Emptiness Formation in Polytropic Quantum Liquids

We study large deviations in interacting quantum liquids with the polytropic equation of state $P(\rho)\sim \rho^\gamma$, where $\rho$ 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 $\gamma$ and the result can be analytically continued to any value of $\gamma\ge 1$. 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.