A Passage to Topological Matter: Colloquium
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 matter, eventually leading to the concept of topological insulator. The topological nature of the integer quantum Hall state is best represented by the ThoulessKohmotoNightingaleden Nijs, or socalled TKNN formula, which connects between the Berry phase and the Hall conductivity. The topological nontriviality 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 interactioninduced 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.
 Publication:

Journal of Korean Physical Society
 Pub Date:
 September 2018
 DOI:
 10.3938/jkps.73.817
 arXiv:
 arXiv:1809.07947
 Bibcode:
 2018JKPS...73..817P
 Keywords:

 Topological matter;
 Berry phase;
 Integer quantum Hall state;
 Chern insulator;
 Topological insulator;
 Topological semimetal;
 Fractional quantum Hall state;
 Fractional Chern insulator;
 Fractional topological insulator;
 Floquet topological insulator;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 16 pages, 8 figures