About the 2D Spin Lattice
The Ising model (a simplification of the Heisenberg model for spin-1/2 particles) places a magnetic spin s = +1 or s = −1 at each site of a 2D square lattice. Nearest-neighbour spins interact via exchange coupling J. The Hamiltonian is H = −J·ΣSᵢSⱼ − μH·ΣSᵢ, where the first sum is over all nearest-neighbour pairs and the second represents coupling to an external field H.
For ferromagnetic coupling (J > 0), parallel spins lower the energy, creating a tendency toward alignment. Below the critical temperature Tc = 2J/(k·ln(1+√2)) ≈ 2.27 J/k, thermal fluctuations are insufficient to disrupt this order and the system develops spontaneous magnetisation — a macroscopic magnetic moment in the absence of any external field. Above Tc, thermal energy overcomes exchange interaction and the system is paramagnetic (disordered). This is the ferromagnetic phase transition.
The simulation uses the Metropolis Monte Carlo algorithm: at each time step, N² random spin-flip attempts are made. Each flip is accepted if it lowers energy, or rejected/accepted with Boltzmann probability if it raises energy. This correctly samples the thermal equilibrium distribution at temperature T. Watch the lattice spontaneously order as you lower T below Tc, and disorder as you raise it above.
Frequently Asked Questions
What is the Heisenberg model of ferromagnetism?
The Heisenberg model describes magnetic interactions in a crystal. Each atom carries a spin (quantum magnetic moment). Neighbouring spins interact via an exchange coupling J: if J > 0 (ferromagnetic), parallel spins lower the energy. The Hamiltonian is H = −J·ΣSᵢ·Sⱼ. In 2D, this is an idealised model of a thin magnetic film.
What is the Metropolis algorithm?
The Metropolis algorithm is a Monte Carlo method for sampling thermal equilibrium configurations. At each step: (1) pick a random spin; (2) propose to flip it; (3) compute ΔE; (4) accept always if ΔE < 0; (5) if ΔE > 0, accept with probability exp(−ΔE/kT). This correctly samples the Boltzmann distribution.
What is the Curie temperature?
The Curie temperature Tc is the critical temperature above which a ferromagnet loses its spontaneous magnetisation. Below Tc, exchange interactions dominate and spins align. Above Tc, thermal fluctuations randomise spins. For the 2D Ising model, Tc ≈ 2.27 J/k. Iron has Tc = 1043 K, nickel 631 K, cobalt 1388 K.
Why do materials become magnetic?
Magnetism arises from quantum mechanical exchange interaction between electrons. When atoms have partially filled d or f orbitals (transition metals like Fe, Co, Ni), orbital overlap leads to exchange energy preferring parallel spin alignment. In iron, this exchange persists up to 1043 K. The macroscopic magnetism we observe is the collective result of ~10²³ aligned electron spins.
What is the difference between ferromagnetism and antiferromagnetism?
In ferromagnetism (J > 0), neighbouring spins prefer to be parallel, leading to spontaneous bulk magnetisation. In antiferromagnetism (J < 0), neighbouring spins prefer to be antiparallel, forming a checkerboard pattern. The net magnetisation of an antiferromagnet is zero. Examples: Fe, Co, Ni are ferromagnetic; MnO, FeO, NiO are antiferromagnetic.
What is a magnetic domain?
A magnetic domain is a region in which all spins are aligned. Domains form because they reduce magnetostatic energy even though they cost exchange energy at domain walls. An unmagnetised iron piece has many cancelling domains. An external field aligns domains; hard magnets trap domains after the field is removed.
What is spin-1/2 and how does it relate to real atoms?
Spin-1/2 means intrinsic angular momentum ℏ/2, with two possible projections: +ℏ/2 (up) or −ℏ/2 (down). Electrons are spin-1/2 particles. In the Ising model, each site has s = +1 or s = −1 representing these states. Real magnetic atoms may have larger spins (e.g., Mn²⁺ has spin-5/2), but spin-1/2 models capture the essential physics.
What is specific heat and why does it peak at Curie temperature?
Specific heat C = dE/dT measures energy absorption per temperature increase. Near a phase transition, large energy fluctuations occur. This causes a peak (divergence in infinite systems) in specific heat at Tc. In real ferromagnets, this λ-shaped peak is observable in calorimetry experiments.
What is the 2D Ising model and is it exactly solvable?
The 2D Ising model was exactly solved by Lars Onsager in 1944. The exact Tc = 2J/(k·ln(1+√2)) and magnetisation M ∝ (1 − sinh⁻⁴(2J/kT))^(1/8) are known analytically. The 3D Ising model is not exactly solvable and requires Monte Carlo or renormalisation group methods.
How is Monte Carlo used in physics beyond spin models?
Monte Carlo methods are widely used in lattice QCD, protein folding, quantum chemistry, nuclear radiation transport, option pricing in finance, and climate modelling. The common thread is random sampling to evaluate high-dimensional integrals or sample from complex probability distributions.