triqs.operators.util.U_matrix.U_matrix_slater

triqs.operators.util.U_matrix.U_matrix_slater(l, radial_integrals=None, U_int=None, J_hund=None, basis='spherical', T=None)

Calculate the full four-index Slater interaction tensor.

\[U^{spherical}_{m1 m2 m3 m4} = \sum_{k=0}^{2l} F_k \alpha(l, k, m1, m2, m3, m4)\]

where \(F_k\) (i.e. \(F_0, F_2, F_4, ...\)) are radial Slater integrals and \(\alpha(l, k, m1, m2, m3, m4)\) are the angular Racah-Wigner coefficients for a spherically symmetric interaction tensor. Either pass the radial integrals \(F_k\) directly, or pass U_int and J_hund — they are then converted via U_J_to_radial_integrals().

The convention for the \(U\)-matrix is fixed by the Hamiltonian

\[\hat{H} = \frac{1}{2} \sum_{ijkl,\sigma \sigma'} U_{ijkl} \hat{c}_{i \sigma}^\dagger \hat{c}_{j \sigma'}^\dagger \hat{c}_{l \sigma'} \hat{c}_{k \sigma}.\]

Parameters:

lint

Angular momentum of the shell being treated (l=2 for a d shell, l=3 for an f shell).

radial_integralslist of float, optional

Slater integrals [F0, F2, F4, ...]. Must be provided if both U_int and J_hund are missing. Used in preference to U_int/J_hund if all three are supplied.

U_intfloat, optional

Value of the screened Hubbard interaction. Must be provided if radial_integrals is not given.

J_hundfloat, optional

Value of the Hund’s coupling. Must be provided if radial_integrals is not given.

basis{‘spherical’, ‘cubic’, ‘other’}, optional

Basis in which the interaction matrix is computed: spherical harmonics (default), cubic harmonics, or an arbitrary basis specified by the transformation matrix \(T\).

Tndarray, optional

Transformation matrix for the basis change. Required when basis='other'; see Notes for the convention.

Returns:

numpy.ndarray

The four-index interaction matrix in the chosen basis.

Notes

The transformation matrix \(T\) is defined so that new creation operators \(\hat{b}^\dagger\) are related to the old ones \(\hat{c}^\dagger\) via

\[\hat{b}_{i \sigma}^\dagger = \sum_j T_{ij} \hat{c}^\dagger_{j \sigma}.\]