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"In and around Abelian anyon models", Zhenghan Wang, Liang Wang, 2020

Reviewed September 6, 2023

Citation: Wang, Liang, and Zhenghan Wang. "In and around Abelian anyon models." Journal of Physics A: Mathematical and Theoretical 53.50 (2020): 505203.

Web: https://arxiv.org/abs/2004.12048

Tags: Abelian-anyons, Expository

This paper gives a nice review of the modern state of the theory of abelian anyon models. The main takeaway is that abelian anyon models are classified by the underlying abelian group of their fusion rules, and a quadratic form giving the twist. Surprisingly, not everything about abelian anyons is well known - there are still open problems and conjectures about their structure.

This paper also gives a nice algorithm method to compute a "K-matrix" for a given abelian model. That is, a matrix which realizes the underlying quadratic form. This matrix is constructed to be maximally nice. In this case, "maximally nice" generally means "positive definite".

One interesting line at the beginning of this paper is a warning: "There is a well-known subtlety in using abelian anyon models for quantum Hall physics: quantum Hall liquids are electron systems, while anyon models are for bosonic/spin systems. The situation is well-understood and we will regard anyon models as the statistics sectors of anyon physics and the constituent electron is modelled by a charge sector. The gluing of the statistics sectors and the charge sectors can be non-trivial, which is important for the detection of anyon statistics via edge current measurement". The reference given for this claim is

> Chamon, C. de C., D. E. Freed, and Xiao-Gang Wen. "Nonequilibrium quantum noise in chiral Luttinger liquids." Physical Review B 53.7 (1996): 4033.

This paper gives a lot of good explicit work for abelian theories, and is definitely a good reference.