these steps, we have $$\begin{aligned}
b^{3}\sum_{i<j}\,\sum_{n>0}a_{-n}(z_{i}^{n}+z_{i}^{n-1}z_{j}+&\cdots&+z_{i}z_{j}^{n-1}+z_{j}^{n})  =  \frac{b}{2}\sum_{n,m>0}a_{-n-m}a_{n}a_{m}\nonumber \\
-\frac{1}{2}b^{2}\sum_{n>0}na_{-n}a_{n} & + & b^{2}(N-1)\sum_{n>0}a_{-n}a_{n\,,}\label{eq:interoper}\end{aligned}$$ The last term of ($\ref{eq:interoper}$) is an irrelevant total energy of free quasi-particles. When acting on a level $n$ state, it gives rise to an eigen-energy $$E_{ir}|n\rangle=b^{2}(N-1)n|n\rangle\,$$ as expected which describes a system of $N-1$ copies of non-interacting bosonic oscillators with the frequency $b^{2}$. Therefore we ignore its contribution and now we obtain the operator formalism of $H_{L}$

$$H_{L}=\sum_{n}\left(1-\frac{b^{2}}{2}\right)na_{-n}a_{n}+\sum_{n,m}b\left(a_{-n}a_{-m}a_{n+m}+\frac{1}{2}a_{-n-m}a_{n}a_{m}\right)\,.\label{eq:HLaughlin}$$

Fermionization of $H_{L}$
-------------------------

Define the deformed bosonic modes as $$\begin{aligned}
2b^{-1}\tilde{a}_{n} & = & a_{n}\, , \quad \frac{1}{2}b\tilde{a}_{-n}=a_{-n\,,}\label{eq:deformLau}\end{aligned}$$ so that the triple-$a$ terms in ($\ref{eq:HLaughlin}$) now becomes $$\begin{aligned}
&&\sum_{n,m>0}\left(\frac{1}{2}b^{2}\tilde{a}_{-n}\tilde{a}_{-m}\tilde{a}_{n+m}+\tilde{a}_{-n-m}\tilde{a}_{n}\tilde{a}_{m}\right)\\
&=&\sum_{n,m>0}\left(\tilde{a}_{-n}\tilde{a}_{-m}\tilde{a}_{n+m}+\tilde{a}_{-n-m}\tilde{a}_{n}\tilde{a}_{m}\right)  -  (1-\frac{b^{2}}{2})\tilde{a}_{-n}\tilde{a}_{-m}\tilde{a}_{n+m}\,.\end{aligned}$$ It is straightforward to write down the fermionic formalism as the case of the Calogero-Sutherland model in previous section. Hence we get $$H_{L}=H_{0}+(1-\frac{b^{2}}{2})H_{d}+(1-\frac{b^{2}}{2})H_{t}\equiv H_{l}+(1-\frac{b^{2}}{2})H_{t}\,.\label{eq:LaughlinHamilton}$$ The eigen-energy for this Hamiltonian is $$E_{\lambda}^{Lau}=\sum_{i=1}^{\lambda_{1}^{t}}(\lambda_{i})^{2}-\frac{b^{2}}{2}\sum_{i=1}^{\lambda_{1}}(\lambda_{i}^{t})^{2}\,.\label{eq:eigenLau}$$ The corresponding excitation state is $$|P_{\lambda}^{Lau}\rangle=D^{Lau}\frac{1}{1-\frac{1-b^{2}/2}{E_{\lambda}^{Lau}-H_{l}}H_{t}}s_{\lambda}(\psi\psi^{*})|vac\rangle\,,$$ where $D^{Lau}$ is the similarity transformation related to ($\ref{eq:deformLau}$). $$D^{Lau}=\exp\bigl(-\log(b/2)(\tilde{q}\tilde{a}_{0}+\sum_{n>0}\tilde{a}_{-n}\tilde{a}_{n}/n)\bigr)\,,$$ and the fermionic expression: $$D_{\pm}^{Lau}=\left(\frac{b}{2}\right)^{\pm\sum_{r>0}(\psi_{-r}\psi_{r}^{*}+\psi_{-r}^{*}\psi_{r})}\,.$$ Notice that the polynomial is not a new polynomial, it is a Jack polynomial with the parameter $\beta=\frac{b^{2}}{2}\,.$

Since the integrability of the Laughlin theory is exactly the same as that of CS model we ignore its analysis here.

Halperin State and 