The Physics Society of Iran Iranian Journal of Physics Research 1682-6957 20 1 2020 05 21 Principles of topology in understanding and development of topological states of matter Principles of topology in understanding and development of topological states of matter 1 22 1606 10.47176/ijpr.20.1.11111 FA M Kargarian Department of physics, Sharif University of Technology, Tehran, Iran Journal Article 2020 04 08 By using concepts of topology in mathematics, quantum mechanics, and their synergetic development during the past few decades, condensed matter physicists have discovered new phases of matter and introduced general frameworks to classify them. The research includes a vast gamut from chemistry of atomic orbitals to material science, promising new applications in the growing technologies. This review article aims to provide a better understanding of these unprecedented electron systems and their underlying topological principles. The article consist of two parts. First, there is a historical review of using topological concepts in condensed matter systems. Then, in the second part, we elaborate on some basics of topology in quantum mechanics and the concept of topological invariants. By using concepts of topology in mathematics, quantum mechanics, and their synergetic development during the past few decades, condensed matter physicists have discovered new phases of matter and introduced general frameworks to classify them. The research includes a vast gamut from chemistry of atomic orbitals to material science, promising new applications in the growing technologies. This review article aims to provide a better understanding of these unprecedented electron systems and their underlying topological principles. The article consist of two parts. First, there is a historical review of using topological concepts in condensed matter systems. Then, in the second part, we elaborate on some basics of topology in quantum mechanics and the concept of topological invariants. geometrical and berry phases topology topological materials Chern invariant https://ijpr.iut.ac.ir/article_1606_495dabfd0ca768a3c3abd672079f48b6.pdf