Official Resources
- Repository: https://github.com/muammar/heisenberg
- License: MIT License
Overview
Heisenberg is a lightweight Python program designed for the exact analysis of quantum spin chains. It constructs the Hamiltonian matrix for the Spin-1/2 Heisenberg model in the $S^z$ basis, diagonalizes it to extract the full spectrum (eigenvalues and eigenvectors), and computes advanced quantities like the Total Position Spread (TPS) tensor, which is relevant for studying localization and insulating behavior.
Scientific domain: Quantum Magnetism, Educational Physics
Target user community: Students and researchers studying small 1D spin systems
Theoretical Methods
- Exact Diagonalization (ED): Full diagonalization of the Hamiltonian matrix $H$.
- Heisenberg Model: $H = \sum_{i} J \mathbf{S}i \cdot \mathbf{S}{i+1}$ (supports Antiferromagnetic/Ferromagnetic couplings).
- Boundary Conditions:
- Open Boundary Conditions (OBC).
- Periodic Boundary Conditions (PBC).
- Position Spread: Calculation of the second moment of the position operator (TPS tensor) to characterize charge/spin fluctuation.
Capabilities
- Hamiltonian: Generation of sparse matrices for $N$ spins.
- Spectrum: Complete set of energy levels and eigenstates.
- Observables:
- Ground state energy.
- TPS tensor (for distinguishing metals/insulators in electronic mappings).
- Simplicity: Pure Python implementation using SciPy sparse matrices.
Key Strengths
- Accessibility: Less than 500 lines of clear Python code, making it an excellent learning resource for understanding how ED works "under the hood."
- TPS Tensor: One of the few simple codes that explicitly implements the Total Position Spread calculation out-of-the-box.
Inputs & Outputs
- Inputs:
- System size $N$.
- Coupling constant $J$.
- Boundary condition flags.
- Outputs:
- Printed eigenvalues.
- Plots of eigenvalues and TPS scaling.
Interfaces & Ecosystem
- Dependencies: NumPy, SciPy, Matplotlib.
- Format: Standalone script.
Performance Characteristics
- Scaling: Exponential $2^N$. Practical limit on a laptop is $N \approx 14-16$ spins.
- Memory: Stores the Hamiltonian, limiting size compared to Lanczos-based codes (like EDLib) that don't need the full matrix.
Comparison with Other Codes
- vs. QuSpin: QuSpin is a professional research framework for ED; Heisenberg is a minimal script. Use QuSpin for serious research; use Heisenberg for learning or quick checks.
- vs. JHeisenbergED: Similar scope, but written in Python instead of Julia.
Application Areas
- Pedagogy: Teaching quantum mechanics of many-body systems.
- Localization: Testing criteria for localization via the TPS tensor.
Community and Support
- Development: Muammar El Khatib.
- Source: GitHub.
Verification & Sources