PAOFLOW.projection.do_minimal#

Attributes#

comm

rank

L S Cederbaum, J Schirmer and H-D Meyer

Functions#

do_minimal(data_controller, first_band)

Remove low-energy bands from the Hamiltonian via block diagonalisation.

Module Contents#

PAOFLOW.projection.do_minimal.comm[source]#
PAOFLOW.projection.do_minimal.rank[source]#

L S Cederbaum, J Schirmer and H-D Meyer J. Phys. A: Mat. Gen. 22, 2427 (1989)

The formula for the block diagonal part of the S matrix is from: B.N. Parlett, The symmetric eigenvalue problem (SIAM’s Classics in Applied Mathematics, 1998) pag. 45, sec. 3.1.1 remark n. 5.

PAOFLOW.projection.do_minimal.do_minimal(data_controller, first_band)[source]#

Remove low-energy bands from the Hamiltonian via block diagonalisation.

Parameters:

  • data_controller (DataController) – Object providing data_arrays and data_attributes. Required array: Hksp (shape (nkpnts, nawf, nawf, nspin)). Required attributes: nawf, nspin, nkpnts. Also reads basis from data_arrays.

  • first_band (int) – Index of the first band to retain. Bands with indices 0, 1, ..., first_band-1 are projected out.

Returns:

Modifies data_controller.data_arrays and data_controller.data_attributes in place:

  • Hksp : np.ndarray, shape (nkpnts, nawf-first_band, nawf-first_band, nspin) — the reduced Hamiltonian after projecting out the lowest first_band bands.

  • Dnm : np.ndarray, shape (nawf-first_band, nawf-first_band, 3) — position-difference matrix for the retained orbitals.

Updates attribute: nawf = nawf - first_band.

Return type:

None

Notes

The block diagonalisation follows the algorithm of Cederbaum, Schirmer, and Meyer (J. Phys. A 22, 2427, 1989). At each k-point the full Hamiltonian is diagonalised, its eigenvector matrix is partitioned into a \((2 \times 2)\) block structure, and a similarity transformation \(T\) is constructed such that \(T^\dagger H T\) is block-diagonal. The lower-right (nawf - first_band) x (nawf - first_band) block is retained as the reduced Hamiltonian.

Small random perturbations (amplitude \(10^{-4}\)) are added to the block matrices to break accidental degeneracies before inversion.