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Emptiness Formation in Polytropic Quantum Liquids

We study large deviations in interacting quantum liquids with the polytropic equation of state $P(ρ)\sim ρ^γ$, where $ρ$ is density and $P$ is pressure. By solving hydrodynamic equations in imaginary time we evaluate the instanton action and calculate the emptiness formation probability (EFP), the probability that no particle resides in a macroscopic interval of a given size. Analytic solutions are found for a certain infinite sequence of rational polytropic indexes $γ$ and the result can be analytically continued to any value of $γ\ge 1$. Our findings agree with (and significantly expand on) previously known analytical and numerical results for EFP in quantum liquids. We also discuss interesting universal spacetime features of the instanton solution.