def _grid_point_group(extent: Sequence[int],
pbc: Sequence[bool]) -> PointGroup:
# axis permutations
# can exchange two axes iff they have the same kind of BC and length
# represent open BC by setting kind[i] = -extent[i], so just have to
# match these
axis_perm = []
axes = np.arange(len(extent), dtype=int)
extent = np.asarray(extent, dtype=int)
kind = np.where(pbc, extent, -extent)
ndim = len(extent)
for perm in permutations(axes):
if np.all(kind == kind[list(perm)]):
axis_perm.append(_perm_symm(perm))
result = PointGroup(axis_perm, ndim=ndim)
# reflections across axes and setting the origin
# OBC axes are only symmetric w.r.t. their midpoint, (extent[i]-1)/2
origin = []
for i in axes:
result = result @ PointGroup(
[Identity(), _axis_reflection(i, ndim)], ndim=ndim)
origin.append(0 if pbc[i] else (extent[i] - 1) / 2)
result = result.elems
result[0] = Identity() # it would otherwise be an equivalent PGSymmetry
return PointGroup(result, ndim=ndim).change_origin(origin)
def _grid_point_group(extent: Sequence[int], pbc: Sequence[bool],
color_edges: bool) -> PointGroup:
"""Point group of `Grid`, made up of axis permutations and flipping each axis."""
ndim = len(extent)
# Cannot exchange two axes if they are colored differently; otherwise,
# can only exchange them if they have the same kind of BC and length.
# Represent open BC by setting kind[i] = -extent[i], so just have to match these
if color_edges:
result = PointGroup([Identity()], ndim=ndim)
else:
axis_perm = []
axes = np.arange(ndim, dtype=int)
extent = np.asarray(extent, dtype=int)
kind = np.where(pbc, extent, -extent)
for perm in permutations(axes):
if np.all(kind == kind[list(perm)]):
if np.all(perm == axes):
axis_perm.append(Identity())
else:
axis_perm.append(_perm_symm(perm))
result = PointGroup(axis_perm, ndim=ndim)
# reflections across axes and setting the origin
# OBC axes are only symmetric w.r.t. their midpoint, (extent[i]-1)/2
origin = []
for i in range(ndim):
result = result @ PointGroup(
[Identity(), _axis_reflection(i, ndim)], ndim=ndim)
origin.append(0 if pbc[i] else (extent[i] - 1) / 2)
return result.change_origin(origin)
def _translations_along_axis(self, axis: int) -> PermutationGroup:
"""
The group of valid translations along an axis as a `PermutationGroup`
acting on the sites of `self.lattice.`
"""
if self.lattice._pbc[axis]:
trans_list = [Identity()]
# note that we need the preimages in the permutation
trans_perm = self.lattice.id_from_position(
self.lattice.positions - self.lattice.basis_vectors[axis])
vector = np.zeros(self.lattice.ndim, dtype=int)
vector[axis] = 1
trans_by_one = Translation(trans_perm, vector)
for _ in range(1, self.lattice.extent[axis]):
trans_list.append(trans_list[-1] @ trans_by_one)
return PermutationGroup(trans_list, degree=self.lattice.n_nodes)
else:
return PermutationGroup([Identity()], degree=self.lattice.n_nodes)
def rotation_group(self) -> PermutationGroup:
"""The group of rotations (i.e. point group symmetries with determinant +1)
as a `PermutationGroup` acting on the sites of `self.lattice`."""
perms = []
for p in self.point_group_.rotation_group():
if isinstance(p, Identity):
perms.append(Identity())
else:
# note that we need the preimages in the permutation
perm = self.lattice.id_from_position(
p.preimage(self.lattice.positions))
perms.append(Permutation(perm, name=str(p)))
return PermutationGroup(perms, degree=self.lattice.n_nodes)