Conformal invariance in out-of-equilibrium Bose-Einstein condensates governed by the Gross-Pitaevskii equation

We study density isolines in quantum turbulence under the Schramm-Loewner framework using direct numerical simulations of the truncated Gross-Pitaevskii equation, in both spherical and cylindrical traps with three-dimensional dynamics. Density isolines develop increasing complexity as turbulence matures. As the systems evolves towards a thermalized regime, it spontaneously develops conformal invariance. In contrast to other systems exhibiting conformal invariance, this system manifests it during the transition towards disorder rather than to self-organization. We discuss a link between this behavior in quantum turbulence and other 4-wave interacting systems.