Topological insulators are new states of quantum matter in which surface states residing in the bulk insulating gap of such systems are protected by time-reversal symmetry. The study of such states was originally inspired by the robustness to scattering of conducting edge states in quantum Hall systems. Recently, such analogies have resulted in the discovery of topologically protected states in two-dimensional and three-dimensional band insulators with large spin-orbit coupling. So far, the only known three-dimensional topological insulator is Bi R ecently, the subject of time-reversal-invariant topological insulators has attracted great attention in condensed-matter physics [1][2][3][4][5][6][7][8][9][10][11][12] . Topological insulators in two or three dimensions have insulating energy gaps in the bulk, and gapless edge or surface states on the sample boundary that are protected by time-reversal symmetry. The surface states of a three-dimensional (3D) topological insulator consist of an odd number of massless Dirac cones, with a single Dirac cone being the simplest case. The existence of an odd number of massless Dirac cones on the surface is ensured by the Z 2 topological invariant 7-9 of the bulk. Furthermore, owing to the Kramers theorem, no time-reversalinvariant perturbation can open up an insulating gap at the Dirac point on the surface. However, a topological insulator can become fully insulating both in the bulk and on the surface if a timereversal-breaking perturbation is introduced on the surface. In this case, the electromagnetic response of three-dimensional (3D) topological insulators is described by the topological θ term of the form S θ = (θ /2π)(α/2π) d 3 x dt E · B, where E and B are the conventional electromagnetic fields and α is the fine-structure constant 10 . θ = 0 describes a conventional insulator, whereas θ = π describes topological insulators. Such a physically measurable and topologically non-trivial response originates from the odd number of Dirac fermions on the surface of a topological insulator.Soon after the theoretical prediction 5 , the 2D topological insulator exhibiting the quantum spin Hall effect was experimentally observed in HgTe quantum wells 6 . The electronic states of the 2D HgTe quantum wells are well described by a 2 + 1-dimensional Dirac equation where the mass term is continuously tunable by the thickness of the quantum well. Beyond a critical thickness, the Dirac mass term of the 2D quantum well changes sign from being positive to negative, and a pair of gapless helical edge states appears inside the bulk energy gap. This microscopic mechanism for obtaining topological insulators by inverting the bulk Dirac gap spectrum can also be generalized to other 2D and 3D systems. The guiding principle is to search for insulators where the

In this paper we give the full microscopic derivation of the model Hamiltonian for the three dimensional topological insulators in the Bi2Se3 family of materials (Bi2Se3, Bi2T e3 and Sb2T e3). We first give a physical picture to understand the electronic structure by analyzing atomic orbitals and applying symmetry principles. Subsequently, we give the full microscopic derivation of the model Hamiltonian introduced by Zhang et al [1] based both on symmetry principles and the k · p perturbation theory. Two different types of k 3 terms, which break the in-plane full rotation symmetry down to three fold rotation symmetry, are taken into account. Effective Hamiltonian is derived for the topological surface states. Both the bulk and the surface models are investigated in the presence of an external magnetic field, and the associated Landau level structure is presented. For more quantitative fitting to the first principle calculations, we also present a new model Hamiltonian including eight energy bands.

The search for topologically non-trivial states of matter has become an important goal for condensed matter physics. Recently, a new class of topological insulators has been proposed. These topological insulators have an insulating gap in the bulk, but have topologically protected edge states due to the time reversal symmetry. In two dimensions the helical edge states give rise to the quantum spin Hall (QSH) effect, in the absence of any external magnetic field. Here we review a recent theory which predicts that the QSH state can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the band structure changes from a normal to an "inverted" type at a critical thickness $d_c$. We present an analytical solution of the helical edge states and explicitly demonstrate their topological stability. We also review the recent experimental observation of the QSH state in HgTe/(Hg,Cd)Te quantum wells. We review both the fabrication of the sample and the experimental setup. For thin quantum wells with well width $d_{QW}< 6.3$ nm, the insulating regime shows the conventional behavior of vanishingly small conductance at low temperature. However, for thicker quantum wells ($d_{QW}> 6.3$ nm), the nominally insulating regime shows a plateau of residual conductance close to $2e^2/h$. The residual conductance is independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance is destroyed by a small external magnetic field. The quantum phase transition at the critical thickness, $d_c= 6.3$ nm, is also independently determined from the occurrence of a magnetic field induced insulator to metal transition.Comment: Invited review article for special issue of JPSJ, 32 pages. For higher resolution figures see official online version when publishe

Following the discovery of the Fe-pnictide superconductors, LDA band structure calculations showed that the dominant contributions to the spectral weight near the Fermi energy came from the Fe 3d orbitals. The Fermi surface is characterized by two hole surfaces around the Γ point and two electron surfaces around the M point of the 2 Fe/cell Brillouin zone. Here, we describe a 2-band model that reproduces the topology of the LDA Fermi surface and exhibits both ferromagnetic and q = (π, 0) spin density wave (SDW) fluctuations. We argue that this minimal model contains the essential low energy physics of these materials.PACS numbers: 71.10. Fd, 71.18.+y, 74.20.Mn, 74.25.Ha, 75.30.Fv Introduction -The recent discovery of superconductivity in a family of Fe-based oxypnictides with large transition temperatures [1,2,3,4,5,6] has led to tremendous activity aimed at identifying the mechanism for superconductivity in these materials. Preliminary experimental results including specific heat [7], point-contact spectroscopy [8] and high-field resistivity [9, 10] measurements suggest the existence of unconventional superconductivity in these materials. Furthermore, transport [11] and neutron scattering [12] measurements have shown the evidence of magnetic order below T = 150K. An experimental determination of the orbital and spin state of the Cooper pairs, however, has not yet been made.