A way to predict how a material behaves at a given temperature by proposing random changes over and over and accepting them with a probability set by their energy – a game of chance, hence the casino’s name. Run long enough, the averages match what an experiment at that temperature would measure.
Going deeper
Monte Carlo sampling turns exchange constants and anisotropies – usually calculated with density functional theory – into a temperature a thermometer could check. For 2D magnets it is the step where the Mermin–Wagner theorem stops being an abstraction.
Sampling instead of counting
A 20 × 20 lattice of up-or-down has 2400, about 10120, configurations – far more than there are atoms in the observable universe, so no thermodynamic average can be computed by enumeration. The Metropolis algorithm sidesteps the problem: propose a small change, accept it outright if it lowers the energy, and otherwise accept it with probability exp(−ΔE/kT). The chain of configurations that results visits each state with exactly the Boltzmann weight it deserves.
What comes out is a set of samples, not an answer, so the usual care applies. Early configurations have to be discarded while the chain equilibrates, successive samples are correlated and do not each count as an independent measurement, and near a both problems get worse as correlation lengths grow. Error bars that ignore autocorrelation are optimistic by a factor nobody can guess from the plot.
From calculated couplings to a Curie temperature
The standard route in 2D magnetism is to compute and from , put them into a Heisenberg model, and sample that model. The is not a detail: in a perfectly isotropic two-dimensional magnet the forbids long-range order at any finite temperature, and a correct simulation reproduces this by losing its order as the simulated sheet is made larger.
Anisotropy gaps the spectrum and rescues the order, but it also makes the magnons interact strongly, which is where the cheaper mean-field and random-phase approximations break down and overestimate the transition temperature badly. Sampling handles that limit correctly, and a fit to simulated results gives a closed expression for the critical temperature in terms of exchange and anisotropy – one that reproduces the 45 K measured for CrI3.
Where the answer comes from
The simulation is only as good as the Hamiltonian it samples. Exchange constants computed with density functional theory depend on the functional and, for correlated 3d elements, on the U chosen; a factor of two in J moves the predicted temperature by roughly the same factor. Classical spins are an approximation that works better for large moments than for spin-½. Itinerant magnets, where the moment itself changes with temperature, are not described by a fixed-moment Heisenberg model at all.
So a predicted is a statement about a model, and it should be quoted that way: which functional, which U, which spin model, which lattice sizes, and how the finite-size trend was extrapolated. Published predictions for the same material routinely differ by a factor of two, and almost always for one of those reasons rather than because of the sampling itself.
For specialists
Stochastic sampling of a thermodynamic ensemble, most often Metropolis sampling of a spin Hamiltonian. In 2D magnetism it is the standard route from exchange and anisotropy parameters – usually computed with DFT – to a Curie or Néel temperature, and it is where Mermin–Wagner physics becomes concrete: without anisotropy the simulated order melts away as the simulated system grows.