The subject of **topological** **insulators** and **topological** superconductors is now one of the most active fields of research in condensed matter physics, developing at a rapid pace. Theorists have systematically classified **topological** states in all dimensions. For future progress on the theoretical side, the most important outstanding problems include interaction and disorder effects, realistic predictions for **topological** Mott insulator materials, a deeper understanding of fractional **topological** **insulators** and realistic predictions for materials realizations of such states, the effective field theory description of the **topological** superconducting state, and realistic materials predictions for **topological** superconductors. On the experimental part, the most important task is to grow materials with sufficient purity so that the bulk insulating behavior can be reached, and to tune the Fermi level close to the Dirac point of the surface state.

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An exciting new development in condensed matter physics over the last decade is the beautiful realization of **topological** field theories [18] in strongly correlated electronic systems, where **topological** field theories are shown to provide a classification of order due to macroscopic entanglement that is independently of symmetry breaking [19]. The FQH state is the first known example of such a quantum state that exhibits no spontaneous broken symmetry, and its properties depend only on its topology rather than geometry, which is topologically distinct from all other quantum states classified by broken symmetry. Recently, the quantum spin Hall (QSH) states and the **topological** **insulators** (TI) have emerged as a new class of **topological** states and have stimulated intense research activities [2]. As described in Section 1, one of the novel properties associated with the TI is the presence of a Dirac spectrum of chiral low-energy excitations, which is a salient feature of the Dirac materials. In the case of 3D-STI, an odd number of massless Dirac cones in their SS are ensured by the Z 2 **topological** invariant of the fully gapped bulk

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In connection to **topological** insulator possessing chiral edge states or surface states without backscatterings, it is one of the central questions that how these topologically protected states behave in the presence of non-magnetic and magnetic impurities. Moreover, as 3d **topological** **insulators** may contain unexpected bulk modes [53] from doping to shift the Fermi energy, how the conventional picture of the Anderson localization applies to the system with helical surface states as well as bulk modes and impurities is a question awaiting resolution. Lastly, physicists numerically found that a trivial insulator can undergo a phase transition into a **topological** insulator by introducing impurities that renormalize the chemical potential as well as the mass term of the Hamiltonian. The Landauer-Buttiker type conductance calculation shows a well-quantized chiral edge transport in 2d system [54], and for the 3d case by Guo, Resenberg et al.[55] the appearance of a strong **topological** insulating phase is verified in terms of quantized conductance and the Witten effect.

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states of matter in classical wave systems [1-5]. The robustness of the **topological** systems against disorders and the associated one-way edge states provide new opportunities for manipulating classical waves. Besides the analogues of conventional **topological** **insulators** [6-24] and **topological** semimetals [25-34], the recent interest on higher-order **topological** **insulators** (HOTIs) [35,36] has opened a new direction of **topological** phases in classical systems [37-53]. Different from the conventional **topological** **insulators**, in which the D-dimensional bulk is gapped and the **topological** invariant counts the number of gapless modes hosted on the (D-1)-dimensional boundaries of the sample, the HOTIs are a new family of **topological** phases of matter that goes beyond the conventional bulk-boundary correspondence principle. For example, unlike conventional two-dimensional (2D) **topological** **insulators**, a 2D second-order HOTI does not exhibit gapless one-dimensional (1D) **topological** edge states, but instead has topologically protected zero-dimensional (0D) corner states. Due to the flexibility in sample design, HOTIs have been soon implemented in mechanical [37,45], photonic [38,40-44], electrical circuits [39,46,47] and acoustic [48-53] systems.

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The discovery and classiﬁcation of distinctive electronic phases of matter has always been an important topic in condensed matter physics. As we know, the behavior of electrons in diﬀerent materials varies dramatically. The electrical insulator, for instance, is one of the most basic electronic phases of matter, characterized by an energy gap for electronic excitations. Recent work has, however, now uncovered a new class of materials termed **topological** **insulators** (TIs) [12, 13, 14, 15, 16]. The most distinguishing feature of these **insulators** is that they can insulate on the inside but conduct on the outside - analogous to a block of wood covered with a layer of copper, except that the material is actually the same throughout. Due to strong spin-orbit coupling, electrons that move along the surface have their spin locked perpendicular to their momentum (spin- momentum locking) (see Fig. 1.2(a) for the band structure of a typical 3D TI [17]). Furthermore, these gapless surface states are topologically protected against disruptions such as defects, chemical passivation, and thermal ﬂuctuations. First predicted and then discovered experimentally in semiconducting alloy Bi 1 − x S x , 3D TI has attracted

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distinguish between the two types of **topological** **insulators**. The experi ment involves placing m agnetic im purities along one of the edges of the two- dimensional sample, for instance, by using the tip of a scamiing tunneling microscope. The im p iu ities’ spin will, in general, be aligned in arb itrary directions. The illum ination with low-frequency polarized infrared light can however induce their alignment. This has l)een dem onstrated, for instance, for Mil im purities in CdTe [58]. The infrared pulse im parts a m om entum to align th e im purity s])ins. which subsequently relax back to th eir random orientations. Tlie Andreev reflection coefficient can then be m easured as a function of tim e, and this can l)c related to the inclination angle 0 of the im purity spin S. The dependence of R-^ on 0 is shown in Fig. 4.5. As the si)in rotates tow ards th e 2 : direction. R ‘^ retu rn s back to the perfectly

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Many solid materials are crystalline in nature, which means that their constituent atoms are ordered in regular and repeating patterns at the microscopic scale. These patterns are often arranged periodically, and hence form a so-called crystal structure. If one wants to describe a sample of a crystalline solid of macroscopic size, it is a powerful idealization to assume that this crystal extends infinitely far in all directions so that the system acquires a certain translational symmetry. Macroscopic samples typically consist of 10 20 − 10 24 atoms, which means that for an electron in the bulk of the sample, this is a very good approximation. On the other hand, if one is interested in effects taking place at the edge of the sample, one has to take a different approach. The electronic structure inside the bulk and at the boundary are not entirely unrelated however, as there is an important theorem for **topological** **insulators** known as the bulk-boundary correspondence. This result roughly states that the Z 2 -index determines the phenomena that take place at the surface, and it

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STM is a surface sensitive technique based on the concept of quantum tunneling. The atomically resolved structure of a surface can be obtained by monitoring the tunneling current in an STM. Besides its unprecedented capability to study surface topography, information on electronic structure at a given location of a sample can also be obtained. This type of measurement is called scanning tunneling spectroscopy (STS) and typically results in a plot of the local density of states (LDOS) as a function of energy of the sample. It has been widely used in the characterization of metals, semiconductors and superconductors. Applying it to study of **topological** surface states, many intriguing properties have recently been unveiled. In this paper, we give an overview on the recent progress of STM study of **topological** **insulators**.

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edge and two right movers at the other, exactly as the integer quantum Hall states. We ﬁnd that for both these cases, the edge modes lead to a perfect Andreev reﬂection for electrons with energy smaller than the superconducting gap. In fact, in both cases the edge modes are perfectly Andreev reﬂected. Normal reﬂection, where an incident particle is reﬂected back without being converted into its antiparticle, is completely suppressed for the edge states as long as the Fermi energy lies in the bulk gap, as we have veriﬁed numerically. These ﬁndings are consistent with recent theoretical and experimental studies for time-reversal symmetric **topological** **insulators**. 22,23 Note that

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These two examples share one common trait. In order for the surface to avoid localization, the disordered ensemble must be invariant under a certain symmetry: translation for a weak TI or time-reversal for a strong TI with a random magnetic field. We show that this property defines a broad class of systems, which we call statistical **topological** **insulators** (STI). An STI is an ensemble of disordered systems belonging to the same symmetry class. This ensemble, as a whole, also has to be invariant under an extra symmetry, which we call statistical symmetry since it is not respected by single ensemble elements. These elements have surfaces pinned to the middle of a **topological** phase transition and protected from localization due to the combined presence of the statistical symmetry and the symmetry of each element, if any. For example, for a weak TI the statistical symmetry is translation, while the symmetry of each element is time reversal.

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共 Received 31 December 2009; revised manuscript received 20 April 2010; published 11 May 2010 兲 Interferometry of non-Abelian edge excitations is a useful tool in **topological** quantum computing. In this paper we present a theory of a non-Abelian edge-state interferometer in a three-dimensional **topological** insulator brought in proximity to an s-wave superconductor. The non-Abelian edge excitations in this system have the same statistics as in the previously studied 5/2 fractional quantum-Hall 共 FQH 兲 effect and chiral p-wave superconductors. There are however crucial differences between the setup we consider and these systems, like the need for a converter between charged and neutral excitations and the neutrality of the non-Abelian excitations. These differences manifest themselves in a temperature scaling exponent of −7 / 4 for the conductance instead of −3 / 2 as in the 5/2 FQH effect.

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A fundamental difference between the quantum Hall and the quan- tum spin Hall effect is that the **topological** invariant is restricted to the values 0 and 1 in the latter case, meaning either a pair of topologically protected edge states or none at all. The emphasize is here on the topo- logical protection - further edge states may exist but would not be stable. In contrast the quantum Hall effect can realize any integer number of topologically protected edge modes, but they all propagate in the same direction. This ensures the **topological** protection: if there was a channel in the opposite direction, left and right moving states could scatter into each other and would hybridize, leading to a gap in the spectrum of edge states. In the quantum spin Hall effect this is prevented for a sin- gle pair of time reversed modes since the so-called Kramers degeneracy of the crossing point in the Brillouin zone (see lower panel of Fig. 1.4) is protected by time-reversal symmetry.

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We develop a unified framework to classify **topological** defects in **insulators** and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H共 k , r 兲 that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The **topological** classes for such defects are identified and explicit formulas for the **topological** invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the **topological** classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majo- rana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

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The **topological** band **insulators** (TBIs) in two and three dimensions have attracted a great deal of interest in both theoretical and experimental condensed matter physics [1,2]. An extensive classification of **topological** phases in free fermion systems with a bulk gap, based on time- reversal symmetry (TRS) and particle-hole symmetry (PHS), has been provided [3], and it defines the so-called tenfold way. This however relies on the spatial continuum limit while TBIs actually need a crystal lattice which, in turn, breaks the translational symmetry. Therefore, the question arises whether the crystal lattice may give rise to an additional subclass of **topological** phases. The simple M - B model introduced for the two-dimensional (2D) HgTe quantum spin Hall insulator [4] gives away a generic wisdom in this regard. Depending on its parameters, this model describes **topological** phases which are in a thermo- dynamic sense distinguishable: their **topological** nature is characterized by a Berry phase Skyrmion lattice in the extended Brillouin zone (BZ), where the sites of this lattice coincide with the reciprocal lattice vectors (‘‘ phase’’) or with the TRS points ( , ) (‘‘ M phase’’).

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cartoons describe a spin-flip scattering event. A right-going electron with up-spin direction (left panel) is inelastically backscattered by the magnetic impurity. In the process both the electron and the impurity spins are reversed (right panel). Note that, given the **topological** nature of the ribbon, spin flip forbids electron transmission as the edge presenting a right-going spin-up state does not possess a right-going spin-down one.

In Section 2.1, we will first discuss the rather simple resistively shunted junction model that has been used to model S/N/S junction. This model does not take MAR into account. The current through a one dimentional S/N/S junction as a result of MAR was first modelled by Averin- Bardas in 1995. Their model is discussed in Section 2.2. As a next step, we are interested in a junction with a **topological** insulator (TI) in the middle. When a TI is brought into contact with a superconductor, its surface becomes superconducting as well via the proximity effect. This combination of spin-momentum locking and superconductivity allows symmetry protected surface states to host Majorana particles; which has been a hot topic for the past decades. The one dimensional S/TI/S junction and its relation to Majorana particles were considered by Badiane, Houzet and Meyer in 2011. A brief review of their work is given in Section 2.3.

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