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In this Book Chapter (invited) we briefly review the basic concepts defining topological insulators and focus on elaborating on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We…

We consider topological invariants describing semimetal (gapless) and insulating (gapped) states of the quantum vacuum of Standard Model and possible quantum phase transitions between these states.

Topological states nurtures the emergence of devices with unprecedented functions in photonics, plasmonics, acoustics and phononics. As one of the recently discovered members, higher-order topological insulators (HOTIs) have been…

Three-dimensional topological insulators represent a new quantum phase of matter with spin-polarized surface states that are protected from backscattering. The static electronic properties of these surface states have been comprehensively…

Following the recent theoretical proposal and experiment on quantum spin Hall effect in HgTe/CdTe quantum wells, we consider a single magnetic impurity localized in the bulk of the system, which we treat as a classical spin. It is shown…

Topological insulators are materials with an insulating bulk interior while maintaining gapless boundary states against back scattering. Bi$_2$Se$_3$ is a prototypical topological insulator with a Dirac-cone surface state around $\Gamma$.…

The surface of a 3D topological insulator is described by a helical electron state with the electron's spin and momentum locked together. We show that in the presence of ferromagnetic fluctuations the surface of a topological insulator is…

A lattice of optical ring resonators can exhibit a topological insulator phase, with the role of spin played by the direction of propagation of light within each ring. Unlike the system studied by Hafezi et al., topological protection is…

Three-dimensional strong topological insulators (TIs) guarantee the existence of a 2-D conducting surface state which completely covers the surface of the TI. The TI surface state necessarily wraps around the TI's top, bottom, and two…

The 2D TI edge states are considered within the Volkov-Pankratov (VP) Hamiltonian. A smooth transition between TI and OI is assumed. The edge states are formed in the total gap of homogeneous 2D material. A pair of these states are of…

Topological insulators are new states of quantum matter with metallic edge/surface states. In this paper, we pointed out that there exists a new type of particle-hole symmetry-protected topological insulator - topological hierarchy…

3D topological insulators are characterized by an insulating bulk and extended surface states exhibiting a helical spin texture. In this work, we investigate the hyperfine interaction between the spin-charge coupled transport of electrons…

The surface of a 3D topological insulator is a new type of two dimensional electron gas (2DEG). The exactly quantized magneto-optical or axionic response of a 3D topological insulator is a direct result of the Berry's phase protected…

Antiferromagnetic topological insulators harbor topological in-gap surface states protected by an anti-unitary $S$ symmetry, which is broken by the inevitable presence of domain walls. Whether an antiferromagnetic topological insulator with…

Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the…

Topological physics opens up a plethora of exciting phenomena allowing to engineer disorder-robust unidirectional flows of light. Recent advances in topological protection of electromagnetic waves suggest that even richer functionalities…

The theory of topological insulators and superconductors has mostly focused on non-interacting and gapped systems. This review article discusses topological phases that are either gapless or interacting. We discuss recent progress in…

The research in topological materials and meta-materials reached maturity and is now gradually entering the phase of practical applications and devices. However, scaling down the experimental demonstrations definitely presents a challenge.…

Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's…

Topological insulators are a novel class of quantum materials in which time-reversal symmetry, relativistic (spin-orbit) effects and an inverted band structure result in electronic metallic states on the surfaces of bulk crystals. These…