A transition with
nothing to point at
Can a system have a genuine phase transition even when there is no such thing as "ordered" for it to become?
Replace the up-or-down magnets with compass needles that can point any direction in the plane, and a theorem takes something away from you. In two dimensions, a system with that kind of continuous freedom cannot settle into a globally aligned state at any temperature above absolute zero. Long-wavelength waves in the needle directions cost almost nothing, and given enough room they always win. So there is no ordered phase, and therefore no order parameter — no single number that is large below some temperature and zero above it.
And yet there is a transition. It just is not about alignment. It is about vortices — points where the needle directions wind all the way around by a full turn as you walk a small loop, like water going down a drain. At low temperature vortices exist only in tightly bound pairs, one spinning each way, each cancelling its partner at any distance. Heat the system past a critical point and the pairs come apart. Free vortices roam, and the system's ability to resist a twist collapses.
That resistance is the thing you can measure. Clamp the needles at one edge, rotate
the clamp at the far edge, and ask how much the system fights back — the helicity
modulus, or spin stiffness. Theory makes an unusually sharp prediction: at the
transition the stiffness does not fade away, it drops off a specific line,
Υ = (2/π)·T. Find where the measured stiffness crosses that line and you
have found the transition, without ever needing an order parameter.
The measured crossing came in at 0.913, against a benchmark of 0.893. The panel below runs the needles. Colour is direction, so the vortices show up as points where the whole colour wheel spins around a single site — watch them pair up as it cools, and tear apart as it heats.
left · every cell is a needle and its
colour is its direction, on a wheel that closes on itself so there is no false
seam. Circles mark vortices: ember for one winding sense, slate for the
other. Cold, they sit almost on top of each other in ± pairs; heat past the crossing
and the pairs visibly come apart and wander off alone. The mean pair separation
in the readout is that unbinding, as a number · right · the stiffness falling with
temperature, crossed by the straight line (2/π)·T. Where they meet is
the transition — no order parameter anywhere in the measurement.
Past the crossing the stiffness is genuinely small, and this short ladder's estimate of a small number is poor — the tail rattles and dips below zero, which is an artefact of 150 samples per temperature rather than physics. The axis is drawn down to −0.2 so you can see that happen instead of having it quietly floored at zero. The crossing itself is measured well before the tail, where the curve is still clean.
The published run — 2026-06-25 — used a 64 × 64 lattice, 26 temperatures across [0.6, 1.1], 40 000 sweeps each with 8 000 discarded to burn-in and one over-relaxation pass per sweep, seed 42, and finished on a GPU in 94 seconds. The over-relaxation is not an optimisation: without it the long-wavelength modes decorrelate so slowly near the transition that the stiffness curve comes out noisy and the crossing is ambiguous.
One number on that receipt deserves to be said out loud rather than quietly omitted. The apparent magnetisation on this finite lattice is not small: it runs from about 0.77 at the cold end to 0.16 at the hot end. A theorem says the magnetisation of the infinite system vanishes at every temperature above zero, and a 64 × 64 lattice will still show you an impressive-looking number that means nothing about the real phase — it would drain away as the lattice grew. That is precisely why the measurement is built on the stiffness instead. The panel prints the apparent magnetisation live so you can watch a plausible, useless number sit there while the real signal is measured somewhere else.
The next rung takes the same prohibition and removes the loophole: give every site an arrow on a sphere instead of a circle and there are no vortices to have a transition with. See M09 — the experiment whose right answer is nothing →