Official Resources
- Homepage: https://github.com/kuansenlin/nested_and_spin_resolved_Wilson_loop
- Repository: https://github.com/kuansenlin/nested_and_spin_resolved_Wilson_loop
- License: MIT License
Overview
nested_wloop is a specialized Python toolkit designed to extend PythTB for the characterization of Higher-Order Topological Insulators (HOTIs) and Fragile Topology. It implements the numerical calculation of Nested Wilson Loops—a hierarchical Berry phase technique required to identify quadrupole and octupole insulators—as well as Spin-Resolved Wilson loops for systems with approximate time-reversal symmetry.
Scientific domain: Higher-Order Topology, Fragile Phases
Target user community: Theorists characterizing HOTIs and Twisted Bilayers
Theoretical Methods
- Nested Wilson Loop: Diagonalization of the standard Wilson loop unitary $W_1$, followed by a second Wilson loop calculation along the remaining direction for the Wannier bands of $W_1$.
- Spin-Resolved Wilson Loop: Projecting the Wilson loop onto spin sectors ($P_{\uparrow} W P_{\uparrow}$) to define Spin Chern numbers in the absence of $S_z$ conservation.
- Multipole Moments: Calculation of bulk quadrupole moments $q_{xy}$ via nested geometric phases.
Capabilities
- HOTI Diagnosis:
- Identifies 2nd order topological insulators (corner states).
- Calculates quadrupole invariants.
- Fragile Topology:
- Detects "Fragile" bands that have trivial Chern number but non-trivial spin winding.
- Workflow:
- Takes a
pythtb.tb_model as input.
- returns nested Wannier charge center spectra.
Key Strengths
- Cutting Edge: One of the few public implementations of the Benalcazar-Bernevig-Hughes (BBH) nested loop procedure.
- Integration: Works directly with PythTB models, meaning users don't need to rewrite their Hamiltonians to use this advanced analysis.
- Fragile Phases: Essential for the modern study of "Wannier Obstructions" in twisted bilayer graphene and similar moiré systems.
Inputs & Outputs
- Inputs: PythTB model object.
- Outputs: Plots of Nested Wannier Spectra ($\nu_y$ vs $k_x$).
Interfaces & Ecosystem
- Dependency: PythTB (must be installed).
- Language: Python 3.
Performance Characteristics
- Complexity: $O(N_k^2)$ due to the nested integration. Slower than standard Berry phase but manageable for typical tight-binding grids ($100 \times 100$).
- Scalability: Serial execution.
Comparison with Other Codes
- vs. Z2Pack: Z2Pack does standard Wilson loops perfectly. nested_wloop adds the specific nested functionality for HOTIs which Z2Pack doesn't natively expose in a "one-shot" function.
- vs. WannierTools: WannierTools computes corner states via creating finite clusters (real space). nested_wloop predicts them from the bulk Bloch functions (momentum space).
Application Areas
- Quadrupole Insulators: Checking the $\mathbb{Z}_2 \times \mathbb{Z}_2$ classification.
- Twisted Bilayers: Diagnosing fragile topology in flat bands.
Community and Support
- Development: Kuan-Sen Lin.
- Source: GitHub.
Verification & Sources