arXiv:1406.1040v2 [cond-mat.mes-hall] 15 Feb 2015 Topological Insulators, Topological Crystalline Insulators, and Topological Kondo Insulators (Review Article) M. Zahid Hasan, 1, 2 Su-Yang Xu, 1 and Madhab Neupane 1 1 Joseph Henry Laboratories: Department of Physics, Princeton University, Princeton, NJ 08544, USA 2 Princeton Institute for the Science and Technology of Materials, School of Engineering and Applied Science, Princeton University, Princeton NJ 08544, USA In this Book Chapter we briefly review the basic concepts defining topological in- sulators and elaborate on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We then present key experimental results that demonstrate magnetism, Kondo insulation, mirror chirality or topological crystalline order and superconductivity in spin-orbit topological insulator settings and how these new phases of matter arise through topological quantum phase transitions from Bloch band insulators via Dirac semimetals at the critical point. CONTENTS I. Introduction 1 II. Z 2 topological insulators 3 III. Topological Kondo Insulator Candidates 8 IV. Topological Quantum Phase Transitions 11 V. Topological Dirac Semimetals 14 VI. Topological Crystalline Insulators 17 VII. Magnetic and Superconducting doped Topological Insulators 20 References 27 INTRODUCTION Topological phases of matter differ from conventional materials in that a topological phase of matter features a nontrivial topological invariant in its bulk electronic wavefunction space which can be realized in a symmetry- protected condition [1–14]. The experimental discoveries of the 2D integer and fractional quantum Hall (IQH and FQH) states [15–19] in the 1980s realize the first two topological phases of matter in nature. These 2D topo- logical systems are insulators in the bulk because the Fermi level is located in the middle of two Landau levels. On the other hand, the edges of these 2D topological insu- lators (IQH and FQH) feature chiral 1D metallic states, leading to remarkable quantized charge transport phe- nomena. The quantized transverse magneto-conductivity σ xy = ne 2 /h (where e is the electric charge and h is Planck’s constant) can be probed by charge transport experiments, which also provides a measure of the topo- logical invariant (the Chern number) n that characterizes these quantum Hall states [20, 21]. In 2005, theoretical advances [22, 23] predicted a third type of 2D topolog- ical insulator, the quantum spin Hall (QSH) insulator. Such a topological state is symmetry protected. A QSH insulator can be viewed as two copies of quantum Hall systems that have magnetic field in the opposite direc- tion. Therefore, no external magnetic field is required for the QSH phase, and the pair of quantum-Hall-like edge modes are related by the time-reversal symmetry (Fig. 1). The QSH phase was experimentally demonstrated in the mercury telluride quantum wells of using charge trans- port by measuring a longitudinal (charge) conductance of about 2e 2 /h (two copies of quantum Hall states) at low temperatures [24]. No spin polarization was mea- sured in this experiment thus spin momentum locking which is essential for the Z 2 topological physics was not known or proven from experiments [24]. It is important to note that the 2D topological (IQH, FQH, and QSH) insulators are only realized at buried in- terfaces of ultraclean semiconductor heterostructures at very low temperatures [24]. Furthermore, their metallic edge states can only be probed by the charge transport method [24]. These facts hinder the systematic stud- ies of many of their important properties, such as their electronic structure, spin polarization texture, tunneling properties, optical properties, as well as their responses under heterostructuring or interfacing with broken sym- metry states. For example, the two counter-propagating edge modes in a QSH insulator is predicted to feature a 1D Dirac band crossing in energy and momentum space [22, 23]. And edge mode moving along the +k direction is expected to carry the opposite spin polarization as com- pared to that of moving to the −k direction [22, 23]. However, neither the Dirac band crossing nor the spin- momentum locking of the edge modes in a QSH insulator are experimentally observed, due to the lack of experi- mental probe that can measure these key properties for a 1D edge mode at a buried interface at mK temperatures, which is challenging. In 2007, it was also theoretically re- alized that the Z 2 topological number can have nontrivial generalization into three-dimensions [11, 12, 25, 26]. In three-dimensions, there exist four (not three) Z 2 topo-