Official Resources
- Repository: https://github.com/zjwang11/ir2tb
- License: Unspecified (Research Code)
Overview
ir2tb is a specialized Python tool developed by Zhijun Wang (author of PyWannier90 and IRVSP) for constructing symmetry-adapted tight-binding models directly from irreducible representations (irreps) of crystallographic space groups. It automates the complex group-theoretical task of constraining the Hamiltonian matrix elements so that they respect the full symmetry of the crystal, enabling the construction of minimal effective models for topological analysis.
Scientific domain: Group Theory, Model Construction
Target user community: Theorists building k·p or minimal lattice models
Theoretical Methods
- Group Theory: Analysis of Little Groups and their Irreducible Representations.
- Method of Invariants: Identifying linear combinations of operators that transform trivially under the group operations.
- Basis Construction: Generating symmetry-adapted basis functions (orbitals).
- Tight-Binding: Constraining onsite energies and hopping amplitudes
t_{ij} based on symmetry.
Capabilities
- Model Generation:
- Input: Space group, Wyckoff positions, Irreps at high-symmetry points.
- Output: Symmetry-constrained Hamiltonian matrix.
- Verification: Can check if a given numerical Hamiltonian breaks any crystal symmetries.
- Topological Analysis: Ensures models used for topological classification have the correct symmetry eigenvalues.
Key Strengths
- Rigorous Symmetrization: Avoids human error in manually deriving Hamiltonian constraints, which is notoriously difficult for non-symmorphic space groups.
- Minimal Models: Helps construct the smallest possible model (fewest bands) that captures the essential topology/physics near the Fermi level.
- Integration: Compalementary to
IRVSP (Irreducible Representations of VASP/WIEN2k), creating a workflow from DFT irreps $\to$ TB model.
Inputs & Outputs
- Inputs:
- Python script specifying the symmetry group and orbitals.
- Outputs:
- Symbolic or numerical Hamiltonian matrices.
Interfaces & Ecosystem
- IRVSP: Often used after determining irreps from DFT using the IRVSP code.
- Python: Integrates with NumPy/SymPy.
Performance Characteristics
- Efficiency: Very fast (algebraic operations). Scaling is determined by the size of the group and the number of basis states.
- Constraint: Primarily a "generator" tool; the heavy lifting is done in solving the resulting model.
Comparison with Other Codes
- vs. TBmodels: TBmodels symmetrizes existing numerical models (from Wannier90). ir2tb constructs the form of the model from group theory principles.
- vs. MagneticTB/MagneticKP: Similar concept but ir2tb focusses on non-magnetic space groups and irreps, whereas the others focus on Magnetic Space Groups.
Application Areas
- Topological Semimetals: Constructing effective models for Dirac/Weyl points protected by symmetry.
- Nodal Lines: Studying symmetry-protected band crossings.
Community and Support
- Development: Zhijun Wang (IOP CAS).
- Source: GitHub.
Verification & Sources