Topological Phases in the Quantum Theory.

Abstract: Extend topological field theory to analyze quantum phase transitions, specifically focusing on the emergence of new topological phases. Core Claim: Extend topological field theory to analyze quantum phase transitions, specifically focusing on the emergence of new topological phases. Core Mechanism: Utilize a continuous, topological field to model the evolution of quantum states, enabling the identification of novel topological phases. Why It Is New: Current field theory approaches often treat quantum phases as discrete. This approach aims to capture the dynamic, topological nature of phase transitions.Abstract: Extend topological field theory to analyze quantum phase transitions, specifically focusing on the emergence of new topological phases. Core Claim: Extend topological field theory to analyze quantum phase transitions, specifically focusing on the emergence of new topological phases. Core Mechanism: Utilize a continuous, topological field to model the evolution of quantum states, enabling the identification of novel topological phases. Why It Is New: Current field theory approaches often treat quantum phases as discrete. This approach aims to capture the dynamic, topological nature of phase transitions.

Sep 6, 20260 citations

Topological Considerations in Quantum Theory

Sami M. Al-Jaber

Jan 1, 19910 citations

Topological quantum field theory for Abelian topological phases and loop braiding statistics in (3+1) -dimensions

Qing-Rui Wang, Meng Cheng, Chenjie Wang +1 more

Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1d$ , it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1d$ and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in $3+1d$ . We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1d$, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1d$ and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in $3+1d$. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

arXiv:1810.13428Physical review. B./Physical review. BJun 18, 201932 citations

Title: Topological Phase Field Theory (TPFT)

Jincheng Zhang

This paper introduces Topological Phase Field Theory (TPFT), a novel approach to quantum computation that leverages the principles of topology to enhance stability and adaptability. Traditional quantum algorithms are limited by the inherent noise of the environment, requiring explicit manipulation of quantum states. TPFT overcomes this limitation by incorporating dynamic, topological structures that actively resist environmental disturbances, allowing for self-adaptive quantum state evolution. We present a framework for designing and implementing quantum circuits that exhibit enhanced robustness and can respond to changing environments. The core mechanism centers around constructing and manipulating topological phases within the quantum state, which provides a fundamentally different approach to quantum state management. This work explores the potential of TPFT to surpass the limitations of traditional quantum algorithms, particularly in applications requiring dynamic adaptation and robust operation.This paper introduces Topological Phase Field Theory (TPFT), a novel approach to quantum computation that leverages the principles of topology to enhance stability and adaptability. Traditional quantum algorithms are limited by the inherent noise of the environment, requiring explicit manipulation of quantum states. TPFT overcomes this limitation by incorporating dynamic, topological structures that actively resist environmental disturbances, allowing for self-adaptive quantum state evolution. We present a framework for designing and implementing quantum circuits that exhibit enhanced robustness and can respond to changing environments. The core mechanism centers around constructing and manipulating topological phases within the quantum state, which provides a fundamentally different approach to quantum state management. This work explores the potential of TPFT to surpass the limitations of traditional quantum algorithms, particularly in applications requiring dynamic adaptation and robust operation.

Sep 1, 20260 citations

Classical Topology and Quantum Phases: Quantum Mechanics

A. P. Balachandran

Elsevier eBooksJan 1, 19897 citations

Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In

2+1d

, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in

3+1d

and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in

3+1d

. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.