Ethar Kareem Mnahi Iraqi Ministry of Education-General Directorate of Education in Thi Qar Governorate *Corresponding Author: ethoo2000@gmail.com Received 28 May. 2025, Accepted 22 Sep. 2025, published 30 Dec. 2025.
AbstractKey wordsDOI
Remarkable changes have occurred in condensed matter physics over the past several years with development of topological materials which offer a new way of thinking about phases of quantum systems in terms of topological invariants and quantum phases of matter beyond the traditional ideas involving the breaking of symmetries. The aim of this review is to give a comprehensive theoretical guide to the foundations of the topological phases including topological insulators semimetals and superconductors. It also describes the mathematical instruments to describe these phases e.g. topological invariants and the theory of the Berry phase and the effective quantum field tools, which are important in determining the stability of these materials to external perturbations. Classification systems including the AltlandZernbauer symmetry classes and the periodic table of topological phases will also be discussed as we attempt to understand the existence underlying relationship between symmetry and topological structure in the generation of unconventional quantum states. We also inform about the most noticeable theoretical frameworks, which lead to this field development such as the Haldane model and the KaneMille model and mention some of their applicable domains in new domains like quantum computing spintronics and energy-efficient technologies. Finally, we put into perspective the main technical issues confronting the field forced to draw such issues as the challenges that surround the preparation of materials and the control of their properties to such challenges as complex electronic interactions and environmental stability.
Condensed Matter Physics, Topological Materials, Topological Invariants, Berry Phase, Quantum Field Theory.
Download full article