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Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

/pdf/topological-insulators-and-superconductors-2nf6p2217g.pdf

Topological insulators and superconductors

Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by ${Z}_{2}$ topological invariants, which characterize the ground state. In two dimensions, there is a single ${Z}_{2}$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four ${Z}_{2}$ invariants that distinguish the ordinary insulator from ``weak'' and ``strong'' topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the ${Z}_{2}$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ as well as $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.

/pdf/topological-insulators-with-inversion-symmetry-1vaykhe1u1.pdf

Topological insulators with inversion symmetry

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Introduction to Quantum Mechanics

Transmission through a Thue-Morse chain.

We study the spectral statistics in the center of the lowest Landau band of a 2D disordered system with smooth potential and strong transverse magnetic field. Due to the finite size of the system, the energy range in which there are extended states is finite as well. The behavior in this range can be viewed as the analogue of the Anderson metal-insulator transition for the case of the Hall system. Accordingly, we verify recent predictions regarding the exponent of the asymptotic power law of $\Sigma^2 (\bar N)$, $\gamma$, and that of the stretched exponential dominating the large $s$ behavior of the spacings distribution, $\alpha$. Both the relations, $\alpha = 1- \gamma$, and $\gamma = 1 - {1\over{\nu d}}$ where $\nu$ is the critical exponent of the localization length and $d$ is the dimension, are found to hold within the accuracy of our computations. However, we find that none of several possible models of the entire spacings distribution correctly describes our situation. Finally, for very large $\bar N$, $\bar N > 60$, we find a new regime in which $\Sigma^2 (\bar N)$ behaves as a power law with an unexpectedly large power, $\gamma_1 = 1.38 \pm 0.02$.

Spectral statistics in the lowest Landau band

We investigate the localization of electrons hopping on quasi-1D strips in the presence of random magnetic field. In the weak-disorder region, by perturbative analytical techniques, we derive scaling laws for the localization length, ${\xi}$, of the form $ \xi \propto \frac{1}{w^{\eta}}$, where $w$ is the size of magnetic disorder and the exponent $\eta$ assumes different values in the various energy ranges. Moreover, in the neighborhood of the energies where a new channel opens a certain rearrangement of the perturbation expansion leads to scaling functions for $\xi$. Although the latter are in general quantitatively wrong, they correctly reproduce the corresponding $\eta$ exponents and the form of the scaling variables and are therefore useful for understanding the behavior of $\xi$.

Localization in quasi-one-dimensional systems with a random magnetic field.

Whereas in the familiar Kondo effect the exchange interaction is dipolar, there are systems in which the exchange interaction is multipolar, as has been realized in a recent experiment. Here, we study multipolar Kondo effect in a Fermi gas of cold $^{173}\mathrm{Yb}$ atoms. Making use of different ac polarizabilities of the electronic ground state $\mathrm{Yb}(^{1}\mathrm{S}_{0})$ and the long-lived metastable state ${\mathrm{Yb}}^{*}(^{3}\mathrm{P}_{2})$, it is suggested that the latter atoms can be localized and serve as a dilute concentration of magnetic impurities while the former ones remain itinerant. The exchange mechanism between the itinerant Yb and the localized ${\mathrm{Yb}}^{*}$ atoms is analyzed and shown to be antiferromagnetic. The quadrupole and octupole interactions act to enhance the Kondo temperature ${T}_{K}$ that is found to be experimentally accessible. The bare exchange Hamiltonian needs to be decomposed into dipole $(\text{d})$, quadrupole $(\text{q})$, and octupole $(\text{o})$ interactions in order to retain its form under renormalization group (RG) analysis, in which the corresponding exchange constants $({\ensuremath{\lambda}}_{\text{d}},{\ensuremath{\lambda}}_{\text{q}}$, and ${\ensuremath{\lambda}}_{\text{o}})$ flow independently. Numerical solution of the RG scaling equations reveals a few finite fixed points. Arguments are presented that the Fermi-liquid fixed point at low temperature is unstable, indicating that the impurity is overscreened, which suggests a non-Fermi-liquid phase. The impurity contributions to the specific heat, entropy, and the magnetic susceptibility are calculated in the weak coupling regime $(T\ensuremath{\gg}{T}_{K})$, and are compared with the analogous results obtained for the standard case of dipolar exchange interaction (the $s\ensuremath{-}d$ Hamiltonian).

Multipolar Kondo effect in a S 0 1 − P 2 3 mixture of Yb 173 atoms

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