A Passage to Topological Matter: Colloquium

TL;DR

This paper provides a pedagogical overview of topological matter, tracing its development from the integer quantum Hall effect to recent advances in topological semimetals and interaction-induced phases, highlighting key concepts and future directions.

Contribution

It offers a comprehensive, accessible review of the main ideas, mathematical frameworks, and recent developments in topological matter, emphasizing the role of the TKNN formula and Dirac monopoles.

Findings

Topological insulators are characterized by Dirac monopoles in Hamiltonian space.

The TKNN formula links Berry phase to Hall conductivity.

Fractional Chern insulators are proposed as interaction-driven topological phases.

Abstract

Topological matter has become one of the most important subjects in contemporary condensed matter physics. Here, I would like to provide a pedagogical review explaining some of the main ideas, which were pivotal in establishing topological matter as such an important subject. Specifically, I explain how the integer quantum Hall state played the role as a prototype for topological insulator, eventually leading to the concept of topological matter in general. The topological nature of the integer quantum Hall state is best represented by the Thouless-Kohmoto-Nightingale-den Nijs, or so-called TKNN formula, which connects between the Berry phase and the Hall conductivity. The topological non-triviality of topological insulator stems from the existence of a Dirac monopole in an appropriate, but often hidden Hamiltonian parameter space. Interestingly, having the identical Dirac monopole structure, the Hamiltonian describing the Rabi oscillation bears the essence of topological insulator. The concept of topological matter has expanded to include topological semimetals such as Weyl and Dirac semimetals. A final frontier in the research of topological matter is the interaction-induced topological phases of matter, namely, the fractional Chern and topological insulators. The existence of the fractional Chern and topological insulators has been proposed theoretically by drawing an analogy from the fractional quantum Hall states. The gist of this proposal is explained along with some of its issues. I conclude this review by discussing some of the future directions in the research of topological matter.