Monte Carlo Physics Simulations

This repository gathers several numerical simulation projects applied to statistical and quantum physics, based on Monte Carlo methods. These implementations were carried out as part of my Master's degree in Physics.

This repository compiles work from two years of study and two distinct courses:

• Master 1 - Monte Carlo Methods (Prof. Torcini):
  • Grade: 20 / 20
  • Content: Introductory scripts in the basics/ folder and the study of the 2D Ising model phase transition (02_ising_2d_phase_transition.ipynb).
• Master 2 - Quantum Monte Carlo (Prof. Honecker):
  • Grade: [en attente de la note] / 20
  • Content: Advanced projects on the 1D Ising model (01_ising_1d_metropolis.ipynb), the Diffusion QMC method (03_diffusion_qmc_ground_state.ipynb), and the Stochastic Series Expansion method (04_sse_heisenberg_chain.ipynb).

Here is an overview of the four main projects. For each model, the physical theory is detailed in the corresponding notebook, and the Monte Carlo algorithms have been implemented in Python to extract fundamental thermodynamic and quantum observables.

File: notebooks/01_ising_1d_metropolis.ipynb

The goal of this project is to simulate the one-dimensional Ising model. Theory predicts the absence of a phase transition at finite temperature. The simulations allow us to visualize the behavior of energy, magnetization, magnetic susceptibility, and specific heat as a function of temperature, confirming a phase change only at $T=0$.

File: notebooks/02_ising_2d_phase_transition.ipynb

Unlike the one-dimensional case, the 2D Ising model exhibits a ferromagnetic-paramagnetic phase transition. This notebook highlights this transition by observing the divergence of the susceptibility and specific heat, allowing us to identify the critical temperature $T_c$.

In 2D, the Ising model underwent a phase transition at Tc.I simulated this phase transition, and here are my results, where Tc can be seen in the magnetization and susceptibility curves:

• magnetisation :
• susceptibility :

I also take the configuration close to Tc, before and after the phase transition:

• T = 1.5 (Before):
• T = 2.304 (close to Tc):
• T = 3.5 (After):

File: notebooks/03_diffusion_qmc_ground_state.ipynb

This project explores the Diffusion QMC method to estimate the ground state energy. By diffusing "walkers" that represent the probability density, we reconstruct the ground state wave function for three distinct potentials: harmonic, cubic anharmonic, and quartic anharmonic. The ground state energies are accurately recovered using an imaginary time extrapolation ($\Delta \tau$).

• Harmonic potential case :
• Extrapolation of the grounds states :

File: notebooks/04_sse_heisenberg_chain.ipynb

This project uses the advanced Stochastic Series Expansion method to simulate a spin chain with the Heisenberg model (ferromagnetic and antiferromagnetic cases). The notebook details the calculation of magnetization, energy, susceptibility, and specific heat for various temperatures down to near absolute zero, thus recovering the ground state energy.

• Antiferromagnetic case result :