Anyons and the quantum Hall effect - a pedagogical review (0711.4697v1)

Abstract: The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined "anyons", may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.

• The paper provides a pedagogical review linking exotic 2D anyon quasi-particles to the fractional quantum Hall effect, exploring their fractional charge, statistical properties, and non-Abelian characteristics.
• It explores how anyons, with fractional charge and unique statistics governed by geometric phase (Aharonov-Bohm effect), are essential for understanding FQHE states.
• The review highlights the potential of non-Abelian anyons, particularly in the ub = 5/2 state, for fault-tolerant topological quantum computation and discusses experimental methods like interferometry for their observation.

Anyons and the Quantum Hall Effect: A Pedagogical Review

The paper authored by Ady Stern presents a detailed examination of anyons and their connection to the fractional quantum Hall effect (FQHE). Rooted in the breakdown of the fundamental dichotomy between bosons and fermions, anyons represent a unique class of quasi-particles that can only exist in two-dimensional systems. This review provides an expert-level insight into the theoretical constructs and experimental implications of these entities, particularly in relation to the quantum Hall effect.

Initially, the paper delineates the conceptual framework establishing fermions and bosons as defined by the symmetry properties of their wavefunctions upon particle interchange. In two-dimensional spaces, however, particles can exhibit fractional statistics, thus forming anyons. These exotic quasi-particles are key to understanding the FQHE, where the resistivity exhibits quantized plateaus at fractional filling factors, such as 1/3 or 1/5.

Key Points and Theoretical Exploration

1. Fractional Charge and Statistical Properties: The fractional charge is fundamental to the concept of anyons in the FQHE. The constructed wavefunctions, pioneered by Laughlin, illustrate how quasi-holes carrying fractional electronic charge — e.g., $e/3$ for the $\nu = 1/3$ state — are induced by magnetic field and density variations in two-dimensional electron gas systems.
2. Aharonov-Bohm Effect and Quantum Mechanics: Through the Aharonov-Bohm effect, which emphasizes the significance of electromagnetic potentials in quantum mechanics, one can understand the profound implications of anyons on phase accumulations during particle enciddling. The induced geometric phase forms the basis for recognizing anyonic statistics.
3. Experimental Interference and Observation: To observe anyons directly, one must leverage the phase interference phenomena in devices such as the Fabry-Perot and Mach-Zehnder interferometers. These devices not only allow for the detection of the theoretical predictions but also unveil the interference patterns crucial for confirming the presence and properties of anyons.
4. Non-Abelian Anyons: A substantial portion of the review focuses on non-Abelian anyons, specifically citing the $\nu = 5/2$ FQHE state, believed to harbor non-Abelian statistics. These are characterized by a degenerate ground state space and transformative behavior under particle exchanges, thereby offering a platform for fault-tolerant quantum computation.
5. Composite Fermion Theory: By employing a Chern-Simons field theory, the paper elucidates how flux attachment leads to composite fermion formation. These are pivotal in describing many fractional quantum Hall states and are extended to conjecture non-Abelian properties through trial wave functions and parafermionic conformal field theories.

Practical and Theoretical Implications

The practical ramifications involve the potential of harnessing non-Abelian anyons in topological quantum computation, where braiding operations can be employed for robust quantum information processing. Theoretically, the paper aligns anyon physics with broader topological quantum field theories and emphasizes the vital role of experimental verification in substantiating these predictions.

Conclusion

Ady Stern's review is an encompassing exploration of anyonic phenomena, providing a robust framework that connects theoretical predictions with experimental capabilities in observing and utilizing anyons. The fractional quantum Hall effect remains a fertile area for both fundamental research and technological advancement, bridging the gap between abstract quantum mechanics and practical computing applications. As research progresses, further experimental insights could propel our understanding, achieving significant breakthroughs in both condensed matter physics and quantum information.