Behind the home page
What you are looking at
The loop behind the home page is a film cooling through a percolation transition and warming back again. Magnetism does not switch on when there is enough of the ferromagnetic phase. It switches on when that phase connects.
- 0–2 s Disorder
An insulating film. Every spin wobbles weakly on its own clock; nothing is magnetically ordered.
- 2–4.9 s Nucleation
As the film cools, puddles of the ferromagnetic phase appear and grow. Inside a puddle the spins tilt and precess together; neighbouring puddles are not in step.
- 4.9 s Threshold
The last link closes and one cluster spans the film from edge to edge. The amber pulse marks that instant, and the only time amber appears.
- 5–8.7 s Coherence
Across the spanning cluster the precession locks into a single travelling spin wave.
- 8.7–12 s Warming
The path breaks, later on the way back than it formed, and the islands shrink back to slate. The last frame is the first.
The picture
Each arrow is one spin on a 64 × 38 square lattice, seen from a camera looking down at 40°. The angle matters: in uniform precession every spin tip sits at the same height, so a spin wave lies flat in the plane of the film and disappears if you look at it edge-on.
Slate arrows are in the insulating phase, where the spins wobble with small, random amplitude and no shared phase. Cyan arrows are in the ferromagnetic phase, where they tilt up and precess together. The faint tiles under the arrows mark the ferromagnetic regions, so the puddles read as areas.
The percolation threshold
Every site carries a threshold drawn from a smoothed random field, which is why the ferromagnetic phase grows as puddles rather than as scattered single sites. A single control parameter p sweeps up and back down; a site is ferromagnetic while p exceeds its threshold.
On every frame the lattice is searched for a cluster that touches both the left and right edges. That cluster is computed, not drawn by hand. For this film it first appears at p ≈ 0.54, below the textbook 0.593 for uncorrelated site percolation on a square lattice, because correlated puddles connect sooner. Just below the threshold most of the film can already be ferromagnetic, yet there is still no path across it.
The way back
The transition in the oxides we study is first order, so it shows hysteresis: on warming, the spanning cluster survives past the point where it formed on cooling, and breaks up later in the cycle.
Why this is the picture we chose
In phase-separated manganite films, we found that magnetic percolation, not composition, governs the magnetization dynamics (Carranza-Celis et al., Phys. Rev. Materials 5, 124413 (2021)). The loop is that result reduced to one image. More in the research line on magnetism dynamics in complex oxides.
What is simplified
This is an illustration built on a real percolation calculation, not a micromagnetic simulation. The cluster geometry and the threshold are computed. The precession frequency, cone angles, wave number and the cooling schedule are chosen to be legible, not taken from a measured material, and there is no damping or exchange interaction between spins.
How it was made
The model is about two hundred lines of JavaScript with a fixed random seed, so every frame can be reproduced exactly. It was rendered frame by frame with HyperFrames: 360 frames at 30 fps, 1920 × 760.