Exact parent Hamiltonians constructed for Gutzwiller ground states of specific
1D spin chains elevated the paradigm from wavefunctions to the models
(one particular eg: Haldane-Shastry model).
The renowned Calogero-Sutherland model (CSM), a many-body system in one spatial
dimension with two-body interaction potential of inverse sine square type, is one among the
simplest integrable continuum microscopic models whose quasiparticle excitations – the
spinons – have abelian fractional statistics [1]. The CSM could be constructed as an exact
parent Hamiltonian with the Laughlin wavefunction as the ground state. We achieve this by obtaining
a destruction operator, calculating its adjoint and forming a non-negative semi-definite
Hermitian operator. Further, inspired by [2] we show the integrability of the model by in-
cluding a two-body particle permutation operator in this destruction operator and thereby
building integrals of motion out of the resulting operator. To proceed further, the Destruction
operators for the Pfaffian continuum wavefunction and the k = 3 Read-Rezayi wavefunction
are derived using CFT null field decoupling technique. Finally, we also construct the parent
Hamiltonian with the Pfaffian wavefunction as the ground state.

[1] Pasquier, V. (1994). A lecture on the Calogero-Sutherland models. In: Alekseev, A., Hi-
etamaki, A., Huitu, K., Morozov, A., Niemi, A. (eds) Integrable Models and Strings. Lecture
Notes in Physics, vol 436. Springer, Berlin, Heidelberg.
[2] Alexios P. Polychronakos, Phys. Rev. Lett. 69, 703 (1992).​