# What is the Fractional Quantum Hall Effect?

The Fractional Quantum Hall Effect (FQHE) is a phenomenon observed in two-dimensional electron gases (2DEGs) subjected to strong magnetic fields and low temperatures. It was discovered in 1982 by Horst Stormer, Daniel Tsui, and Robert Laughlin, who were awarded the Nobel Prize in Physics in 1998 for their work on this topic. Unlike the conventional Quantum Hall Effect, which involves only integral multiples of the fundamental magnetic flux quantum, FQHE exhibits fractional values of this quantity.

The FQHE emerges due to the strong interactions between electrons in the 2DEG, which can form quasi-particles with fractional charges and statistics. These quasi-particles, called anyons, can move around the sample’s edge without being scattered, leading to quantized conductance values. The FQHE is a remarkably robust effect, insensitive to impurities, disorder, and temperature, and has been observed in various materials, including gallium arsenide, graphene, and topological insulators.

# Understanding the Physics Behind FQHE

The physics behind the FQHE is rooted in the concept of topological order, which characterizes the properties of a material that are robust against local perturbations. In FQHE, the topological order arises from the unique non-Abelian statistics of anyons, which can exchange positions in a non-commutative way. This exchange leads to a geometric phase factor that depends only on the path of the exchange, not on the details of the anyons’ trajectories.

The topological order in FQHE can be described by a mathematical framework called the topological quantum field theory (TQFT), which assigns a set of topological invariants to the system. These invariants can be computed using techniques from group theory, representation theory, and category theory. The TQFT approach provides a powerful tool for understanding the properties of FQHE and predicting new phenomena.

# Applications of FQHE in Modern Science

The FQHE has important applications in various areas of modern science, including condensed matter physics, quantum computing, and metrology. In condensed matter, FQHE systems have been used to study the properties of fractional quasi-particles and their interactions, such as the emergence of non-Abelian anyons and the fractional statistics of excitations. In quantum computing, FQHE has been proposed as a platform for topological quantum computation, which is more robust against decoherence than conventional quantum computing.

In metrology, the FQHE has led to the development of the most precise methods for measuring the fine-structure constant and the von Klitzing constant, which are fundamental constants of nature. These measurements rely on the accurate determination of the quantized Hall conductance of FQHE systems, which can be used as a calibration standard for resistance measurements. The FQHE-based metrology has accuracy and stability at the level of parts per billion, making it a valuable tool for metrological applications.

# Example of FQHE in Real-World Research

One example of FQHE in real-world research is the recent discovery of a new type of FQHE in graphene. In 2020, a group of researchers led by Andrea Young at the University of California, Santa Barbara, reported the observation of a surprising FQHE state in graphene in a high magnetic field. The state was characterized by a fractional charge of e/3, which is a third of the elementary charge, and was attributed to the formation of a novel type of anyon called composite fermions.

The discovery of this new FQHE state in graphene represents a significant step towards the understanding of the rich physics of topological order and anyonic excitations in 2D materials. It also has potential applications in quantum computing and metrology, as graphene is a promising material for implementing topologically protected qubits and resistance standards. The study of FQHE in graphene and other 2D materials is a rapidly evolving field that promises to reveal new fundamental physics and practical applications.