The Laughlin wave function at filling fraction \(\nu=\frac{1}{m}\)is

\begin{equation}

\Psi_m=\prod_{i<j}(z_i-z_j)^m e^{-\sum|z_i|^2/4l_B^2}

\end{equation}

It is claimed in section 7.2.3 of Wen's book that the wave function of a quasi-hole excitation on the top of this state is

\begin{equation}

\Psi^h(\xi,\xi^*)=\sqrt{C(\xi,\xi^*)}\prod_i(\xi-z_i)\Psi_m

\end{equation}

I am wondering why this is a wave function of a quasi-hole, it looks more like the wave function of a quasi-particle to me. If this is indeed the wave function for a quasi-hole instead of a quasi-particle, what is the wave function of a quasi-particle?

Later in section 7.2.4, the author discusses a way to construct generalization of Laughlin states by adding quasi-holes or quasi-particles on the top of Laughlin states, and when the density of quasi-holes or quasi-particles reaches certain value, they form a Laughlin state by itself. And there are two examples: by condensing quasi-holes of $\nu=1/3$ Laughlin state, we get a $\nu=2/7$ FQH state with wave function:

\begin{equation}

\Psi=\int\prod_id^2\xi_i\Psi_3\prod(\xi_i-z_j)(\xi_i^*-\xi_j^*)e^{-\frac{1}{4l_B^2}\frac{1}{3}|\xi_i|^2}

\end{equation}

and by condensing quasi-particles of $\nu=1/3$ Laughlin state, we get a $\nu=2/5$ FQH state with wave function:

\begin{equation}

\Psi=\int\prod_id^2\xi_i\prod_{i<j}(\xi_i-\xi_j)^2(\xi_i^*-2\partial_{z_i})\Psi_3

\end{equation}

My understanding is, the original electrons and added excitations form a Laughlin state with a common filling fraction, then by charge conservation, we should have

\begin{equation}

{\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}={\rm resulting\ filling}

\end{equation}

However, these two examples are telling me (notice the factor $\frac{1}{2}$):

\begin{equation}

{\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}\times\frac{1}{2}={\rm resulting\ filling}

\end{equation}

(The first example fits as

\begin{equation}

\frac{1}{3}-\frac{1}{3}\times\frac{2}{7}\times\frac{1}{2}=\frac{2}{7}

\end{equation}

and the second example fits as

\begin{equation}

\frac{1}{3}+\frac{1}{3}\times\frac{2}{5}\times\frac{1}{2}=\frac{2}{5}.)

\end{equation}

My questions are:

1) Why should there by a $\frac{1}{2}$ factor?

2) Is it related to the power of $(\xi_i-\xi_j)^2$, namely $2$?

3) If it is related to that $2$, can I change that $2$ into another integer to get another FQH state?