Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators

Abstract

IntroductionTopological phases of matter differ from conventional materials in that the former feature a nontrivial topological invariant in their bulk electronic wavefunction space [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The experimental discoveries of the two-dimensional (2D) integer and fractional quantum Hall (IQH and FQH) states [15][16][17][18][19] in the 1980s realized the first two topological phases of matter in nature. These 2D topological systems are insulators in the bulk because the Fermi level is located in the middle of two Landau levels. On the other hand, the edges of these 2D topological insulators (TIs) (IQH and FQH) feature chiral 1D metallic states, leading to remarkable quantized charge transport phenomena. The quantized transverse magneto-conductivity = ne 2 ∕h (where e is the electric charge and h is the Planck constant) can be probed by charge transport experiements, which also provides a measure of the topological invariant (the Chern number) n that characterizes these quantum Hall states [20,21]. In 2005, theoretical advances [22, 23] predicted a third type of 2D TI, the quantum spin Hall (QSH) insulator. Such a topological state is symmetry-protected. A QSH insulator can be viewed as two copies of quantum Hall systems that have magnetic fields in the opposite direction. Therefore, no external magnetic field is required for the QSH phase, and the pair of quantum-Hall-like edge modes are related by the time-reversal (TR) symmetry (Figure 4.1). In 2007, the QSH phase was experimentally demonstrated in the (Hg, Cd)Te quantum wells using charge transport by measuring a longitudinal conductance of about 2e 2 ∕h (two copies of quantum Hall currents) at millikelvin temperatures [24]. The topological number that describes the QSH phase is a Z 2 invariant (ν). The Z 2 invariant can only take two values, 0 or 1, where ν = 0(1) is topologically trivial (nontrivial).However, it is important to note that the 2D topological (IQH, FQH, and QSH) insulators are realized only at buried interfaces of ultraclean semiconductor

Bloch state tomography using Wilson lines

“…For example, the eigenvalues of Wilson-Zak loops (i.e., Wilson lines closed by a reciprocal lattice vector) can be used to formulate the Z 2 invariant of topological insulators [7] and identify topological orders protected by lattice symmetries [8,9]. Although experiments have accessed the geometry of isolated bands through various methods including transport measurements [3,12,13], interferometry [14,15], and angle-resolved photoemission spectroscopy [5,16], Wilson lines have thus far remained a theoretical construct [7][8][9][10].…”

Massive Dirac Fermion Observed in Lanthanide-Doped Topological Insulator Thin Films

“…It is important to note, however, that disorder alone is an unlikely cause of the observed gap as it is only observed for Dy-doped films, whereas Ho and Gd doping, which leads to very similar structural and magnetic properties, fails to open up a gap91011. Further experimental studies using spin-resolved ARPES may provide conclusive evidence of the existence of the topological phase in the Dy-doped samples2425. Accompanying first-principles calculations are also urgently needed, which are complicated by the highly correlated, atomic-like nature of the 4 f electrons in Ln systems26.…”

Emerging Dirac materials for THz plasmonics

“…They hold promise for the design of THz metamaterials as discussed in Section 3. Nonetheless, a new generation of topological materials has shown up with the advent of the so-called Weyl and Dirac semimetals (WSMs and DSMs, respectively) [94][95][96]. These are 3D systems with linearly dispersed bands hosting massless fermions that are described by the Dirac and Weyl Hamiltonians.…”