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)[source]

Calculate the full four-index U matrix

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

where \(F_k\) [\(F_0, F_2, F_4, ...\)] are radial Slater integrals and \(\alpha(l, k, m1, m2, m3, m4)\) denote angular Racah_Wigner numbers for a spherical symmetric interaction tensor. The user can either specify directly the radial integral \(F_k\), or U_int / J_hund are given using the function U_J_to_radial_integrals() to convert back to radial integrals.

The convetion for the U matrix is given by the definition of the following Hamiltonian:

\[H = \frac{1}{2} \sum_{ijkl,\sigma \sigma'} U_{ijkl} a_{i \sigma}^\dagger a_{j \sigma'}^\dagger a_{l \sigma'} a_{k \sigma}.\]
Parameters:

  • l (integer) –

    Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell). radial_integrals : list, optional

    Slater integrals [F0,F2,F4,..]. Must be provided if U_int and J_hund are not given. Preferentially used to compute the U_matrix if provided alongside U_int and J_hund.

  • U_int (scalar, optional) – Value of the screened Hubbard interaction. Must be provided if radial_integrals are not given.

  • J_hund (scalar, optional) – Value of the Hund’s coupling. Must be provided if radial_integrals are not given.

  • basis (string, optional) –

    The basis in which the interaction matrix should be computed. Takes the values

    • ’spherical’: spherical harmonics,

    • ’cubic’: cubic harmonics,

    • ’other’: other basis type as given by the transformation matrix T.

  • T (real/complex numpy array, optional) –

    Transformation matrix for basis change. Must be provided if basis=’other’. The transformation matrix is defined such that new creation operators \(b^\dagger\) are related to the old ones \(a^\dagger\) as

    \[b_{i \sigma}^\dagger = \sum_j T_{ij} a^\dagger_{j \sigma}.\]

Returns:

U_matrix – The four-index interaction matrix in the chosen basis.

Return type:

float numpy array