Building a 2D Model Using the Ising Model Concept with Monte Carlo Implementation

In our course project, we first implemented a two-dimensional model based on the Ising model concept. We then employed importance sampling and Monte Carlo methods with a Metropolis-Hastings algorithm to simulate ferromagnetic-paramagnetic phase transitions. The simulation code calculates thermodynamic properties including average energy (Ev), heat capacity (Cv), magnetization (M), and magnetic susceptibility (X) for paramagnetic materials through statistical averaging over Monte Carlo steps.

We investigated the temperature dependence of these properties by generating Ev-T, Cv-T, M-T, and X-T plots using data visualization libraries. Our implementation revealed that:

- The internal energy of paramagnetic materials initially increases with temperature before converging to a stable value, as computed through Boltzmann distribution sampling.

- Both heat capacity Cv and magnetic susceptibility X show non-monotonic behavior, peaking before decreasing with rising temperature, calculated using energy fluctuations and magnetization variance respectively.

- Magnetization M drops sharply to zero at the transition temperature Tc, determined through spin alignment analysis in the Monte Carlo simulation.

- The ferromagnetic phase transition temperature Tc was numerically determined to be approximately 2.35 (in reduced units), identified from the magnetization curve inflection point.

These computational results demonstrate significant changes in thermodynamic properties with temperature, particularly the rapid disappearance of magnetization at Tc. The findings provide valuable insights into phase transition mechanisms, validated through our Monte Carlo implementation with proper boundary conditions and equilibrium checks.