Simulation of the Blume-Capel model based on Metropolis algorithm: after initialization the system moves around typical configurations by randomly flipping spins and accepting/rejecting the configuration based on the relative difference in energies (i.e. ratio of probabilties).
The Blume-Capel model is an Ising model with vacancies: spins takes values {±1, 0} with probabilty \(\mathbb{P}[\sigma] \propto e^{-{1\over T} H[\sigma]}\) where the Hamiltonian is given by \[ H[\sigma] = -\sum_{i\sim j}\sigma_i\sigma_j-T \Delta\sum_i\sigma_i^2\] The Temperature \( T\) and the Fugacity \( \Delta \) can be adjusted either with the sliders or by clicking on the phase diagram - their default values are set close to the numerically conjectured value of the critic point (for the square lattice) \(T_c\sim.610\) and \(\Delta_c \sim3.222\).
Simulations can be performed for several lattice sizes and with different boundary conditions:

• Free (no condition on the spins on the boundary);
• Homogeneous (black (+1) spins on the whole boundary);
• Dobrushin (mixed) +/- boundary conditions with +1 spins on one half of the boundary and -1 spins on the other half;
• and Dobrushin +/free boundary conditions with +1 spins on one half of the boundary and no condition on the other half.