home | literature reviews

"Wavefunctionology: The special structure of certain fractional quantum Hall wavefunctions", Steven Simon, 2020

Reviewed October 7, 2023

Citation: Simon, Steven H. "Wavefunctionology: The special structure of certain fractional quantum Hall wavefunctions." Fractional Quantum Hall Effects: New Developments. 2020. 377-434.

Web: https://arxiv.org/abs/2107.00437

Tags: Expository, Pedagogical, Quantum-hall-effect, Conformal-field-theory, TQFT

This paper details the beautiful theory of how to mathematically classify the fractional quantum hall system at different filling values. The beautiful fact is that, unlike most condensed matter systems, the wavefunctions in FQHE can be explicitly written down, and are thus highly amenable to detailed mathematical study. This is a review article of the major approaches towards this goal, as well as a great mathematical introduction to the FQHE.

The main focus of this article is on those filling levels in which wavefunctions can be explicitly written down. Such descriptions are not present at every filling level, and Simon spends a lot of time talking about why this is the case.

Unlike other reviews in this area, Simon uses a minimal amount of conformal field theory.

Mathematically, the approach I find the most interest is the "pattern of zeroes" approach pioneered by Zhenghan Wang:

> Wen, Xiao-Gang, and Zhenghan Wang. "Pattern-of-zeros approach to Fractional quantum Hall states and a classification of symmetric polynomial of infinite variables." Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research. Springer Berlin Heidelberg, 2014.

However, seeing as this review is one of four papers that cite it I can assume that it is not very popular among the wider community. In Simon's words, "the pattern of zeros approach needs to be supplemented with additional information in order to uniquely define a particular wavefunction".