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 four-wave interacting systems.

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• Bose-Einstein condensates
• Spatiotemporal chaos
• Turbulence
• Atomic gases
• Conformal symmetry