U matrix construction

Tools to generate the U-matrices used with Hamiltonian-construction functions are provided in the triqs.operators.util.U_matrix module.

Functions to construct Coulomb tensors

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 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

triqs.operators.util.U_matrix.reduce_4index_to_2index(U_4index)

Reduces the four-index matrix to two-index matrices for parallel and anti-parallel spins.

Parameters:

U_4index (float numpy array) – The four-index interaction matrix.

Returns:

• U (float numpy array) – The two-index interaction matrix for parallel spins.

• Uprime (float numpy array) – The two-index interaction matrix for anti-parallel spins.

triqs.operators.util.U_matrix.U_matrix_kanamori(n_orb, U_int, J_hund, Up_int=None, full_Uijkl=False, Jc_hund=None)

Calculate the Kanamori two-index interaction matrix for parallel spins:

\[U_{m m'}^{\sigma \sigma} \equiv U_{m m' m m'} - J_{m m'}\]

with:

\[J_{m m'} \equiv U_{m m' m' m} ,\]

and the two-index interaction matrix for anti-parallel spins:

\[U_{m m'}^{\sigma \bar{\sigma}} \equiv U_{m m' m m'}\]

If full_Uijkl=True is specified the full four index Uijkl tensor is returned instead:

\[\begin{split}U_{m m m m} = U, \\ U_{m m' m m'} = U', \\ U_{m m' m' m} = J, \\ U_{m m m' m'} = J_C,\end{split}\]

with \(m \neq m'\).

Parameters:

• n_orb (integer) – Number of orbitals in basis.

• U_int (float) – Value of the screened Hubbard interaction.

• J_hund (float) – Value of the Hund’s coupling.

• Up_int (float, optional) – Value of the screened U prime parameter defaults to U_int-2*J_hund if not given. (fully rotationally invariant form)

• full_Uijkl (bool, optional) – retunr instead the full four-index Uijkl tensor default is False

• Jc_hund (foat, optional) – only used if full_Uijkl=True, defaults to J_hund

Returns:

• U (float numpy array) – The two-index interaction matrix for parallel spins or the four-index Uijkl tensor if full_Uijkl=True

• Uprime (float numpy array) – The two-index interaction matrix for anti-parallel spins.

triqs.operators.util.U_matrix.t2g_submatrix(U, convention='triqs')

Extract the t2g submatrix of the full d-manifold two- or four-index U matrix.

Parameters:

• U (float numpy array) – Two- or four-index interaction matrix.

• convention (string, optional) –

  The basis convention. Takes the values

  • ’triqs’: basis ordered as (“xy”,”yz”,”z^2”,”xz”,”x^2-y^2”),

  • ’vasp’: same as ‘triqs’,

  • ’wien2k’: basis ordered as (“z^2”,”x^2-y^2”,”xy”,”yz”,”xz”),

  • ’wannier90’: basis order as (“z^2”, “xz”, “yz”, “x^2-y^2”, “xy”),

  • ’qe’: same as ‘wannier90’.

Returns:

U_t2g – The t2g component of the interaction matrix.

Return type:

float numpy array

triqs.operators.util.U_matrix.eg_submatrix(U, convention='triqs')

Extract the eg submatrix of the full d-manifold two- or four-index U matrix.

Parameters:

• U (float numpy array) – Two- or four-index interaction matrix.

• convention (string, optional) –

  The basis convention. Takes the values

  • ’triqs’: basis ordered as (“xy”,”yz”,”z^2”,”xz”,”x^2-y^2”),

  • ’vasp’: same as ‘triqs’,

  • ’wien2k’: basis ordered as (“z^2”,”x^2-y^2”,”xy”,”yz”,”xz”),

  • ’wannier90’: basis order as (“z^2”, “xz”, “yz”, “x^2-y^2”, “xy”),

  • ’qe’: same as ‘wannier90’.

Returns:

U_eg – The eg component of the interaction matrix.

Return type:

float numpy array

triqs.operators.util.U_matrix.transform_U_matrix(U_matrix, T)

Transform a four-index interaction matrix into another basis. The transformation matrix is defined such that new creation operators \(b^\dagger\) are related to the old ones \(a^\dagger\) as

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

Parameters:

• U_matrix (float numpy array) – The four-index interaction matrix in the original basis.

• T (real/complex numpy array, optional) – Transformation matrix for basis change. Must be provided if basis=’other’.

Returns:

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

Return type:

float numpy array

triqs.operators.util.U_matrix.spherical_to_cubic(l, convention='triqs')

Get the spherical harmonics to cubic harmonics transformation matrix.

Parameters:

• l (integer) – Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).

• convention (string, optional) –

  The basis convention. Takes the values

  • ’triqs’: basis ordered as (“xy”,”yz”,”z^2”,”xz”,”x^2-y^2”),

  • ’vasp’: same as ‘triqs’,

  • ’wien2k’: basis ordered as (“z^2”,”x^2-y^2”,”xy”,”yz”,”xz”),

  • ’wannier90’: basis order as (“z^2”, “xz”, “yz”, “x^2-y^2”, “xy”),

  • ’qe’: same as ‘wannier90’.

Returns:

T – Transformation matrix for basis change.

Return type:

real/complex numpy array

triqs.operators.util.U_matrix.cubic_names(l)

Get the names of the cubic harmonics.

Parameters:

l (integer or string) – Angular momentum of shell being treated. Also takes ‘t2g’ and ‘eg’ as arguments.

Returns:

cubic_names – Names of the orbitals.

Return type:

tuple of strings

triqs.operators.util.U_matrix.U_J_to_radial_integrals(l, U_int, J_hund)

Determine the radial integrals F_k from U_int and J_hund.

Parameters:

• l (integer) – Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).

• U_int (scalar) – Value of the screened Hubbard interaction.

• J_hund (scalar) – Value of the Hund’s coupling.

Returns:

radial_integrals – Slater integrals [F0,F2,F4,..].

Return type:

list

triqs.operators.util.U_matrix.radial_integrals_to_U_J(l, F)

Determine U_int and J_hund from the radial integrals.

Parameters:

• l (integer) – Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).

• F (list) – Slater integrals [F0,F2,F4,..].

Returns:

• U_int (scalar) – Value of the screened Hubbard interaction.

• J_hund (scalar) – Value of the Hund’s coupling.

triqs.operators.util.U_matrix.angular_matrix_element(l, k, m1, m2, m3, m4)

Calculate the angular matrix element

\[\begin{split}(2l+1)^2 \begin{pmatrix} l & k & l \\ 0 & 0 & 0 \end{pmatrix}^2 \sum_{q=-k}^k (-1)^{m_1+m_2+q} \begin{pmatrix} l & k & l \\ -m_1 & q & m_3 \end{pmatrix} \begin{pmatrix} l & k & l \\ -m_2 & -q & m_4 \end{pmatrix}.\end{split}\]

Parameters:

• l (integer)

• k (integer)

• m1 (integer)

• m2 (integer)

• m3 (integer)

• m4 (integer)

Returns:

ang_mat_ele – Angular matrix element.

Return type:

scalar

triqs.operators.util.U_matrix.three_j_symbol(jm1, jm2, jm3)

Calculate the three-j symbol

\[\begin{split}\begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix}.\end{split}\]

Parameters:

• jm1 (tuple of integers) – (j_1 m_1)

• jm2 (tuple of integers) – (j_2 m_2)

• jm3 (tuple of integers) – (j_3 m_3)

Returns:

three_j_sym – Three-j symbol.

Return type:

scalar

triqs.operators.util.U_matrix.clebsch_gordan(jm1, jm2, jm3)

Calculate the Clebsh-Gordan coefficient

\[\begin{split}\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{j_1-j_2+m_3} \sqrt{2 j_3 + 1} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix}.\end{split}\]

Parameters:

• jm1 (tuple of integers) – (j_1 m_1)

• jm2 (tuple of integers) – (j_2 m_2)

• jm3 (tuple of integers) – (j_3 m_3)

Returns:

cgcoeff – Clebsh-Gordan coefficient.

Return type:

scalar

triqs.operators.util.U_matrix.subarray(a, idxlist, n=None)

Extract a subarray from a matrix-like object.

Parameters:

• a (matrix or array)

• idxlist (list of tuples) – Columns that need to be extracted for each dimension.

Returns:

subarray

Return type:

matrix or array

Examples

idxlist = [(0),(2,3),(0,1,2,3)] gives

• column 0 for 1st dim,

• columns 2 and 3 for 2nd dim,

• columns 0, 1, 2 and 3 for 3rd dim.