[Paper Review] Fractional Statistics and Chern-Simons Field Theory in 2+1 Dimensions

Avinash Khare|ArXiv.org|Aug 4, 1999

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper provides a comprehensive field-theoretic analysis of fractional statistics and anyons in 2+1 dimensions using Chern-Simons (CS) field theory. It establishes that fractional statistics arises naturally in two spatial dimensions due to topological properties of the CS term, which acts as a gauge field mass term, and demonstrates that anyons emerge either as solitonic solutions or as fundamental quanta with fractional statistics, unifying two distinct realizations within a single framework.

ABSTRACT

The question of anyons and fractional statistics in field theories in 2+1 dimensions with Chern-Simons (CS) term is discussed in some detail. Arguments are spelled out as to why fractional statistics is only possible in two space dimensions. This phenomenon is most naturally discussed within the framework of field theories with CS term, hence as a prelude to this discussion I first discuss the various properties of the CS term. In particular its role as a gauge field mass term is emphasized. In the presence of the CS term, anyons can appear in two different ways i.e. either as soliton of the corresponding field theory or as a fundamental quanta carrying fractional statistics and both approaches are elaborated in some detail.

Motivation & Objective

• To clarify why fractional statistics is exclusively possible in two spatial dimensions.
• To analyze the role of the Chern-Simons term as a gauge field mass term in 2+1D field theories.
• To unify two realizations of anyons: as solitonic solutions and as fundamental quanta with fractional statistics.
• To provide a field-theoretic foundation for anyons in topological field theories relevant to condensed matter systems.
• To establish the theoretical basis for anyonic statistics within the framework of Chern-Simons gauge theories.

Proposed method

• Use of Chern-Simons field theory as the primary theoretical framework to describe anyons in 2+1 dimensions.
• Analysis of the CS term's role in generating effective mass for gauge fields and enabling topological interactions.
• Derivation and examination of the statistical properties of anyons via path integral and canonical quantization approaches.
• Construction of solitonic solutions (e.g., vortex-like configurations) that carry fractional statistics.
• Investigation of fundamental quanta carrying fractional statistics in the presence of the CS term.
• Comparison of the solitonic and fundamental anyon pictures within a consistent field-theoretic formulation.

Experimental results

Research questions

• RQ1 Why is fractional statistics only possible in two spatial dimensions?
• RQ2 How does the Chern-Simons term act as a gauge field mass term in 2+1 dimensions?
• RQ3 What is the field-theoretic mechanism that leads to fractional statistics in anyons?
• RQ4 How do solitonic solutions of the CS theory realize anyonic statistics?
• RQ5 Can fundamental quanta in CS theory carry fractional statistics, and how does this compare to solitonic anyons?

Key findings

• Fractional statistics is fundamentally tied to the topological nature of the Chern-Simons term in 2+1 dimensions, which is absent in higher dimensions.
• The Chern-Simons term generates a mass for the gauge field, providing a dynamical mechanism that supports anyonic statistics.
• Anyons can emerge as solitonic solutions (e.g., vortices) carrying fractional statistics due to the topological flux attachment mechanism.
• Fundamental quanta in the presence of the CS term also carry fractional statistics, demonstrating a dual realization of anyons.
• The field-theoretic framework unifies solitonic and fundamental anyons under a single gauge-theoretic description.
• The paper establishes that the CS term is essential for realizing fractional statistics in a consistent quantum field theory in 2+1D.