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Physica Scripta

Topological field theory and the discovery of topological materials

Shoucheng Zhang

Published Jan 1, 2012• 5 citations
Publisher

Abstract

Topological field theory (TFT) describes universal low-energy properties of topological states of quantum matter. It also predicts exact quantization effects that can be experimentally measured. Recently, TFT has directly led to the theoretical prediction of topological materials. In this paper, I shall briefly review these recent developments.

DOI
10.1088/0031-8949/2012/t146/014022

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Topological field theory (TFT) describes universal low-energy properties of topological states of quantum matter. It also predicts exact quantization effects that can be experimentally measured. Recently, TFT has directly led to the theoretical prediction of topological materials. In this paper, I shall briefly review these recent developments.
This paper explores the potential of Topological Field Theory (TFT) to revolutionize quantum field theory by introducing dynamic topological structures within the framework of quantum field theory. We argue that current limitations in precisely modeling quantum states, particularly in the context of entanglement, necessitate a novel approach that leverages topological properties. This work proposes a new TFT model predicated on the principles of topology and its application to quantum state calculation, offering a pathway towards more accurate and robust predictions of quantum phenomena. The core mechanism centers around dynamically constructing and manipulating topological landscapes, enabling a deeper understanding of quantum state dynamics and improved mitigation of quantum entanglement challenges.
This paper presents an innovative framework for Topological Field Theory (TFT) based on quantum entanglement, allowing for dynamic adjustment of topological field configurations based on measurement. We introduce the concept of "Quantum Entanglement Topology" – a novel mechanism that leverages quantum entanglement to induce localized, adaptive transformations in the topological field. This approach offers a significant departure from traditional TFT approaches, potentially unlocking deeper geometric insights and providing a pathway to precise prediction of topological structure. The core mechanism relies on establishing a quantum entanglement connection between the field and the measurement process, enabling the field to subtly adjust its shape in response to external influences. This work aims to go beyond the limitations of static geometric transformations in TFT, providing a framework for complex geometric manipulation and enhanced predictive capabilities. We explore the implications of this dynamic approach on the fundamental nature of topological fields.
We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter physics that issuing TSM. Fiber bundle theory is a powerful concept to describe the non-trivial topology properties underlying the physical system. So we briefly present some motivation of fiber bundle theory and following that several effective topological methods have been introduced to judge whether a fiber bundle is trivial or not. Next, we give some topological invariants that characterizes the non-trivial TSM in the non-interacting systems in all dimensions, which is called topological band theory. Following that, we review and generalize the topological response using topological field theory called Chern-Simons effective theory. Finally, the classification of free-fermion systems have been studied by loop space and K-theory.
In recent years an increasing amount of attention has been devoted to quantum materials with topological characteristics that are robust against disorder and other perturbations. In this context it was discovered that topological materials can be classified with respect to their dimension and symmetry properties. This review provides an overview of the classification schemes of both fully gapped and gapless topological materials and gives a pedagogical introduction into the field of topological band theory.