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Topological insulators are new states of quantum matter which can not 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…
Fractional topological insulators are electronic systems that carry fractionally charged excitations, conserve charge and are symmetric to reversal of time. In this review we introduce the basic essential concepts of the field, and then…
Topological insulators are new class of materials which are characterized by a bulk band gap like ordinary band insulator but have protected conducting states on their edge or surface. These states emerge out due to the combination of…
Topological electric quadrupole is a recently proposed concept that extends the theory of electric polarization of crystals to higher orders. Such a quadrupole phase supports topological states localized on both edges and corners. In this…
Conventional topological insulators support boundary states that have one dimension lower than the bulk system that hosts them, and these states are topologically protected due to quantized bulk dipole moments. Recently, higher-order…
A two-dimensional topological crystalline insulator (TCI) with a single unit cell (u.c.) thickness is demonstrated here. To that end, one first shows that tetragonal ($C_4$ in-plane) symmetry is not a necessary condition for the creation of…
This paper proposes a quantitative description of the low energy edge states at the interface between two-dimensional topological insulators. They are modeled by continuous Hamiltonians as systems of Dirac equations that are amenable to a…
The topology of the Brillouin zone, foundational in topological physics, is always assumed to be a torus. We theoretically report the construction of Brillouin real projective plane ($\mathrm{RP}^2$) and the appearance of quadrupole…
We consider the energy spectrum and the spin-parity structure of the eigenstates for a quantum dot made of a strong topological insulator. Using the effective low-energy theory in a finite-length cylinder geometry, numerical calculations…
Recently there is a surge of interests in the so-called topologically protected corner states in 2D and 3D systems. Such systems are considered as high order topological insulators. Wannier centers are used as topological invariants to…
Topological phases of matter are generally characterized by topological properties of energy bands of a system. Their transitions under preserved symmetries occur through closing a gap of energy bands, leading to topologically protected…
Topologically gapless edge states, characterized by topological invariants and Berry's phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons and phonons. Recently, a…
We study the boundary physics of bulk insulators by considering three coupled Hubbard chains in a linear confining potential. In the Hartree-Fock approximation, the ground state at and slightly off the particle-hole symmetric point remains…
The bulk-edge correspondence characterizes topological insulators and superconductors. We generalize this concept to the bulk-corner correspondence and the edge-corner correspondence in two dimensions. In the bulk-corner (edge-corner)…
We identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two and three dimensional crystalline HOT insulators, accommodating topological corner states, we…
Topological phases of matter have been extensively investigated in solid state materials and classical wave systems with integer dimensions. However, topological states in non-integer dimensions remain largely unexplored. Fractals, being…
Topological phases with insulating bulk and gapless surface or edge modes have attracted much attention because of their fundamental physics implications and potential applications in dissipationless electronics and spintronics. In this…
Topological insulators exhibit gapless edge or surface states that are topologically protected by time-reversal symmetry. However, several promising candidates for topologically insulating materials (such as Bi$_2$Se$_3$ and HgTe) contain…
Topological insulators feature a number of topologically protected boundary modes linked to the value of their bulk invariant. While in one-dimensional systems the boundary modes are zero dimensional and localized, in two-dimensional…
The topological edge states of two-dimensional topological insulators with large energy gap furnish ideal conduction channels for dissipationless current transport. Transition metal tellurides XTe5 (X=Zr, Hf) are theoretically predicted to…