quasicrystalline lattice. We demonstrate that the higher-order topological insulators are robust against weak …
The topological Anderson and Mott insulators are two phases that have so far been separately and widely explored beyond topological band insulators. Here we combine the two seemingly different topological phases into a system of spin-1/2 interacting fermionic atoms in a disordered optical lattice. We find that the topological Anderson and Mott insulators in the noninteracting and clean limits can be adiabatically connected without gap closing in the phase diagram of our model. Lying between the two phases, we uncover a disordered correlated topological insulator, which is induced from a trivia…
Clean topological insulators exposed to a magnetic field develop Landau levels accompanied by a nonzero Hall conductivity for the infinite slab geometry. In this work we consider the case of disordered topological insulator nanowires and find, in contrast, that a zero Hall plateau emerges within a broad energy window close to the Dirac point. We numerically calculate the conductance and its distribution for a statistical ensemble of disordered nanowires, and use the conductance fluctuations to study the dependence of the insulating phase on system parameters, such as the nanowire length, disor…
We discuss the effects of disorder in time-reversal invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field theory analysis (the large-$N_f$ expansion where $N_f$ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, a…
Abstract: Using ab initio calculations supported by experimental transport measurements, we present the first credible candidate for the realization of a disorder-induced Topological Anderson Insulator in a real material system. High energy reactive ball-milling produces a polymorph of Cu2ZnSnS4 with high cation disorder, which shows an inverted ordering of bands at the Brillouin zone center, in contrast to its ordered phase. Adiabatic continuity arguments establish that this disordered Cu2ZnSnS4 can be connected to the closely related Cu2ZnSnSe4, previously predicted to be a 3D topological in…
Topological insulators are characterized by specially protected conduction on their outer boundaries. We show that the protected edge conduction exhibited by 2-D topological insulators (and also Chern insulators) is independent of non-magnetic boundary disorder. In particular, the edge states residing inside the bulk gap remain conducting even when edge state inhomogeneities destroy the characteristic linear Dirac relation between energy and momentum. The main effects of boundary disorder on the in-gap states are to decrease the Fermi velocity, increase the density of states, pull the states i…
We study the effect of electrostatic disorder on the conductivity of a three-dimensional antiferromagnetic insulator (a stack of quantum anomalous Hall layers with staggered magnetization). The phase diagram contains regions where the increase of disorder first causes the appearance of surface conduction (via a topological phase transition), followed by the appearance of bulk conduction (via a metal-insulator transition). The conducting surface states are stabilized by an effective time-reversal symmetry that is broken locally by the disorder but restored on long length scales. A simple self-c…
We theoretically study the spin and charge generation along with the electron transport on a disordered surface of a doped three-dimensional topological insulator/magnetic insulator junction by using Green's function techniques. We find that the spin and charge current are induced by not only local but also nonlocal magnetization dynamics through nonmagnetic impurity scattering on the disordered surface of the doped topological insulator. We also clarify that the spin current as well as charge density are induced by spatially inhomogeneous magnetization dynamics, and the spin current diffusive…
We numerically demonstrate a practical means of systematically controlling topological transport on the surface of a three dimensional topological insulator, by introducing strong disorder in a layer of depth $d$ extending inward from the surface of the topological insulator. The dependence on $d$ of the density of states, conductance, scattering time, scattering length, diffusion constant, and mean Fermi velocity are investigated. The proposed control via disorder depth $d$ requires that the disorder strength be near the large value which is necessary to drive the TI into the non-topological …
Topological behavior can be masked when disorder is present. A topological insulator, either intrinsic or interaction induced, may turn gapless when sufficiently disordered. Nevertheless, the metallic phase that emerges once a topological gap closes retains several topological characteristics. By considering the self-consistent disorder-averaged Green function of a topological insulator, we derive the condition for gaplessness. We show that the edge states survive in the gapless phase as edge resonances and that, similar to a doped topological insulator, the disordered topological metal also h…
We investigate disorder-driven topological phase transitions in quantized electric quadrupole insulators. We show that chiral symmetry can protect the quantization of the quadrupole moment $q_{xy}$, such that the higher-order topological invariant is well-defined even when disorder has broken all crystalline symmetries. Moreover, nonvanishing $q_{xy}$ and consequent corner modes can be induced from a trivial insulating phase by disorder that preserves chiral symmetry. The critical points of such topological phase transitions are marked by the occurrence of extended boundary states even in the …
Impurities and defects are ubiquitous in topological insulators (TIs) and thus understanding the effects of disorder on electronic transport is important. We calculate the distribution of the random conductance fluctuations $P(G)$ of disordered 2D TI wires modeled by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian with realistic parameters. As we show, the disorder drives the TIs into different regimes: metal (M), quantum spin-Hall insulator (QSHI), and ordinary insulator (OI). By varying the disorder strength and Fermi energy, we calculate analytically and numerically $P(G)$ across the entire pha…
We investigate the effect of spatially correlated disorder on two-dimensional topological insulators and on the quantum spin Hall effect which the helical edge states in these systems give rise to. Our work expands the scope of previous investigations which found that uncorrelated disorder can induce a nontrivial phase called the topological Anderson insulator (TAI). In extension of these studies, we find that spatial correlations in the disorder can entirely suppress the emergence of the TAI phase. We show that this phenomenon is associated with a quantum percolation transition and quantify i…
Motivated by the recent experimental realization of the topological Anderson insulator and research interest on the topological quasicrystal lattices, we investigate the effects of disorder on topological properties of a two-dimensional Penrose-type quasicrystal lattice that supports the quantum spin Hall insulator (QSHI) and normal insulator (NI) phases in the clean limit. It is shown that the helical edge state of the QSHI phase is robust against weak disorder. Most saliently, it is found that disorder can induce a phase transition from NI to QSHI phase in the quasicrystal system. The numeri…
Algebraic and geometric mean density of states in disordered systems may reveal properties of electronic localization. In order to understand the topological phases with disorder in two dimensions, we present the calculated density of states for disordered Bernevig-Hughes-Zhang model. The topological phase is characterized by a perfectly quantized conducting plateau, carried by helical edge states, in a two-terminal setup. In the presence of disorder, the bulk of the topological phase is either a band insulator or an Anderson insulator. Both of them can protect edge states from backscattering.…
Recent studies of disorder or non-Hermiticity induced topological insulators inject new ingredients for engineering topological matter. Here we consider the effect of purely non-Hermitian disorders, a combination of these two ingredients, in a 1D coupled-cavity array with disordered gain and loss. Topological photonic states can be induced by increasing gain-loss disorder strength with topological invariants carried by localized states in the complex bulk spectra. The system showcases rich phase diagrams and distinct topological states from Hermitian disorders. The non-Hermitian critical behav…
Three-dimensional topological insulators feature Dirac-like surface states which are topologically protected against the influence of weak quenched disorder. Here we investigate the effect of surface disorder beyond the weak-disorder limit using large-scale numerical simulations. We find two qualitatively distinct regimes: Moderate disorder destroys the Dirac cone and induces diffusive metallic behavior at the surface. Even more remarkably, for strong surface disorder a Dirac cone reappears, as new weakly disordered "surface" states emerge in the sample beneath the disordered surface layer, wh…
We investigate the effects of periodic fields and disorder on topological properties of a Weyl-semimetal thin film. The two periodic fields, i.e., a periodic magnetic field and elliptically polarized light, are discussed respectively. By use of the Floquet theory, we find that both the two periodic drives can resonantly induce the topological transitions from normal insulator (NI) phases to Floquet topological insulator (FTI) phases. The Floquet topological transitions are characterized by variation of Chern number. Moreover, we show that the Floquet topological transitions can be explained by…
We study the magneto-transport properties on the disordered surface of a topological insulator attached with a ferromagnet/ferromagnet junction. Since, in the surface Dirac Hamiltonian, out-of-plane magnetization induces a mass gap, while in-plane magnetization has a role of the effective vector potential, the mechanism of magneto-transport is different between these two cases. The former is similar to the conventional one in ferromagnetic metals, while the latter is due to the shift of Fermi circles in momentum space. Our numerical calculations show that the magnetoconductance in in-plane con…
We examine the transport properties of magnetically doped topological insulator (TI) thin films subject to correlated non-magnetic disorder. For the disorder we choose a quasi-periodic potential with a random phase. We restrict the disorder to a central region, which is coupled to two leads in a clean quantum spin Hall insulator (QSHI) state and concentrate on different orientations of the quasi-periodicity in the two-dimensional central region. In the case of a diagonally oriented or purely longitudinal quasi-periodicity we find different topological Anderson insulator (TAI) phases, with a qu…
We develop a minimal theory for the recently observed metal-insulator transition (MIT) in two-dimensional (2D) moiré multilayer transition metal dichalcogenides (mTMD) using Coulomb disorder in the environment as the underlying mechanism. In particular, carrier scattering by random charged impurities leads to an effective 2D MIT approximately controlled by the Ioffe-Regel criterion, which is qualitatively consistent with the experiments. We find the necessary disorder to be around $5$-$10\times10^{10}$cm$^{-2}$ random charged impurities in order to quantitatively explain much, but not all, of …
The interplay among topology, disorder, and non-Hermiticity can induce some exotic topological and localization phenomena. Here we investigate this interplay in a two-dimensional non-Hermitian disordered Chern-insulator model with two typical kinds of non-Hermiticities, the nonreciprocal hopping and on-site gain-and-loss effects. The topological phase diagrams are obtained by numerically calculating two topological invariants in the real space, which are the disorder-averaged open-bulk Chern number and the generalized Bott index, respectively. We reveal that the nonreciprocal hopping (the gain…
In this numerical study we investigate the influence and interplay of disorder, spin-orbit coupling and magnetic field on the edge-transport in HgTe/CdTe quantum wells in the framework of coherent elastic scattering. We show that the edge states remain unaffected by the combined effect of moderate disorder and a weak magnetic field at realistic spin-orbit coupling strengths. Agreement with the experimentally observed linear magnetic field dependence for the conductance of long samples is obtained when considering the existence of charge puddles.
We investigate a transition between a two-dimensional topological insulator conduction state, characterized by a conductance $G=2$ (in fundamental units $e^2/h$) and a Chern insulator with $G=1$, induced by polarized magnetic impurities. Two kinds of coupling, ferro and antiferromagnetic, are considered with the electron and hole subbands. We demonstrate that for strong disorder, a phase $G=1$ exists even for ferromagnetic order, in contrast with the prediction of the mean field approximation. This result is supported by direct numerical computations using Landauer transport formula, and by an…
In an ultrathin topological insulator (TI) film, a hybridization gap opens in the TI surface states, and the system is expected to become either a trivial insulator or a quantum spin Hall insulator when the chemical potential is within the hybridization gap. Here we show, however, that these insulating states are destroyed by the presence of a large and long-range-correlated disorder potential, which converts the expected insulator into a metal. We perform transport measurements in ultrathin, dual-gated topological insulator films as a function of temperature, gate voltage, and magnetic field,…
The generalized bulk-boundary correspondence predicts the existence of the chiral hinge states in three-dimensional second-order topological insulators (3DSOTIs), resulting in a quantized Hall effect in three dimensions. Chiral hinge states in Hermitian 3DSOTIs are characterized by the quantized transmission coefficients with zero fluctuations, even in the presence of disorders. Here, we show that chiral hinge transport in disordered non-Hermitian systems deviates from the paradigm of the Hermitian case. Our numerical calculations prove the robustness of hinge states of disordered non-Hermitia…
Two-dimensional Na$_3$Bi is a dual topological insulator protected by time-reversal and mirror symmetry, resulting in a promising platform for devices design. However, in reality, the design of topological devices is hindered by a sensitivity against disorder and temperature. We study the topological properties of Na$_3$Bi in the presence of intrinsic defects, investigating the robustness of the edge states and the resulting transport properties. We apply a recursive Green's function technique enabling the study of disordered systems with lengths comparable to experimentally synthesized materi…
Disorder effects on three-dimensional second-order topological insulators (3DSOTIs) are investigated numerically and analytically. The study is based on a tight-binding Hamiltonian for non-interacting electrons on a cubic lattice with a reflection symmetry that supports a 3DSOTI in the absence of disorder. Interestingly, unlike the disorder effects on a topological trivial system that can only be either a diffusive metal (DM) or an Anderson insulator (AI), disorders can sequentially induce four phases of 3DSOTIs, three-dimensional first-order topologicalinsulators (3DFOTIs), DMs and AIs. At a …
We explore the physics of a Chern insulator subjected to a two step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by presence and chirality of edge-modes in the two bulk gaps of the Floquet quasienergy spectrum, around $0$ and $π$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically non-tri…
The disorder-driven metal-insulator transition in the quantum spin Hall systems is studied by scaling analysis of the Thouless conductance $g$. Below a critical disorder strength, the conductance is independent of the sample size $M$, an indication of critically delocalized electron states. The calculated beta function $β=d\ln g/d\ln M$ indicates that the metal-insulator transition is Kosterlitz-Thouless (KT) type, which is characterized by bounding and unbounding of vortex-antivortex pairs of the local currents. The KT like metal-insulator transition is a basic characteristic of the quantum s…
Topological insulators in three dimensions are characterized by a Z2-valued topological invariant, which consists of a strong index and three weak indices. In the presence of disorder, only the strong index survives. This paper studies the topological invariant in disordered three-dimensional system by viewing it as a super-cell of an infinite periodic system. As an application of this method we show that the strong index becomes non-trivial when strong enough disorder is introduced into a trivial insulator with spin-orbit coupling, realizing a strong topological Anderson insulator. We also nu…
We investigate the scattering and localization properties of edge and bulk states in a disordered two-dimensional topological insulator when they coexist at the same fermi energy. Due to edge-bulk backscattering (which is not prohibited \emph{a priori} by topology or symmetry), Anderson disorder makes the edge and bulk states localized indistinguishably. Two methods are proposed to effectively decouple them and to restore robust transport. The first kind of decouple is from long range disorder, since edge and bulk states are well separated in $k$ space. The second one is from an edge gating, o…
We study theoretically the role of quenched magnetic disorder at the surface of a topological insulator by numerical simulation and scaling analysis. It is found that all the surface states are localized while the transverse conductivity is quantized to be $\pm {{e^2} \over {2 h}}$ as long as the Fermi energy is within the bulk gap. This greatly facilitates the realization of the topological magnetoelectric effect proposed by Qi {\it et al.} (Phys. Rev. B\textbf{78}, 195424 (2008)) with the surface magnetization direction being controlled by the simultaneous application of magnetic and electri…
