# How to write Fractional Quantum Hall States with Symmetric Polynomials?

Is there a link between Fractional Quantum Hall States and Symmetric Polynomials. In papers of Xiao Gang Wen [1], [2] work out a few examples:

• $\Phi_{1/2} = \prod_{i < j} (z_i - z_j)^2$ is the 1/2 Laughlin State and similarly $\Phi_{1/4} = \prod_{i < j} (z_i - z_j)^4$

• the rest of the paper [2] searches for arithmetic conditions on the exponent to get more polynomials

The more recent paper 1 writes down an explicit state:

$$\prod_{I; i < j} (z^I_i - z^I_j)^{K_{II}} \times \prod_{i; I < J} (z_i^I - z_i^J)^{K_{IJ}} \times e^{- \frac{1}{4}\sum_{i,I} |z_i^I|^2}$$

this time with the Gaussian. Apologies if the letters are getting too small... is corresonds to Chern Simons theory with $\mathcal{L} = \frac{K_{IJ}}{4\pi} a_{I \mu} \partial_\nu a_{J \lambda} \epsilon^{\mu \nu \lambda}$. This is part of Wen's "new mathematical language"

I would like to understand better how to write down these Abelian $U(1)$ Chern-Simons states, and how they relate to Wen's other approaches, in particular:

• symmetric polynomials
• pattern of zeros

There are numerous other references which I haven't had time to parse, written in the same neo-classical style.

• What is your actual question? The proposed wavefunction corresponding to a generic U(1) Chern-Simons theory is already written down... – Meng Cheng Mar 31 '16 at 18:01
• @MengCheng do these symmetric functions change in the same way as Anyons supposedly do? do they exhibit braiding? – john mangual Mar 31 '16 at 18:02
• I found this paper [3] and this question. – john mangual Mar 31 '16 at 18:23
• @MengCheng XGW says in various places is his own theory is incomplete in regards to computing fractional statistics for Abelian anyons – john mangual Mar 31 '16 at 19:32
• See also 'Jack Polynomials' arxiv.org/abs/0707.3637. – Brioschi May 14 '18 at 13:35