• analogue gravity
• correlation function
• effective field theory
• Hawking radiation

We develop a novel method for building a gravitational analog model for a flowing Bose-Einstein condensate. The analogue metric is obtained using effective field theory methods, integrating out the heavy radial fluctuations. In this way, we also obtain interaction terms up to the quartic order in the fields. The microscopic Lagrangian describes a complex massive scalar field, with a global U(1) symmetry, that is spontaneously broken.
From the quadratic effective Lagrangian, we obtain a dispersion law for phonons in presence of an acoustic horizon generated by the background flow. We observe that the phonon dispersion relation may exhibit a non-monotonic behaviour determined by the Lagrangian’s parameters. In this case, a non-trivial minimum appears associated with a characteristic length scale, indicating the breaking of translation symmetry. Then, we determine the constraints on the Lagrangian’s parameters to make this happen.
Finally, we design an original procedure to calculate transparently the correlation functions through field-theory tools. We apply this method to compute the density-density correlation function, reproducing known results from other methods, with the future perspective of analyzing open problems. In the case of a non-monotonic phonon dispersion law, we find the existence of long-range order, opening the perspective of uncovering intriguing connections between the behavior of horizons and quantum phase transitions.