def S_op(component,
spin_names,
orb_names,
off_diag=None,
map_operator_structure=None):
r"""
Create a component of the spin vector operator.
.. math::
\hat S_{x,y,z} = \frac{1}{2}\sum_{i\sigma\sigma'} a^\dagger_{i\sigma} \mathbf{\tau}^{x,y,z}_{\sigma\sigma'} a_{i\sigma'},
\quad\hat S_\pm = \hat S_x \pm i \hat S_y.
Parameters
----------
component : string
Component to be created, one of 'x', 'y', 'z', '+', or '-'.
*Warning*: y-component is not supported at the moment!
spin_names : list of strings
Names of the spins, e.g. ['up','down'].
orb_names : list of strings or int
Names of the orbitals, e.g. [0,1,2] or ['t2g','eg'].
off_diag : boolean
Do we have (orbital) off-diagonal elements?
If yes, the operators and blocks are denoted by ('spin', 'orbital'),
otherwise by ('spin_orbital',0).
map_operator_structure : dict
Mapping of names of GF blocks names from one convention to another,
e.g. {('up', 0): ('up_0', 0), ('down', 0): ('down_0',0)}.
If provided, the operators and blocks are denoted by the mapping of ``('spin', 'orbital')``.
Returns
-------
S : Operator
The component of the spin vector operator.
"""
# FIXME
assert component != 'y', "We cannot construct operators with complex coefficients at the moment. Sorry for that!"
mkind = get_mkind(off_diag, map_operator_structure)
pm = pauli_matrix[component]
S = Operator()
spin_range = range(len(spin_names))
for n1, n2 in product(spin_range, spin_range):
for on in orb_names:
S += 0.5 * c_dag(*mkind(spin_names[n1], on)) * pm[n1, n2] * c(
*mkind(spin_names[n2], on))
return S
def S_op(component, spin_names, orb_names, off_diag = None, map_operator_structure = None):
r"""
Create a component of the spin vector operator.
.. math::
\hat S_{x,y,z} = \frac{1}{2}\sum_{i\sigma\sigma'} a^\dagger_{i\sigma} \mathbf{\tau}^{x,y,z}_{\sigma\sigma'} a_{i\sigma'},
\quad\hat S_\pm = \hat S_x \pm i \hat S_y.
Parameters
----------
component : string
Component to be created, one of 'x', 'y', 'z', '+', or '-'.
*Warning*: y-component is not supported at the moment!
spin_names : list of strings
Names of the spins, e.g. ['up','down'].
orb_names : list of strings or int
Names of the orbitals, e.g. [0,1,2] or ['t2g','eg'].
off_diag : boolean
Do we have (orbital) off-diagonal elements?
If yes, the operators and blocks are denoted by ('spin', 'orbital'),
otherwise by ('spin_orbital',0).
map_operator_structure : dict
Mapping of names of GF blocks names from one convention to another,
e.g. {('up', 0): ('up_0', 0), ('down', 0): ('down_0',0)}.
If provided, the operators and blocks are denoted by the mapping of ``('spin', 'orbital')``.
Returns
-------
S : Operator
The component of the spin vector operator.
"""
# FIXME
assert component != 'y', "We cannot construct operators with complex coefficients at the moment. Sorry for that!"
mkind = get_mkind(off_diag,map_operator_structure)
pm = pauli_matrix[component]
S = Operator()
spin_range = range(len(spin_names))
for n1, n2 in product(spin_range,spin_range):
for on in orb_names:
S += 0.5 * c_dag(*mkind(spin_names[n1],on)) * pm[n1,n2] * c(*mkind(spin_names[n2],on))
return S
def L_op(component,
spin_names,
orb_names,
off_diag=None,
map_operator_structure=None,
basis='spherical',
T=None):
r"""
Create a component of the orbital momentum vector operator.
.. math::
\hat L_{z,+,-} &= \sum_{ii'\sigma} a^\dagger_{i\sigma} L^{z,+,-}_{ii'} a_{i'\sigma},\\
\hat L_x &= \frac{1}{2}(\hat L_+ + \hat L_-),\ \hat L_y = \frac{1}{2i}(\hat L_+ - \hat L_-),\\
L^z_{ii'} &= i\delta_{i,i'}, \
L^+_{ii'} = \delta_{i,i'+1}\sqrt{l(l+1)-i'(i'+1)}, \
L^+_{ii'} = \delta_{i,i'-1}\sqrt{l(l+1)-i'(i'-1)}.
Parameters
----------
component : string
Component to be created, one of 'x', 'y', 'z', '+', or '-'.
*Warning*: y-component is not supported at the moment!
spin_names : list of strings
Names of the spins, e.g. ['up','down'].
orb_names : list of strings or int
Names of the orbitals, e.g. [0,1,2] or ['t2g','eg'].
off_diag : boolean
Do we have (orbital) off-diagonal elements?
If yes, the operators and blocks are denoted by ('spin', 'orbital'),
otherwise by ('spin_orbital',0).
map_operator_structure : dict
Mapping of names of GF blocks names from one convention to another,
e.g. {('up', 0): ('up_0', 0), ('down', 0): ('down_0',0)}.
If provided, the operators and blocks are denoted by the mapping of ``('spin', 'orbital')``.
basis : string, optional
The basis in which the interaction matrix should be computed.
Takes the values
- 'spherical': spherical harmonics,
- 'cubic': cubic harmonics (valid only for the integer orbital momenta, i.e. for odd sizes of orb_names),
- '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'.
Returns
-------
L : Operator
The component of the orbital momentum vector operator.
"""
# FIXME
assert component != 'y', "We cannot construct operators with complex coefficients at the moment. Sorry for that!"
l = (len(orb_names) - 1) / 2.0
L_melem_dict = {
'z':
lambda m, mp: m if np.isclose(m, mp) else 0,
'+':
lambda m, mp: np.sqrt(l * (l + 1) - mp * (mp + 1))
if np.isclose(m, mp + 1) else 0,
'-':
lambda m, mp: np.sqrt(l * (l + 1) - mp * (mp - 1))
if np.isclose(m, mp - 1) else 0,
'x':
lambda m, mp: 0.5 * (L_melem_dict['+'](m, mp) + L_melem_dict['-']
(m, mp)),
'y':
lambda m, mp: -0.5j * (L_melem_dict['+'](m, mp) - L_melem_dict['-']
(m, mp))
}
L_melem = L_melem_dict[component]
orb_range = range(int(2 * l + 1))
L_matrix = np.array([[L_melem(o1 - l, o2 - l) for o2 in orb_range]
for o1 in orb_range])
# Transform from spherical basis if needed
if basis == "cubic":
if not np.isclose(np.mod(l, 1), 0):
raise ValueError(
"L_op: cubic basis is only defined for the integer orbital momenta."
)
T = spherical_to_cubic(l)
if basis == "other" and T is None:
raise ValueError("L_op: provide T for other bases.")
if T is not None:
L_matrix = np.einsum("ij,jk,kl", np.conj(T), L_matrix, np.transpose(T))
mkind = get_mkind(off_diag, map_operator_structure)
L = Operator()
for sn in spin_names:
for o1, o2 in product(orb_range, orb_range):
L += c_dag(*mkind(sn, orb_names[o1])) * L_matrix[o1, o2] * c(
*mkind(sn, orb_names[o2]))
return L
def ls_op(spin_names,
orb_names,
off_diag=None,
map_operator_structure=None,
basis='spherical',
T=None):
"""one-body spin-orbit coupling operator"""
spin_range = range(len(spin_names))
l = (len(orb_names) - 1) / 2.0
orb_range = range(int(2 * l + 1))
pauli_matrix = {
'x': np.array([[0, 1], [1, 0]]),
'y': np.array([[0, -1j], [1j, 0]]),
'z': np.array([[1, 0], [0, -1]]),
'+': np.array([[0, 2], [0, 0]]),
'-': np.array([[0, 0], [2, 0]])
}
L_melem_dict = {
'z':
lambda m, mp: m if np.isclose(m, mp) else 0,
'+':
lambda m, mp: np.sqrt(l * (l + 1) - mp * (mp + 1))
if np.isclose(m, mp + 1) else 0,
'-':
lambda m, mp: np.sqrt(l * (l + 1) - mp * (mp - 1))
if np.isclose(m, mp - 1) else 0,
'x':
lambda m, mp: 0.5 * (L_melem_dict['+'](m, mp) + L_melem_dict['-']
(m, mp)),
'y':
lambda m, mp: -0.5j * (L_melem_dict['+'](m, mp) - L_melem_dict['-']
(m, mp))
}
# define S matrix
S_matrix = {}
for component in ['z', '+', '-']:
pm = pauli_matrix[component]
S_matrix[component] = np.array(
[[0.5 * pm[n1, n2] for n2 in spin_range] for n1 in spin_range])
# define L matrix
L_matrix = {}
for component in ['z', '+', '-']:
L_melem = L_melem_dict[component]
L_matrix[component] = np.array(
[[L_melem(o1 - l, o2 - l) for o2 in orb_range]
for o1 in orb_range])
# Transform from spherical basis if needed
if basis == "cubic":
if not np.isclose(np.mod(l, 1), 0):
raise ValueError(
"L_op: cubic basis is only defined for the integer orbital momenta."
)
T = spherical_to_cubic(l)
if basis == "other" and T is None:
raise ValueError("L_op: provide T for other bases.")
if T is not None:
L_matrix = np.einsum("ij,jk,kl", np.conj(T), L_matrix,
np.transpose(T))
# LS_matrix[n1,n2,o1,o2] = sum_x S_matrix[x][n1,n2] * L_matrix[x][o1,o2]
LS_matrix = np.einsum("ij,kl", S_matrix['z'], L_matrix['z'])\
+ np.einsum("ij,kl", S_matrix['+'], L_matrix['-']) * 0.5\
+ np.einsum("ij,kl", S_matrix['-'], L_matrix['+']) * 0.5
mkind = get_mkind(off_diag, map_operator_structure)
ls = Operator()
for n1, n2 in product(spin_range, spin_range):
for o1, o2 in product(orb_range, orb_range):
ls += c_dag(*mkind(spin_names[n1], orb_names[o1])) * LS_matrix[
n1, n2, o1, o2] * c(*mkind(spin_names[n2], orb_names[o2]))
return ls
def L_op(component, spin_names, orb_names, off_diag = None, map_operator_structure = None, basis='spherical', T=None):
r"""
Create a component of the orbital momentum vector operator.
.. math::
\hat L_{z,+,-} &= \sum_{ii'\sigma} a^\dagger_{i\sigma} L^{z,+,-}_{ii'} a_{i'\sigma},\\
\hat L_x &= \frac{1}{2}(\hat L_+ + \hat L_-),\ \hat L_y = \frac{1}{2i}(\hat L_+ - \hat L_-),\\
L^z_{ii'} &= i\delta_{i,i'}, \
L^+_{ii'} = \delta_{i,i'+1}\sqrt{l(l+1)-i'(i'+1)}, \
L^+_{ii'} = \delta_{i,i'-1}\sqrt{l(l+1)-i'(i'-1)}.
Parameters
----------
component : string
Component to be created, one of 'x', 'y', 'z', '+', or '-'.
spin_names : list of strings
Names of the spins, e.g. ['up','down'].
orb_names : list of strings or int
Names of the orbitals, e.g. [0,1,2] or ['t2g','eg'].
off_diag : boolean
Do we have (orbital) off-diagonal elements?
If yes, the operators and blocks are denoted by ('spin', 'orbital'),
otherwise by ('spin_orbital',0).
map_operator_structure : dict
Mapping of names of GF blocks names from one convention to another,
e.g. {('up', 0): ('up_0', 0), ('down', 0): ('down_0',0)}.
If provided, the operators and blocks are denoted by the mapping of ``('spin', 'orbital')``.
basis : string, optional
The basis in which the interaction matrix should be computed.
Takes the values
- 'spherical': spherical harmonics,
- 'cubic': cubic harmonics (valid only for the integer orbital momenta, i.e. for odd sizes of orb_names),
- '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'.
Returns
-------
L : Operator
The component of the orbital momentum vector operator.
"""
l = (len(orb_names)-1)/2.0
L_melem_dict = {'z' : lambda m,mp: m if np.isclose(m,mp) else 0,
'+' : lambda m,mp: np.sqrt(l*(l+1)-mp*(mp+1)) if np.isclose(m,mp+1) else 0,
'-' : lambda m,mp: np.sqrt(l*(l+1)-mp*(mp-1)) if np.isclose(m,mp-1) else 0,
'x' : lambda m,mp: 0.5*(L_melem_dict['+'](m,mp) + L_melem_dict['-'](m,mp)),
'y' : lambda m,mp: -0.5j*(L_melem_dict['+'](m,mp) - L_melem_dict['-'](m,mp))}
L_melem = L_melem_dict[component]
orb_range = range(int(2*l+1))
L_matrix = np.array([[L_melem(o1-l,o2-l) for o2 in orb_range] for o1 in orb_range])
# Transform from spherical basis if needed
if basis == "cubic":
if not np.isclose(np.mod(l,1),0):
raise ValueError("L_op: cubic basis is only defined for the integer orbital momenta.")
T = spherical_to_cubic(int(l))
if basis == "other" and T is None: raise ValueError("L_op: provide T for other bases.")
if T is not None: L_matrix = np.einsum("ij,jk,kl",np.conj(T),L_matrix,np.transpose(T))
mkind = get_mkind(off_diag,map_operator_structure)
L = Operator()
for sn in spin_names:
for o1, o2 in product(orb_range,orb_range):
L += c_dag(*mkind(sn,orb_names[o1])) * L_matrix[o1,o2] * c(*mkind(sn,orb_names[o2]))
return L
from pytriqs.archive import *
from pytriqs.applications.impurity_solvers.cthyb import AtomicProblem
f = HDFArchive('dca_beta20_np975_histogram.out.h5','r')
h_diag = f['h_diag-0']
fc = HDFArchive('check.h5','w')
fc['h_diag-0'] = h_diag
A = PostProcess(f['density_matrix-0'],f['h_loc-0'])
CHEC = HDFArchive('densmat.h5','w')
CHEC['density_matrix'] = f['density_matrix-0']
c_dag_0_up = ( c_dag("00-up",0) + c_dag("10-up",0) + c_dag("01-up",0) + c_dag("11-up",0))/2
c_dag_1_up = ( c_dag("00-up",0) - c_dag("10-up",0) + c_dag("01-up",0) - c_dag("11-up",0))/2
c_dag_2_up = ( c_dag("00-up",0) - c_dag("10-up",0) - c_dag("01-up",0) + c_dag("11-up",0))/2
c_dag_3_up = ( c_dag("00-up",0) + c_dag("10-up",0) - c_dag("01-up",0) - c_dag("11-up",0))/2
c_dag_0_down = ( c_dag("00-down",0) + c_dag("10-down",0) + c_dag("01-down",0) + c_dag("11-down",0))/2
c_dag_1_down = ( c_dag("00-down",0) - c_dag("10-down",0) + c_dag("01-down",0) - c_dag("11-down",0))/2
c_dag_2_down = ( c_dag("00-down",0) - c_dag("10-down",0) - c_dag("01-down",0) + c_dag("11-down",0))/2
c_dag_3_down = ( c_dag("00-down",0) + c_dag("10-down",0) - c_dag("01-down",0) - c_dag("11-down",0))/2
ops_0hole = [
(c_dag_0_up*c_dag_3_down-c_dag_0_down*c_dag_3_up)*(c_dag_1_up*c_dag_2_down-c_dag_1_down*c_dag_2_up)/2,
(c_dag_0_up*c_dag_1_down-c_dag_0_down*c_dag_1_up)*(c_dag_2_up*c_dag_3_down-c_dag_2_down*c_dag_3_up)/2
]
ops_1hole = [
from pytriqs.operators.operators import Operator, c, c_dag, n, dagger
from pytriqs.archive import *
from pytriqs.applications.impurity_solvers.cthyb import AtomicProblem
f = HDFArchive('dca_beta20_np975_histogram.out.h5', 'r')
h_diag = f['h_diag-0']
fc = HDFArchive('check.h5', 'w')
fc['h_diag-0'] = h_diag
A = PostProcess(f['density_matrix-0'], f['h_loc-0'])
CHEC = HDFArchive('densmat.h5', 'w')
CHEC['density_matrix'] = f['density_matrix-0']
c_dag_0_up = (c_dag("00-up", 0) + c_dag("10-up", 0) + c_dag("01-up", 0) +
c_dag("11-up", 0)) / 2
c_dag_1_up = (c_dag("00-up", 0) - c_dag("10-up", 0) + c_dag("01-up", 0) -
c_dag("11-up", 0)) / 2
c_dag_2_up = (c_dag("00-up", 0) - c_dag("10-up", 0) - c_dag("01-up", 0) +
c_dag("11-up", 0)) / 2
c_dag_3_up = (c_dag("00-up", 0) + c_dag("10-up", 0) - c_dag("01-up", 0) -
c_dag("11-up", 0)) / 2
c_dag_0_down = (c_dag("00-down", 0) + c_dag("10-down", 0) +
c_dag("01-down", 0) + c_dag("11-down", 0)) / 2
c_dag_1_down = (c_dag("00-down", 0) - c_dag("10-down", 0) +
c_dag("01-down", 0) - c_dag("11-down", 0)) / 2
c_dag_2_down = (c_dag("00-down", 0) - c_dag("10-down", 0) -
c_dag("01-down", 0) + c_dag("11-down", 0)) / 2
c_dag_3_down = (c_dag("00-down", 0) + c_dag("10-down", 0) -
