its gradation, gives the effective “gauge” potential. Therefore, the Hamiltonian could be written as a Landau-Ginzberg pseudo-potential form.

Inspired by these observations and a previous work [@Wu:2011tt], we use the same method for Laughlin and Halperin states. Then we obtain complicated Hamiltonians with non-linear interactions. However, they are all exact solvable. The resolving strategy is as follows: firstly, we interpret the ground state as correlation function in CFT. Secondly, by Jastrow transformation we drop the contribution of ground state and obtain a relative simple Hamiltonian. Thirdly, the eigen-equation of the new Hamiltonian can be transformed into an operator equation acting on the coherent basis. Fourthly, it turns out that the operator formalism is exactly triangulated. Therefore we can extract the spectrum as well as the state in a recursive way. Finally, to analyze the integrability closely, we derive the fermionization for the bosonic theory. Hence the integrability is clearly determined by free fermions and the explicit triangularization.

We find, interestingly, the integrability behind Laughlin state, is the same as the famous Calogero-Sutherland model. Hence the excitations are those of Jack polynomials[@macdonald1995symmetric]. During last two decades, people claimed that ground states of some FQHEs have the same properties as those of 