# ================================== LICENSE ===================================
# Magnopy - Python package for magnons.
#
# Copyright (C) 2023 Magnopy Team
#
# e-mail: anry@uv.es, web: magnopy.org
#
# This program is free software: you can redistribute it and/or modify it under
# the terms of the GNU General Public License as published by the Free Software
# Foundation, either version 3 of the License, or (at your option) any later
# version.
#
# This program is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
# FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License along with
# this program. If not, see <https://www.gnu.org/licenses/>.
# ================================ END LICENSE =================================
import numpy as np
from magnopy._parameters._interaction_parameters import _InteractionParametersIterator
from magnopy._parameters._renormalization import _renormalized_parameters
from magnopy._local_rf import span_local_rfs
from magnopy._spinham._convention import Convention
__all__ = ["is_eigenstate"]
[docs]
def is_eigenstate(spinham, spin_directions, energy_tolerance=1e-8):
r"""
Checks all eigenstate conditions as described in the supplementary note 4 of
|paper-2026|_ (eqs. S.40-S.50).
Parameters
----------
spinham : :py:class:`magnopy.SpinHamiltonian`
Spin Hamiltonian
spin_directions : (M, 3) |array-like|_
Directional vectors. M is an amount of atoms in ``spinham.magnetic_atoms``.
energy_tolerance : float, default 1e-8
Numerical accuracy for comparing energy values to zero. Given in the same units as
``spinham.units``.
Returns
-------
result : bool
``True`` if ``spin_directions`` describe an eigenstate of ``spinham``. ``False``
otherwise.
"""
spin_values = np.array(spinham.magnetic_atoms.spins, dtype=float)
x, y, z = span_local_rfs(directional_vectors=spin_directions, hybridize=False)
p = x + 1j * y
convention = Convention(
spin_normalized=False,
multiple_counting=True,
c1=1,
c21=1,
c22=1,
c31=1,
c32=1,
c33=1,
c41=1,
c42=1,
c43=1,
c44=1,
c45=1,
)
initial_convention = spinham.convention
spinham.convention = convention
parameters = spinham._parameters.copy()
spinham.convention = initial_convention
for i, ((n, p_n, nus, alphas), _) in enumerate(parameters._container):
alphas = tuple(spinham.map_to_magnetic[alpha] for alpha in alphas)
parameters._container[i][0] = (n, p_n, nus, alphas)
renormalized_parameters = _renormalized_parameters(
parameters=parameters,
convention=convention,
spin_directions=spin_directions,
spin_values=spin_values,
)
if not _check_S_40(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_41(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_42(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_43(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_44(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_45(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_46(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_47(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_48(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_49(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
if not _check_S_50(
renormalized_parameters=renormalized_parameters,
p=p,
z=z,
spin_values=spin_values,
energy_tolerance=energy_tolerance,
):
return False
return True
def _check_S_50(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.50 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=5
):
alpha_1, alpha_2, alpha_3, alpha_4 = alphas
u_11 = np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_2],
p[alpha_3],
p[alpha_4],
)
if abs(u_11) > energy_tolerance:
return False
return True
def _check_S_49(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.49 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=4
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
alpha_1, alpha_2, alpha_3 = alphas[1:]
if round(2 * spin_values[alpha_1]) > 1:
u_10 = np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_2],
p[alpha_3],
)
if abs(u_10) > energy_tolerance:
return False
return True
def _check_S_48(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.48 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=3
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
alpha_1 = alphas[0]
alpha_2 = alphas[2]
if (
round(2 * spin_values[alpha_1]) > 1
and round(2 * spin_values[alpha_2]) > 1
):
u_9 = np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_2],
p[alpha_2],
)
if abs(u_9) > energy_tolerance:
return False
return True
def _check_S_47(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.47 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=2
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]) and ((0, 0, 0), alphas[0]) == (
nus[1],
alphas[2],
):
alpha_1 = alphas[0]
alpha_2 = alphas[3]
if round(2 * spin_values[alpha_1]) > 2:
u_8 = np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_1],
p[alpha_2],
)
if abs(u_8) > energy_tolerance:
return False
return True
def _check_S_46(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.46 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 3:
u_7 = np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_1],
p[alpha_1],
)
if abs(u_7) > energy_tolerance:
return False
return True
def _check_S_45(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.45 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_6 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=3
):
nu_2, nu_3 = nus
alpha_1, alpha_2, alpha_3 = alphas
key = (nu_2, nu_3, alpha_1, alpha_2, alpha_3)
if key not in u_6:
u_6[key] = 0
u_6[key] += np.einsum(
"ijk,i,j,k",
parameter,
p[alpha_1],
p[alpha_2],
p[alpha_3],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=4
):
if (nus[1], alphas[2]) == (nus[2], alphas[3]):
nu_2 = nus[0]
nu_3 = nus[1]
alpha_1, alpha_2, alpha_3 = alphas[:3]
key = (nu_2, nu_3, alpha_1, alpha_2, alpha_3)
if key not in u_6:
u_6[key] = 0
u_6[key] -= 6 * np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_2],
z[alpha_3],
p[alpha_3],
)
for key in u_6:
if abs(u_6[key]) > energy_tolerance:
return False
return True
def _check_S_44(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.44 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_5 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=2
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
nu_2 = nus[1]
alpha_1 = alphas[0]
alpha_2 = alphas[2]
if round(2 * spin_values[alpha_1]) > 1:
key = (nu_2, alpha_1, alpha_2)
if key not in u_5:
u_5[key] = 0
u_5[key] += np.einsum(
"ijk,i,j,k",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=2
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]) and ((0, 0, 0), alphas[0]) == (
nus[1],
alphas[2],
):
nu_2 = nus[2]
alpha_1 = alphas[0]
alpha_2 = alphas[3]
if round(2 * spin_values[alpha_1]) > 1:
key = (nu_2, alpha_1, alpha_2)
if key not in u_5:
u_5[key] = 0
medium_matrix = 2 * np.einsum("i,j->ij", z[alpha_1], p[alpha_1])
medium_matrix += np.einsum("i,j->ij", p[alpha_1], z[alpha_1])
u_5[key] -= 4 * np.einsum(
"ijkl,ij,k,l",
parameter,
medium_matrix,
p[alpha_1],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=3
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
nu_2 = nus[1]
alpha_1 = alphas[0]
alpha_2 = alphas[2]
if round(2 * spin_values[alpha_1]) > 1:
key = (nu_2, alpha_1, alpha_2)
if key not in u_5:
u_5[key] = 0
u_5[key] -= 6 * np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
z[alpha_2],
p[alpha_2],
)
for key in u_5:
if abs(u_5[key]) > energy_tolerance:
return False
return True
def _check_S_43(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.43 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_4 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 2:
key = alpha_1
if key not in u_4:
u_4[key] = 0
u_4[key] += np.einsum(
"ijk,i,j,k",
parameter,
p[alpha_1],
p[alpha_1],
p[alpha_1],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 2:
key = alpha_1
if key not in u_4:
u_4[key] = 0
medium_matrix = 3 * np.einsum(
"i,j,k->ijk", z[alpha_1], p[alpha_1], p[alpha_1]
)
medium_matrix += 2 * np.einsum(
"i,j,k->ijk", p[alpha_1], z[alpha_1], p[alpha_1]
)
medium_matrix += np.einsum("i,j,k->ijk", p[alpha_1], p[alpha_1], z[alpha_1])
u_4[key] -= np.einsum("ijkl,ijk,l", parameter, medium_matrix, p[alpha_1])
for key in u_4:
if abs(u_4[key]) > energy_tolerance:
return False
return True
def _check_S_42(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.42 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_3 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=2, p_n=2
):
nu_2 = nus[0]
alpha_1, alpha_2 = alphas
key = (nu_2, alpha_1, alpha_2)
if key not in u_3:
u_3[key] = 0
u_3[key] += np.einsum(
"ij,i,j",
parameter,
p[alpha_1],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=2
):
if (nus[0], alphas[1]) == (nus[1], alphas[2]):
nu_2 = nus[0]
alpha_1, alpha_2 = alphas[:2]
key = (nu_2, alpha_1, alpha_2)
if key not in u_3:
u_3[key] = 0
u_3[key] -= 3 * np.einsum(
"ijk,i,j,k",
parameter,
p[alpha_1],
z[alpha_2],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=2
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]) and ((0, 0, 0), alphas[0]) == (
nus[1],
alphas[2],
):
nu_2 = nus[2]
alpha_1, alpha_2 = alphas[0], alphas[3]
key = (nu_2, alpha_1, alpha_2)
if key not in u_3:
u_3[key] = 0
medium_matrix = np.einsum("i,j->ij", z[alpha_1], z[alpha_1])
if round(2 * spin_values[alpha_1]) > 1:
medium_matrix += (spin_values[alpha_1] - 0.5) * np.einsum(
"i,j->ij", np.conjugate(p[alpha_1]), p[alpha_1]
)
medium_matrix += (
0.5
* spin_values[alpha_1]
* np.einsum("i,j->ij", p[alpha_1], np.conjugate(p[alpha_1]))
)
u_3[key] += 4 * np.einsum(
"ijkl,ij,k,l",
parameter,
medium_matrix,
p[alpha_1],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=3
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
nu_2 = nus[1]
alpha_1, alpha_2 = alphas[0], alphas[2]
key = (nu_2, alpha_1, alpha_2)
if key not in u_3:
u_3[key] = 0
u_3[key] += 3 * np.einsum(
"ijkl,i,j,k,l",
parameter,
z[alpha_1],
p[alpha_1],
z[alpha_2],
p[alpha_2],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=4
):
if (nus[1], alphas[2]) == (nus[2], alphas[3]):
nu_2 = nus[0]
alpha_1, alpha_2, alpha_3 = alphas[:3]
key = (nu_2, alpha_1, alpha_2)
if key not in u_3:
u_3[key] = 0
u_3[key] += (
3
* spin_values[alpha_3]
* np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_2],
np.conjugate(p[alpha_3]),
p[alpha_3],
)
)
for key in u_3:
if abs(u_3[key]) > energy_tolerance:
return False
return True
def _check_S_41(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.41 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_2 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=2, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 1:
key = alpha_1
if key not in u_2:
u_2[key] = 0
u_2[key] += p[alpha_1] @ parameter @ p[alpha_1]
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 1:
key = alpha_1
if key not in u_2:
u_2[key] = 0
medium_matrix = 2 * np.einsum("i,j->ij", z[alpha_1], p[alpha_1])
medium_matrix += np.einsum("i,j->ij", p[alpha_1], z[alpha_1])
u_2[key] -= np.einsum(
"ijk,ij,k",
parameter,
medium_matrix,
p[alpha_1],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=1
):
alpha_1 = alphas[0]
if round(2 * spin_values[alpha_1]) > 1:
key = alpha_1
if key not in u_2:
u_2[key] = 0
medium_matrix = 4 * np.einsum(
"i,j,k->ijk", z[alpha_1], z[alpha_1], p[alpha_1]
)
medium_matrix += 2 * np.einsum(
"i,j,k->ijk", z[alpha_1], p[alpha_1], z[alpha_1]
)
medium_matrix += np.einsum("i,j,k->ijk", p[alpha_1], z[alpha_1], z[alpha_1])
medium_matrix += (
0.5
* spin_values[alpha_1]
* np.einsum(
"i,j,k->ijk", p[alpha_1], p[alpha_1], np.conjugate(p[alpha_1])
)
)
medium_matrix += (spin_values[alpha_1] - 0.5) * np.einsum(
"i,j,k->ijk", p[alpha_1], np.conjugate(p[alpha_1]), p[alpha_1]
)
medium_matrix += (
1.5
* (spin_values[alpha_1] - 1)
* np.einsum(
"i,j,k->ijk", np.conjugate(p[alpha_1]), p[alpha_1], p[alpha_1]
)
)
u_2[key] += np.einsum(
"ijkl,ijk,l",
parameter,
medium_matrix,
p[alpha_1],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=3
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
alpha_1 = alphas[0]
alpha_2 = alphas[2]
if round(2 * spin_values[alpha_1]) > 1:
key = alpha_1
if key not in u_2:
u_2[key] = 0
u_2[key] += (
3
* spin_values[alpha_2]
* np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
p[alpha_1],
np.conjugate(p[alpha_2]),
p[alpha_2],
)
)
for key in u_2:
if abs(u_2[key]) > energy_tolerance:
return False
return True
def _check_S_40(
renormalized_parameters, p, z, spin_values, energy_tolerance=1e-8
) -> bool:
r"""
Check eigenstate condition from eq. S.40 of the supplementary material of |paper-2026|_.
Parameters
----------
renormalized_parameters
Renormalized interaction parameters.
p : (M, 3) :numpy`ndarray`
Complex spin components of the local reference frame.
z : (M, 3) :numpy`ndarray`
Directional vectors of the local reference frame.
spin_values : (M, 3) :numpy`ndarray`
Spin values.
energy_tolerance : float, default 1e-8
Tolerance for energy differences.
"""
u_1 = {}
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=1, p_n=1
):
alpha_1 = alphas[0]
key = alpha_1
if key not in u_1:
u_1[key] = 0
u_1[key] += parameter @ p[alpha_1]
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=2, p_n=1
):
alpha_1 = alphas[0]
key = alpha_1
if key not in u_1:
u_1[key] = 0
u_1[key] -= z[alpha_1] @ parameter @ p[alpha_1]
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=1
):
alpha_1 = alphas[0]
key = alpha_1
if key not in u_1:
u_1[key] = 0
medium_matrix = np.einsum("i,j->ij", z[alpha_1], z[alpha_1])
medium_matrix += (
0.5
* spin_values[alpha_1]
* np.einsum("i,j->ij", p[alpha_1], np.conjugate(p[alpha_1]))
)
if round(2 * spin_values[alpha_1]) > 1:
medium_matrix += (spin_values[alpha_1] - 0.5) * np.einsum(
"i,j->ij", np.conjugate(p[alpha_1]), p[alpha_1]
)
u_1[key] += np.einsum(
"ijk,ij,k",
parameter,
medium_matrix,
p[alpha_1],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=3, p_n=2
):
if (nus[0], alphas[1]) == (nus[1], alphas[2]):
alpha_1 = alphas[0]
alpha_2 = alphas[1]
key = alpha_1
if key not in u_1:
u_1[key] = 0
u_1[key] += (
1.5
* spin_values[alpha_2]
* np.einsum(
"ijk,i,j,k",
parameter,
p[alpha_1],
np.conjugate(p[alpha_2]),
p[alpha_2],
)
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=1
):
alpha_1 = alphas[0]
key = alpha_1
if key not in u_1:
u_1[key] = 0
medium_matrix = np.einsum("i,j,k->ijk", z[alpha_1], z[alpha_1], z[alpha_1])
medium_matrix += (
0.5
* spin_values[alpha_1]
* (
np.einsum("i,j,k->ijk", p[alpha_1], p[alpha_1], z[alpha_1])
+ np.einsum("i,j,k->ijk", z[alpha_1], p[alpha_1], p[alpha_1])
)
)
if round(2 * spin_values[alpha_1]) > 1:
medium_matrix += (spin_values[alpha_1] - 0.5) * (
2
* np.einsum(
"i,j,k->ijk", np.conjugate(p[alpha_1]), z[alpha_1], p[alpha_1]
)
+ np.einsum(
"i,j,k->ijk", z[alpha_1], np.conjugate(p[alpha_1]), p[alpha_1]
)
+ np.einsum(
"i,j,k->ijk", np.conjugate(p[alpha_1]), p[alpha_1], z[alpha_1]
)
)
u_1[key] -= np.einsum(
"ijkl,ijk,l",
parameter,
medium_matrix,
p[alpha_1],
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=2
):
if (nus[0], alphas[1]) == (nus[1], alphas[2]) and (nus[0], alphas[1]) == (
nus[2],
alphas[3],
):
alpha_1 = alphas[0]
alpha_2 = alphas[1]
key = alpha_1
if key not in u_1:
u_1[key] = 0
u_1[key] += (
2
* spin_values[alpha_2]
* np.einsum(
"ijkl,i,j,k,l",
parameter,
p[alpha_1],
np.conjugate(p[alpha_2]),
z[alpha_2],
p[alpha_2],
)
)
for nus, alphas, parameter in _InteractionParametersIterator(
parameters=renormalized_parameters, n=4, p_n=3
):
if ((0, 0, 0), alphas[0]) == (nus[0], alphas[1]):
alpha_1 = alphas[0]
alpha_2 = alphas[2]
key = alpha_1
if key not in u_1:
u_1[key] = 0
u_1[key] += (
3
* spin_values[alpha_2]
* np.einsum(
"ijkl,i,j,k,l",
parameter,
z[alpha_1],
p[alpha_1],
np.conjugate(p[alpha_2]),
p[alpha_2],
)
)
for key in u_1:
if abs(u_1[key]) > energy_tolerance:
return False
return True
