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Topological insulators (TIs) host novel states of quantum matter, distinguished from trivial insulators by the presence of nontrivial conducting boundary states connecting the valence and conduction bulk bands. Up to date, all the TIs…

The surface of a 3+1d topological insulator hosts an odd number of gapless Dirac fermions when charge conjugation and time-reversal symmetries are preserved. Viewed as a purely 2+1d system, this surface theory would necessarily explicitly…

Topological insulators, featuring bulk-boundary correspondence, have been realized on a large number of noncrystalline materials, among which amorphous network, quasicrystals and fractal lattices are the most prominent ones. By contrast,…

We introduce exotic gapless states---`composite Dirac liquids'---that can appear at a strongly interacting surface of a three-dimensional electronic topological insulator. Composite Dirac liquids exhibit a gap to all charge excitations but…

The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological (SPT) phases, especially for interacting systems. In this paper, we systematically…

We propose that the noncentrosymmetric LiGaGe-type hexagonal $ABC$ crystal SrHgPb realizes a new type of topological semimetal that hosts both Dirac and Weyl points in momentum space. The symmetry-protected Dirac points arise due to a band…

Most topological insulators discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the…

Gapless Dirac surface states are protected at the interface of topological and normal band insulators. In a binary superlattice bearing such interfaces, we establish that valley-dependent dimerization of symmetry-unrelated Dirac surface…

Crystalline symmetries play an important role in the classification of band structures, and the rich variety of spatial symmetries in solids leads to various topological crystalline phases (TCPs). However, compared with topological…

In this tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focussing on the low energy description of higher-order topological insulators in terms of the Dirac equation. Our aim…

Topological phases arise from the elegant mathematical structures imposed by the interplay between symmetry and topology1-5. From gapped topological insulators to gapless semimetals, topological materials in both quantum and classical…

Symmetry and topology are two fundamental aspects of many quantum states of matter. Recently, new topological materials, higher-order topological insulators, were discovered, featuring, e.g., bulk-edge-corner correspondence that goes beyond…

In the recently discovered topological crystalline insulators (TCIs), topology and crystal symmetry intertwine to create surface states with a unique set of characteristics. Among the theoretical predictions for TCIs is the possibility of…

Three dimensional topological insulator crystals consist of an insulating bulk enclosed by metallic surfaces, and detailed theoretical predictions about the surface state band topology and spin texture are available. While several…

Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of the bulk wave functions. This review presents a pedagogical account on…

One of key challenges in current material research is to search for new topological materials with inverted bulk-band structure. In topological insulators, the band inversion caused by strong spin-orbit coupling leads to opening of a band…

We study the properties of a family of anti-pervoskite materials, which are topological crystalline insulators with an insulating bulk but a conducting surface. Using ab-initio DFT calculations, we investigate the bulk and surface topology…

We discuss the proximate phases of a three-dimensional system with Dirac-like dispersion. Using the cubic lattice with plaquette $\pi$-flux as a model, we find, among others phases, a chiral topological insulator and singlet topological…

The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are…

We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological…