A crowd of people who can move freely but suddenly start acting as one: that's how a superconductor carries current without resistance. In cuprate superconductors (materials based on copper and oxygen), this ability behaves strangely: the density of the "superfluid" that carries current first grows with the addition of charge carriers (doping), then falls. As if the crowd becomes more and more cohesive, and then suddenly scatters.

In the new work, scientists started with a model where electrons can hop between atoms but avoid double occupancy, like bus passengers who prefer to sit alone. They added disorder: random atomic-scale bumps that should impede motion. It turned out that the superfluid density barely notices these obstacles in underdoped samples (where charge carriers are fewer than optimal). It grows in direct proportion to the total number of charges, and each new passenger immediately joins the common rhythm.

But in overdoped samples (with excess carriers), the picture changes: superfluidity falls even though the total number of charges keeps growing. This contrast points to a hidden boundary — a transition from a metal where electrons still feel strong mutual repulsion to an ordinary "liquid" metal where they are almost free. It is here, at the end of the pseudogap (an energy step that disappears at a certain doping), that the regime change occurs. Thus, the theory confirms that the superconducting "glue" in cuprates works differently in the two worlds on either side of this point. And so the crowd that seemed unified actually lives by different rules — depending on how many participants it has.