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We calculate the Emptiness Formation Probability (EFP) in the spin-CalogeroModel (sCM) and Haldane-Shastry Model (HSM) using their hydrodynamicdescription. The EFP is the probability that a region of space is completelyvoid of particles in the ground state of a quantum many body system. Wecalculate this probability in an instanton approach, by considering the moregeneral problem of an arbitrary depletion of particles (DFP). In the limit oflarge size of depletion region the probability is dominated by a classicalconfiguration in imaginary time that satisfies a set of boundary conditions andthe action calculated on such solution gives the EFP/DFP with exponentialaccuracy. We show that the calculation for sCM can be elegantly performed byrepresenting the gradientless hydrodynamics of spin particles as a sum of twospin-less Calogero collective field theories in auxiliary variables.Interestingly, the result we find for the EFP can be casted in a formreminiscing of spin-charge separation, which should be violated for anon-linear effect such as this. We also highlight the connections between sCM,HSM and $\lambda=2$ spin-less Calogero model from a EFP/DFP perspective.

Emptiness and Depletion Formation Probability in spin models with inverse square interaction