U matrix construction

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

pytriqs.operators.util.U_matrix.U_matrix(l, radial_integrals=None, U_int=None, J_hund=None, basis='spherical', T=None)[source]

Calculate the full four-index U matrix being given either radial_integrals or U_int and J_hund. The convetion for the U matrix is that used to construct the Hamiltonians, namely:

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

The four-index interaction matrix in the chosen basis.

pytriqs.operators.util.U_matrix.reduce_4index_to_2index(U_4index)[source]

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.

pytriqs.operators.util.U_matrix.U_matrix_kanamori(n_orb, U_int, J_hund)[source]

Calculate the Kanamori U and Uprime matrices.

Parameters:

n_orb : integer

Number of orbitals in basis.

U_int : scalar

Value of the screened Hubbard interaction.

J_hund : scalar

Value of the Hund’s coupling.

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.

pytriqs.operators.util.U_matrix.t2g_submatrix(U, convention='')[source]

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

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

U_t2g : float numpy array

The t2g component of the interaction matrix.

pytriqs.operators.util.U_matrix.eg_submatrix(U, convention='')[source]

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

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

U_eg : float numpy array

The eg component of the interaction matrix.

pytriqs.operators.util.U_matrix.transform_U_matrix(U_matrix, T)[source]

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 : float numpy array

The four-index interaction matrix in the new basis.

pytriqs.operators.util.U_matrix.spherical_to_cubic(l, convention='')[source]

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

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

T : real/complex numpy array

Transformation matrix for basis change.

pytriqs.operators.util.U_matrix.cubic_names(l)[source]

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 : tuple of strings

Names of the orbitals.

pytriqs.operators.util.U_matrix.U_J_to_radial_integrals(l, U_int, J_hund)[source]

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 : list

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

pytriqs.operators.util.U_matrix.radial_integrals_to_U_J(l, F)[source]

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.

pytriqs.operators.util.U_matrix.angular_matrix_element(l, k, m1, m2, m3, m4)[source]

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 : scalar

Angular matrix element.

pytriqs.operators.util.U_matrix.three_j_symbol(jm1, jm2, jm3)[source]

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 : scalar

Three-j symbol.

pytriqs.operators.util.U_matrix.clebsch_gordan(jm1, jm2, jm3)[source]

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 : scalar

Clebsh-Gordan coefficient.

pytriqs.operators.util.U_matrix.subarray(a, idxlist, n=None)[source]

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 : 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.