Condensed Matter Model

2D Ising Model Simulation using the Metropolis Algorithm

Overview

This repository contains a 2D Ising model simulation implemented in Python. The simulation uses the Metropolis-Hastings method to evolve a grid of spins (values ±1). A Matplotlib animation shows how the spin configuration changes over time.

Features

• Random Initial Lattice: Creates an (N \times N) array of spins (\pm 1).
• Local Energy Updates: Computes the energy cost of flipping a single spin using its four nearest neighbors (periodic boundary conditions).
• Metropolis Sweeps: Performs a full sweep by visiting every lattice site in random order and deciding whether or not to flip the spin.
• Interactive Animation: Uses matplotlib.animation.FuncAnimation to visualize the evolution of the spin lattice.
• User-Configurable: Lattice size N, temperature T, total sweeps steps, and animation interval plot_interval are all adjustable.

Requirements

• Python 3.x
• NumPy
• Matplotlib

Install via:

pip install numpy matplotlib

How to Run

1. Clone or download this repository.
2. Open a terminal in the repository folder.
3. Run the Python script:

python ising.py

or (if you modify it to accept command-line args):

python ising.py --N 50 --T 2.2 --steps 1000 --plot_interval 50

A Matplotlib window will appear, showing the spins (red/blue) as the simulation proceeds.

Code Explanation

1. initialize_lattice(N)
   Creates an (N \times N) NumPy array of ±1 spins.

2. energy_change(lattice, i, j, J=1.0)
   Calculates the energy difference (\Delta E) if we flip the spin at ((i, j)), using periodic boundary conditions.

3. metropolis_step(lattice, beta)

• Visits each site once in random order.
• Computes (\Delta E).
• Decides whether to flip the spin based on the Metropolis criterion ( \exp(-\beta \Delta E)).

4. run_simulation(N=50, T=2.2, steps=1000, plot_interval=50)

• Initializes the lattice.
• Creates a Matplotlib figure and uses FuncAnimation to periodically update the lattice and refresh the plot.
• Returns the final configuration after the animation completes.

Customization

• Lattice Size (N): Change to see finite-size effects.
• Temperature (T): Explore the phase transition near T≈2.27 in 2D.
• Sweeps (steps): Longer runs allow the system to equilibrate.
• Plot Interval: Controls how frequently the animation updates.

License

Feel free to use, modify, and distribute this code for educational or research purposes. A link back here is always appreciated.

Enjoy exploring the 2D Ising model!