Colloquium: Topological band theory

Abstract

The first-principles band theory paradigm has been a key player not only in the process of discovering new classes of topologically interesting materials, but also for identifying salient characteristics of topological states, enabling direct and sharpened confrontation between theory and experiment. We begin this review by discussing underpinnings of the topological band theory, which basically involves a layer of analysis and interpretation for assessing topological properties of band structures beyond the standard band theory construct. Methods for evaluating topological invariants are delineated, including crystals without inversion symmetry and interacting systems. The extent to which theoretically predicted properties and protections of topological states have been verified experimentally is discussed, including work on topological crystalline insulators, disorder/interaction driven topological insulators (TIs), topological superconductors, Weyl semimetal phases, and topological phase transitions. Successful strategies for new materials discovery process are outlined. A comprehensive survey of currently predicted 2D and 3D topological materials is provided. This includes binary, ternary and quaternary compounds, transition metal and f-electron materials, Weyl and 3D Dirac semimetals, complex oxides, organometallics, skutterudites and antiperovskites. Also included is the emerging area of 2D atomically thin films beyond graphene of various elements and their alloys, functional thin films, multilayer systems, and ultra-thin films of 3D TIs, all of which hold exciting promise of wide-ranging applications. We conclude by giving a perspective on research directions where further work will broadly benefit the topological materials field.

Evolution of the topologically protected surface states in superconductorβ-Bi2Pd from the three-dimensional to the two-dimensional limit

“…The prediction and discovery of topological insulators (TIs) [1][2][3][4][5][6][7][8][9][10][11][12][13] have triggered the quest for other classes of materials that host exotic quantum states, such as topological superconductors [14][15][16] and Weyl [17][18][19][20][21], Dirac [22][23][24][25][26], and nodal-line [27,28] semimetals. Similar to TIs, topological superconductors (TSCs) are characterized by a full paring gap in the bulk and topologically protected gapless states on the edge or surface that can support massless Majorana fermions [14,15,[29][30][31].…”

Topological Hopf and Chain Link Semimetal States and Their Application to Co2MnGa

“…Using first-principles band structure calculations, we predict the ferromagnetic full Heusler compound Co2MnGa as a candidate. Both Hopf link and chain-like bulk band crossings and unconventional topological surface states are identified.Since the discovery of Dirac and Weyl semimetals [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], topological semimetals have emerged as an active frontier in condensed matter physics. Their unique topological properties are predicted to give rise to a wide range of exotic transport and optical phenomena [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Tunable current partition at zero-line intersection of quantum anomalous Hall topologies

“…The presence of edge states that are topologically protected from backscattering is one of the striking hall-marks of topologically nontrivial insulators [1][2][3][4][5]. According to the rigorous bulk-edge correspondence rule [6,7], edge states appear at the boundary of two-dimensional topological systems, like quantum Hall effect [8], quantum anomalous Hall effect [9,10], and quantum spin-Hall effect (or two-dimensional topological insulators) [11,12].…”