Topological States of Quantum Condensed Matter

Laura Ortiz Martín

Wei-Feng Tsai, Hsin Lin, Arun Bansil

Shun-Qing Shen

Abstract The discovery of topological insulators and superconductors is an important advance in condensed matter physics. Topological phases reflect global properties of the quantum states in materials, and the boundary states are the characteristic of the materials. Such phases constitute a new branch in condensed matter physics. Here a historic development is briefly introduced, and the known family of phases in condensed matter are summarized.

N.-C. Yeh

The research field of topological phases and materials is one of the most exciting frontiers in modern condensed matter physics [1]; [2]; [3]; [4]; [5]; [6] ; [7]. In contrast to the classification of conventional condensed matter phases according to their symmetries, different topological states of matter generally do not involve broken symmetries. Rather, they are associated with certain fundamental physical properties that are insensitive to smooth changes in materials parameters. Therefore, a topological state cannot change into another unless the system undergoes a quantum phase transition [6] ; [7].

Thomas Klein Kvorning

Mehdi Kargarian

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.