A superfluid flows without friction, like water that never stops. But when you spin it, it doesn't just rotate: each tiny vortex remembers exactly how it was spun. This 'trace' is not a metaphor but a real geometric imprint, which physicists call the Berry phase (the phase acquired by a wave as it travels along a closed path). Michael Berry showed that such phases appear everywhere — from pendulums to quantum systems.

The authors built a theory where this effect is embedded directly into the description of a superfluid with nonzero angular momentum density. The equations contain a single term with an integer coefficient, like a quantum counter of rotations. From it follow three concrete consequences. First, the relativistic Mermin–Ho relation, which links the fluid's vorticity to the arrangement of vortices. Second, the anomalous Ettingshausen effect: if you create a temperature difference, a particle flow appears in the transverse direction. Third, the anomalous Hall viscosity: the fluid resists shear in an unusual way, depending on the direction of rotation.

Such a theory is not an abstraction. It is needed to describe the interiors of neutron stars, where matter is in a superfluid state and rotates rapidly. It is also relevant for experiments with ultracold atomic gases, where quantum vortices are literally visible under a microscope. One term, and so much physics.