Water boils at one hundred degrees — everyone knows that. But why exactly one hundred? Why not smoothly, but abruptly: a second ago it was liquid, and suddenly — bubbles, steam, bubbling. Such sharp changes are called phase transitions, and they have a rigorous mathematical foundation.
In 1952, Tsung-Dao Lee and Chen Ning Yang proved a theorem that explains when a jump is possible and when it isn't. They worked with a magnet model: tiny magnetic arrows (spins) on a crystal lattice. It turned out that if each arrow individually obeys a certain rule (called the Lee-Yang property), then the whole magnet will also have a sharp phase transition. It's like bricks: if every brick is even, the wall will be straight.
But what if the bricks are uneven? The Blume-Capel model describes exactly such a case. In it, each magnetic arrow has an internal "laziness"—it prefers not to magnetize, to stay at rest. And when this laziness is too strong, the Lee-Yang rule for a single arrow stops working. It would seem that the whole magnet also cannot undergo a true phase transition.
The authors of the new work discovered something surprising. When such "lazy" arrows come together and start interacting, they help each other. Their collective behavior, like a choir where each voice adjusts to its neighbors, unexpectedly restores the Lee-Yang property to the system. Even if the soloist is off-key, the choir as a whole can sound clean.
This is the first result of its kind. Previously, it was thought that if the foundation is crooked, the house will be crooked too. It turns out that connections between floors can straighten the whole structure. This perspective is important not only for the physics of magnets: it hints at how, in very different systems, from alloys to neural networks, order emerges from seemingly disordered parts.
The Lee-Yang theorem was originally proved for magnets, but later it was applied to describe the behavior of liquids, polymers, and even some problems in information theory.