Official Resources
- Repository: https://github.com/ryanlevy/HubbardModel2D
- License: MIT License
Overview
HubbardModel2D is a specialized C++ code for performing Exact Diagonalization (ED) of the single-orbital Fermi-Hubbard model on 1D chains and 2D square lattices. It uses the Lanczos algorithm to compute ground state energies, wavefunctions, and spectral properties. While primarily designed for small clusters due to the exponential scaling of ED, it serves as a transparent and efficient tool for benchmarking and studying strong correlation effects on finite lattices.
Scientific domain: Strongly Correlated Electrons, Quantum Magnetism
Target user community: Students and researchers characterizing the Hubbard model
Theoretical Methods
- Fermi-Hubbard Model: $H = -t \sum_{\langle ij \rangle \sigma} (c_{i\sigma}^\dagger c_{j\sigma} + h.c.) + U \sum_i n_{i\uparrow} n_{i\downarrow}$.
- Exact Diagonalization: Construction of the full sparse Hamiltonian in the occupation number basis for fixed particle number and spin.
- Lanczos Algorithm: Iterative method to find the lowest eigenvalues and eigenvectors (Ground State).
- Conserved Quantities: Exploits $N_{\uparrow}$ and $N_{\downarrow}$ conservation (but not spatial symmetries).
Capabilities
- Simulations:
- Ground State Energy $E_0$.
- Wavefunction amplitues.
- Spectral Functions $A(\omega)$ (via Spectra library).
- Geometries:
- 1D Chains (PBC/OBC).
- 2D Square Clusters ($2 \times 2$, $2 \times 4$, $4 \times 4$, etc.).
- Observables:
- Spin-spin correlations $\langle S_i \cdot S_j \rangle$.
- Charge correlations $\langle n_i n_j \rangle$.
Key Strengths
- Simplicity: Codebase is relatively small and uses the highly readable
Eigen library for linear algebra, making it easy to modify.
- Backend Options: Can switch between
ietl (classic) and Spectra (modern) for the Lanczos implementation.
- Benchmarking: Ideal for generating exact reference data for verifying approximate methods like VMC or DMRG on small systems.
Inputs & Outputs
- Inputs: C++ main file modification (recompilation required for changes) to set $t$, $U$, lattice size.
- Outputs:
- Console output of energies.
- Text files for correlation functions.
Interfaces & Ecosystem
- Dependencies: Eigen3, Spectra (header-only).
- Ecosystem: Standalone C++ tool.
Performance Characteristics
- Scaling: Exponential Hilbert space growth. Feasible for up to $\sim 16$ sites (e.g., $4 \times 4$ at half-filling is pushing memory limits).
- Parallelism: OpenMP shared-memory parallelism for matrix-vector multiplication.
Comparison with Other Codes
- vs. ALPS: ALPS handles full spatial symmetries, extending the reach to slightly larger systems ($\sim 20$ sites). HubbardModel2D is simpler but restricted to smaller sizes.
- vs. EDLib: Similar scope; HubbardModel2D is an application for the Hubbard model, while EDLib is a library.
Application Areas
- Mott Physics: Studying the metal-insulator transition in small clusters.
- Pairing Correlations: Checking for d-wave pairing signatures in the 2D Hubbard ground state.
Community and Support
- Development: Ryan Levy.
- Source: GitHub.
Verification & Sources