Wannierization

Wannierization

What get_wannier_hamiltonian does

h.get_wannier_hamiltonian(bands=[a,b], nk=...) (src/pyqula/wanniertk/wannierize.py) Wannierizes a fixed, contiguous range of h's bands -- 0-indexed, both ends inclusive, Wannierized jointly as one group -- and returns a new, smaller multicell Hamiltonian whose real-space hoppings exactly reproduce that band subspace on the wannierization k-mesh.

The default initial guess

Without trial_vectors=, the minimization starts from num_wann of h's own orbitals, the ones the SCDM column selection (Damle, Lin and Ying, arXiv:1507.03354) picks for the selected bands on the mesh (_default_trial_vectors), in ascending orbital order (the identity for a full manifold). It is deterministic, so repeated calls give the same Wannier functions. Do not go back to a random draw: for the gapped honeycomb valence band it stopped in a local minimum (spread 2.5 instead of 0.32) in a quarter of the calls (tests/wannier/test_default_trial_vectors.py). Disentanglement picks from the frozen window instead (_default_disentanglement_trial_vectors), and Nambu Hamiltonians keep their own defaults (identity for the full manifold, electron-hole-paired orbitals otherwise).

Where the hoppings sit

The mesh fixes each hopping only up to a translation by the nk supercell, so _mesh_to_real_space puts them on the cells of the Wigner-Seitz cell of that supercell (real lattice metric, _wigner_seitz_cells), each divided by its degeneracy ndegen: Wannier90's hamiltonian_wigner_seitz construction, with the same cells as the bundled port wannierpy/_engine/ws_vectors.py but found class by class, so it also covers a skewed, anisotropic supercell (nk=[12,2] on the honeycomb lattice) where Wannier90's two-supercell search, and the port, fail. The cell set is inversion symmetric, so H_wan(k) is Hermitian at every k. Do not go back to a plain fftfreq box of cells: for an even nk it holds R=-nk/2 without +nk/2, and H_wan(k) off the mesh is not Hermitian (tests/wannier/test_wigner_seitz_hoppings.py).

wannier_functions is different on purpose (_mesh_to_wannier_functions): a Wannier function is a function, so every class of cells appears once (the fftfreq box), since splitting an amplitude over degenerate images would break its normalization. Wannier90's use_ws_distance refinement (the Wigner-Seitz test on R + tau_n - tau_m per pair of Wannier functions) is not implemented; it only matters for several Wannier functions centred far apart within the cell.

How far the interpolation between mesh points is converged depends on the Wannier functions, not on these cells: a band group that comes close to the rest of the spectrum somewhere (the low-energy BdG pair of a chain with a small gap to the next band) has wide Wannier functions and needs a dense mesh, and its wannier_spread_total keeps growing with nk until the mesh resolves that region.