Interplay of Electronic Orders in Topological Quantum Materials

Abstract

Topological quantum materials hold great promise for future technological applications. Their unique electronic properties, such as protected surface states and exotic quasi-particles, offer opportunities for designing novel electronic and spintronics devices and allow quantum information processing. The origin of the interplay between various electronic orders in topological quantum materials, such as superconductivity and magnetism, remains unclear, particularly whether these electronic orders cooperate, compete, or simply coexist. Since the 2000s, the combination of topology and matter has sparked a tremendous surge of interest among theoreticians and experimentalists alike. Novel theoretical descriptions and predictions as well as complex experimental setups confirming or refuting these theories continuously appear in renowned journals. This review aims to provide conceptual tools to understand the fundamental concepts of this ever-growing field. Superconductivity and its historical development will serve as a second pillar alongside topological materials. While the main focus of this review is topological materials such as topological insulators and semimetals, topological superconductors will be explained phenomenologically.

Figure 1

Topological phenomena and materials. In the topmost arrow-bar, the year of the paper of the theoretical prediction is given, followed by the experimental discovery (in topological terms). Additionally, the table gives insight into the currently most interesting characteristic topological phenomena or materials, including diagrams showing characteristic conductance diagrams (only applies to QHE and QAHE) or the energy dispersion relations, qualitatively. Furthermore, exemplary applications and current topics of research are provided. Data taken from refs (, , , −32).

Figure 2

Qualitative characteristic functions of the QHE dependent on magnetic field strength B. (a) Schematic characteristic functions of the transverse (red) and longitudinal (green) resistivity. (b) Quantized Landau levels under a high magnetic field B. Adapted from ref (30). Copyright 2019, Purdue University.

Figure 3

Schematic band structures (green = valence band, blue = conduction band, and red = surface states) of (a) a regular insulator, (b) a Dirac cone with a Dirac point (in yellow), and (c) a TI. The TI exhibits surface states, indicated in red, as well as band inversion due to strong SOC. Adapted with permission from ref (64). Copyright 2021, Springer Nature.

Figure 4

Three-dimensional schematic band structures of (a) DSM and (b) WSM. In DSM, the conduction (blue) and valence (green) bands are connected at a Dirac point (yellow). When TRS or inversion symmetry is broken, the Dirac point splits into two Weyl points of opposite chirality, which are connected at the surface through a Fermi arc (pink). Modified from ref (31). Copyright 2018, Technische Universität Dresden.

Figure 5

Chiral crystals. (a) Example of inversion and rotation symmetry breaking leading to right and left handed materials. (b) Chiral fermions responsible for topologically nontrivial Chern numbers. Reprinted with permission from ref (10). Copyright 2018, Springer Nature.

Figure 6

Schematic magnetization curves of type-I and type-II SCs. Adapted with permission from ref (89). Copyright 2019, Elsevier.

Figure 7

Schematic comparison of a perfect conductor and SC. Reprinted with permission from ref (99). Copyright 2015, Elsevier.

Figure 8

Schematic of electron transport and dispersion relations of chiral and helical SCs opposing their nonsuperconducting quantum Hall and quantum spin Hall relatives, including their dispersion relations. Reprinted with permission from ref (145). Copyright 2009, American Physical Society.