关键词: 量子Hall 效应；Berry 相位；拓扑绝缘体；拓扑半金属；拓扑超导体；Dirac 电子； 拓扑性激发；量子输运

Abstract: Exploring the topologically nontrivial states in condensed matter, especially their electronic structure and transport property, is very important and timely. Most topics discussed in this paper are relevant to the topological properties of electronic wavefunctions. Besides a brief introduction, the paper contains five sections, including topological quantum phenomena, various topological phases, unusual transport properties of topological quasiparticles, topological collective excitations and coupled excitations, and ongoing research in topological quantum states. These presentations expound a variety of aspects in the studies of topological quantum states. With all these aspects together, a comprehensive understanding of the topological quantum states in condensed matter physics can be achieved.

Key words: Quantum Hall effect; Berry phase; Topological insulator; Topological semimetal; Topological superconductor; Dirac electron; Topological excitation; Quantum transport