Quantum Geometric Tensor and Berry Curvature Visualization and Sample MonteCarlo Simulation of Biopolymer

Overview

This project computes and visualizes the quantum geometric tensor for a simple moiré superlattice model. The quantum geometric tensor includes the quantum metric and Berry curvature, which are essential in understanding geometric and topological properties of electronic wavefunctions in condensed matter physics.

Description

The script calculates the quantum geometric tensor components (g_xx, g_yy, g_xy) and the Berry curvature over a grid of momentum space values (kx, ky). It then visualizes these components as color maps using matplotlib.

How It Works

1. Define the Hamiltonian:

   • The model considers a 2x2 Hamiltonian matrix parameterized by kx, ky and potential V.

2. Compute the Quantum Geometric Tensor:

   • The eigenvalues and eigenvectors of the Hamiltonian are obtained.
   • The ground state wavefunction is numerically differentiated to estimate derivatives.
   • Using these derivatives, the quantum metric tensor and Berry curvature are computed.

3. Visualization:

   • The computed tensor components are plotted over the momentum space grid.
   • A matplotlib heatmap is generated for each component.

Dependencies

Ensure you have the following Python libraries installed:

pip install numpy matplotlib

Usage

Run the script using Python:

python quantum_geometry.py

This will generate and display the visualizations.

Expected Output

The script will display four heatmaps corresponding to:

1. g_xx - Quantum metric tensor component (xx direction)
2. g_yy - Quantum metric tensor component (yy direction)
3. g_xy - Off-diagonal component of the quantum metric tensor
4. Berry Curvature - Topological property of the wavefunction

Each heatmap will be color-coded and labeled accordingly.

Applications

• Studying the geometry of electronic wavefunctions in moiré superlattices.
• Analyzing topological properties of condensed matter systems.
• Exploring Berry curvature effects in quantum Hall systems.

2. Biopolymer Monte Carlo Simulation

Overview

The Biopolymer Monte Carlo Simulation models the behavior of polymer chains by randomly perturbing monomer positions and accepting changes based on the Metropolis criterion.

Features

Implements random-walk initialization of polymer chains.

Uses a Lennard-Jones-like potential for energy calculations.

Employs the Metropolis algorithm for system evolution.

Usage

from biopolymer_monte_carlo import BiopolymerMonteCarlo

num_monomers = 50
num_steps = 10000
temperature = 1.0

sim = BiopolymerMonteCarlo(num_monomers, num_steps)
sim.run_simulation(temperature)
sim.plot_results()

Results

The simulation produces:

Energy vs. Monte Carlo Steps: Tracks energy fluctuations over time.

Final Polymer Configuration: 3D visualization of polymer structure.

Author

Janak Sitaula