Topological quantum field theories (TQFTs) and topological invariants have made significant contributions to our understanding of the quantum states of matter, particularly in the context of topological phases and quantum many-body systems. Here are some key aspects of their contributions:
Characterizing topological phases: Topological quantum field theories provide a powerful framework for characterizing and classifying topological phases of matter. These phases are characterized by global or topological properties that are insensitive to local details and can exhibit robust and exotic phenomena such as fractionalization, anyonic statistics, and protected edge or surface states. TQFTs offer a mathematical description of these topological aspects and provide a systematic approach to understanding their properties and classification.
Topological invariants: Topological invariants are mathematical quantities that capture the topological properties of quantum systems. They are often associated with symmetries or global features of the system and remain unchanged under continuous deformations. These invariants play a crucial role in classifying and distinguishing different topological phases. Examples of such invariants include the Chern number, the TKNN invariant, the topological index, and the winding number. These invariants can provide insights into the structure and behavior of topological phases and are experimentally measurable quantities.
Protected edge or surface states: Topological phases often exhibit unique boundary states that are robust against local perturbations. These protected edge or surface states arise due to the topological properties of the bulk and can exhibit novel phenomena such as chiral currents, Majorana modes, or non-Abelian anyons. TQFTs and topological invariants help elucidate the relationship between the bulk and boundary states, providing a theoretical framework for understanding the origin and stability of these protected states.
Universal properties and critical behavior: Topological quantum field theories also contribute to our understanding of universal properties and critical behavior in quantum many-body systems. By capturing the essential topological and global aspects of the system, TQFTs provide insights into the behavior of the system near critical points, phase transitions, and quantum criticality. They can reveal universal scaling laws, emergent symmetries, and other critical phenomena that arise due to the topological properties of the underlying quantum state.
Overall, topological quantum field theories and topological invariants have revolutionized our understanding of quantum states of matter by providing powerful tools for characterizing and classifying topological phases, describing their properties, and elucidating their emergence and behavior in various quantum systems. They have led to the discovery of new phases of matter and have paved the way for exploring topological quantum computation and other technological applications.