This simulation models a one-dimensional chain of 32 coupled spins as classical 3-vectors evolving under the Landau–Lifshitz equation, ds/dt = −s × H − α s × (s × H), where the effective field H comes from nearest-neighbour Heisenberg exchange plus an optional transverse field. You see each spin drawn as a coloured arrow, a heatmap and profile curve of its z-component, and you can flip individual spins or kick the centre to launch spin waves that ripple down the lattice.
The Heisenberg model is the foundation of modern magnetism: ferromagnetic coupling (J > 0) aligns neighbouring spins, while antiferromagnetic coupling (J < 0) makes them alternate. The same physics governs magnetic ordering, spin-wave (magnon) excitations in real crystals, and the read/write behaviour of magnetic storage and emerging spintronic devices.
What is the Heisenberg spin chain?
It is a row of magnetic moments (spins) that interact only with their nearest neighbours through an exchange coupling J. Despite its simplicity, the Heisenberg chain captures the essential physics of magnetic ordering and spin waves, and is one of the most-studied models in condensed-matter physics.
What is the difference between ferromagnetic and antiferromagnetic coupling?
With ferromagnetic coupling (J > 0) neighbouring spins lower their energy by pointing the same way, so the chain orders into a uniform magnetised state. With antiferromagnetic coupling (J < 0) neighbours prefer to be opposite, producing an alternating up–down–up pattern called a Néel state.
What is a spin wave?
A spin wave is a collective, wave-like disturbance of the spin directions that propagates through the lattice, much like a ripple on water. Its quantised unit is called a magnon. Kicking the central spin in the simulation launches exactly such a wave, which you can watch travel along the chain.
Each spin follows the Landau–Lifshitz equation ds/dt = −s × H − α s × (s × H). The first term makes the spin precess around the local effective field H, and the second is a damping term (controlled by α) that gradually relaxes the spin toward alignment with H.
J sets the strength of the exchange interaction between neighbouring spins. Larger J means stiffer coupling: spin waves travel faster and ordering is more robust against the transverse field and thermal-like disturbances you introduce by clicking.
B⊥ applies an external magnetic field along the z-axis. It biases the spins toward alignment with the field and adds an extra precession frequency. Increasing it competes with the exchange coupling and changes the character of the spin-wave motion.
Damping (α in the Landau–Lifshitz equation) controls how quickly energy leaves the system. Low damping lets spin waves persist and bounce for a long time; high damping quickly settles the chain into its ordered ground state.
Colour encodes the z-component of each spin: blue marks spins pointing up (+z) and red marks spins pointing down (−z), with intermediate hues in between. This lets you read the magnetic ordering and spot domain walls at a glance, alongside the arrow direction.
No — it treats each spin as a classical unit vector evolving by the Landau–Lifshitz equation, which is the standard semiclassical description of magnetisation dynamics. It faithfully reproduces spin waves, domain walls and ordering, but not genuinely quantum effects such as entanglement between spins.
The Heisenberg chain underlies the magnetism of ferromagnets and antiferromagnets, spin-wave (magnon) propagation used in spintronics and magnonic devices, and the physics of magnetic data storage. Studying such chains also informs research into quantum magnets and one-dimensional materials.