qpms/qpms/symmetries.py

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from sympy.combinatorics import Permutation, PermutationGroup
Permutation.print_cyclic = True
import cmath
from cmath import exp, pi
from math import sqrt
import numpy as np
np.set_printoptions(linewidth=200)
import numbers
import re
ň = None
from .tmatrices import zflip_tyty, xflip_tyty, yflip_tyty, zrotN_tyty, WignerD_yy_fromvector, identity_tyty, apply_ndmatrix_left
from .cyquaternions import IRot3
from .cycommon import get_mn_y
s3long = np.sqrt(np.longdouble(3.))
def grouprep_try(tdict, src, im, srcgens, imgens, immultop = None, imcmp = None):
tdict[src] = im
for i in range(len(srcgens)):
new_src = src * srcgens[i]
new_im = (im * imgens[i]) if (immultop is None) else immultop(im, imgens[i])
if new_src not in tdict.keys():
grouprep_try(tdict, new_src, new_im, srcgens, imgens, immultop, imcmp)
elif ((new_im != tdict[new_src]) if (imcmp is None) else (not imcmp(new_im, tdict[new_src]))): # check consistency
print(src, ' * ', srcgens[i], ' --> ', new_src)
print(im)
print(' * ')
print(imgens[i])
print(' --> ')
print(new_im)
print(' != ')
print(tdict[new_src])
raise ValueError("Homomorphism inconsistency detected")
return
def group_dps_try(elemlist, elem, gens):
'''Deterministic group depth-first search'''
elemlist.append(elem)
for i in range(len(gens)):
newelem = elem * gens[i]
if newelem not in elemlist:
group_dps_try(elemlist, newelem, gens)
return
class SVWFPointGroupInfo: # only for point groups, coz in svwf_rep() I use I_tyty, not I_ptypty or something alike
def __init__(self,
name,
permgroupgens, # permutation group generators
irrepgens_dict, # dictionary with irrep generators,
svwf_rep_gen_func, # function that generates a tuple with svwf representation generators
rep3d_gens = None, # 3d (quaternion) representation generators of a point group: sequence of qpms.irep3 instances
):
self.name = name
self.permgroupgens = permgroupgens
self.permgroup = PermutationGroup(*permgroupgens)
self.irrepgens_dict = irrepgens_dict
self.svwf_rep_gen_func = svwf_rep_gen_func
self.irreps = dict()
for irrepname, irrepgens in irrepgens_dict.items():
is1d = isinstance(irrepgens[0], (int,float,complex))
irrepdim = 1 if is1d else irrepgens[0].shape[0]
self.irreps[irrepname] = generate_grouprep(self.permgroup,
1 if is1d else np.eye(irrepdim),
permgroupgens, irrepgens,
immultop = None if is1d else np.dot,
imcmp = None if is1d else np.allclose
)
self.rep3d_gens = rep3d_gens
self.rep3d = None if rep3d_gens is None else generate_grouprep(
self.permgroup,
IRot3(),
permgroupgens, rep3d_gens,
immultop = None, imcmp = (lambda x, y: x.isclose(y))
)
def deterministic_elemlist(self):
thelist = list()
group_dps_try(thelist, self.permgroup.identity, self.permgroupgens)
return thelist
def svwf_rep(self, lMax, *rep_gen_func_args, **rep_gen_func_kwargs):
'''
This method generates full SVWF (reducible) representation of the group.
'''
svwfgens = self.svwf_rep_gen_func(lMax, *rep_gen_func_args, **rep_gen_func_kwargs)
my, ny = get_mn_y(lMax)
nelem = len(my)
I_tyty = np.moveaxis(np.eye(2)[:,:,ň,ň] * np.eye(nelem), 2,1)
return generate_grouprep(self.permgroup, I_tyty, self.permgroupgens, svwfgens, immultop = mmult_tyty, imcmp = np.allclose)
def svwf_irrep_projectors(self, lMax, *rep_gen_func_args, **rep_gen_func_kwargs):
return gen_point_group_svwfrep_projectors(self.permgroup, self.irreps, self.svwf_rep(lMax, *rep_gen_func_args, **rep_gen_func_kwargs))
# alternative, for comparison and testing; should give the same results
def svwf_irrep_projectors2(self, lMax, *rep_gen_func_args, **rep_gen_func_kwargs):
return gen_point_group_svwfrep_projectors2(self.permgroup, self.irreps, self.svwf_rep(lMax, *rep_gen_func_args, **rep_gen_func_kwargs))
def svwf_irrep_projectors2_w_bases(self, lMax, *rep_gen_func_args, **rep_gen_func_kwargs):
return gen_point_group_svwfrep_projectors2_w_bases(self.permgroup, self.irreps, self.svwf_rep(lMax, *rep_gen_func_args, **rep_gen_func_kwargs))
def generate_c_source(self):
'''
Generates a string with a chunk of C code with a definition of a qpms_finite_group_t instance.
See also groups.h.
'''
permlist = self.deterministic_elemlist()
order = len(permlist)
permindices = {perm: i for i, perm in enumerate(permlist)} # 'invert' permlist
identity = self.permgroup.identity
s = "{\n"
# char *name
s += ' "%s", // name\n' % self.name
# size_t order;
s += ' %d, // order\n' % order
# qpms_gmi_t idi
s += ' %d, // idi\n' % permindices[identity]
# qpms_gmi_t *mt
s += ' (qpms_gmi_t[]) { // mt\n'
for i in range(order):
ss = ', '.join([str(permindices[permlist[i]*permlist[j]]) for j in range(order)])
s += ' ' + ss + ',\n'
s += ' },\n'
# qpms_gmi_t *invi
s += ' (qpms_gmi_t[]) { // invi\n'
s += ' ' + ', '.join([str(permindices[permlist[j]**-1]) for j in range(order)])
s += '\n },\n'
# qpms_gmi_t *gens
s += ' (qpms_gmi_t[]) {' + ', '.join([str(permindices[g]) for g in self.permgroupgens]) + '}, // gens\n'
# int ngens
s += ' %d, // ngens\n' % len(self.permgroupgens)
# qpms_permutation_t permrep[]
s += ' (qpms_permutation_t[]){ // permrep\n'
for i in range(order):
s += ' "%s",\n' % str(permlist[i])
s += ' },\n'
# char **elemlabels
s += ' NULL, // elemlabels\n'
# int permrep_nelem
s += ' %d, // permrep_nelem\n' % self.permgroup.degree
# qpms_irot3_t rep3d[]
if self.rep3d is None:
s += ' NULL, // rep3d TODO!!!\n'
else:
s += ' (qpms_irot3_t[]) { // rep3d\n'
for i in range(order):
s += ' ' + self.rep3d[permlist[i]].crepr() + ',\n'
s += ' },\n'
# int nirreps
s += ' %d, // nirreps\n' % len(self.irreps)
# struct qpms_finite_grep_irrep_t irreps[]
s += ' (struct qpms_finite_group_irrep_t[]) { // irreps\n'
for irname in sorted(self.irreps.keys()):
irrep = self.irreps[irname]
s += ' {\n'
is1d = isinstance(irrep[identity], (int, float, complex))
dim = 1 if is1d else irrep[identity].shape[0]
# int dim
s += ' %d, // dim\n' % dim
# char name[]
s += ' "%s", //name\n' % re.escape(irname)
# complex double *m
if (is1d):
s += ' (complex double []) {' + ', '.join([str(irrep[permlist[i]]) for i in range(order)]) + '} // m\n'
else:
s += ' (complex double []) {\n'
for i in range(order):
s += ' // %s\n' % str(permlist[i])
for row in range(dim):
s += ' '
for col in range(dim):
s += '%s, ' % re.sub('j', '*I', str(irrep[permlist[i]][row,col]))
s += '\n'
mat = irrep[permlist[i]]
s += ' }\n'
#s += ' %d, // dim\n' %
s += ' },\n'
s += ' } // end of irreps\n'
s += '}'
return s
# srcgroup is expected to be PermutationGroup and srcgens of the TODO
# imcmp returns True if two elements of the image group are 'equal', otherwise False
def generate_grouprep(srcgroup, im_identity, srcgens, imgens, immultop = None, imcmp = None):
sz = srcgens[0].size
for g in srcgens:
if g.size != sz:
raise ValueError('All the generators must have the same "size"')
tdict = dict()
grouprep_try(tdict, Permutation(sz-1), im_identity, srcgens, imgens, immultop = immultop, imcmp = imcmp)
if(srcgroup.order() != len(tdict.keys())): # basic check
raise ValueError('The supplied "generators" failed to generate the preimage group: ',
srcgroup.order(), " != ", len(tdict.keys()))
return tdict
# matrices appearing in 2d representations of common groups as used in Bradley, Cracknell p. 61 (with arabic names instead of greek, because lambda is a keyword)
epsilon = np.eye(2)
alif = np.array(((-1/2,-s3long/2),(s3long/2,-1/2)))
bih = np.array(((-1/2,s3long/2),(-s3long/2,-1/2)))
kaf = np.array(((0,1),(1,0)))
lam = np.array(((1,0),(0,-1)))
ra = np.array(((0,-1),(1,0)))
mim = np.array(((-1/2,-s3long/2),(-s3long/2,1/2)))
nun = np.array(((-1/2,s3long/2),(s3long/2,1/2)))
def mmult_tyty(a, b):
return(apply_ndmatrix_left(a, b, (-4,-3)))
def mmult_ptypty(a, b):
return(apply_ndmatrix_left(a, b, (-6,-5,-4)))
def gen_point_group_svwfrep_irreps(permgroup, matrix_irreps_dict, sphrep_full):
'''
Gives the projection operators $P_kl('\Gamma')$ from Dresselhaus (4.28)
for all irreps $\Gamma$ of D3h.;
as an array with indices [k,l,t,y,t,y]
Example of creating last argument:
sphrep_full = generate_grouprep(D3h_permgroup, I_tyty, D3h_srcgens, [C3_tyty, vfl_tyty, zfl_tyty],
immultop = mmult_tyty, imcmp = np.allclose)
'''
order = permgroup.order()
sphreps = dict()
nelem = sphrep_full[permgroup[0]].shape[-1] # quite ugly hack
for repkey, matrixrep in matrix_irreps_dict.items():
arepmatrix = matrixrep[permgroup[0]] # just one of the matrices to get the shape etc
if isinstance(arepmatrix, numbers.Number):
dim = 1 # repre dimension
preprocess = lambda x: np.array([[x]])
elif isinstance(arepmatrix, np.ndarray):
if(len(arepmatrix.shape)) != 2 or arepmatrix.shape[0] != arepmatrix.shape[1]:
raise ValueError("Arrays representing irrep matrices must be of square shape")
dim = arepmatrix.shape[0]
preprocess = lambda x: x
else:
raise ValueError("Irrep is not a square array or number")
sphrep = np.zeros((dim,dim,2,nelem,2,nelem), dtype=complex)
for i in permgroup.elements:
sphrep += preprocess(matrixrep[i]).conj().transpose()[:,:,ň,ň,ň,ň] * sphrep_full[i]
sphrep *= dim / order
# clean the nonexact values here
for x in [0, 0.5, -0.5, 0.5j, -0.5j]:
sphrep[np.isclose(sphrep,x)]=x
sphreps[repkey] = sphrep
return sphreps
def gen_point_group_svwfrep_projectors(permgroup, matrix_irreps_dict, sphrep_full):
'''
The same as gen_point_group_svwfrep_irreps, but summed over the kl diagonal, so
one gets single projector onto each irrep space and the arrays have indices
[t, y, t, y]
'''
summedprojs = dict()
for repi, W in gen_point_group_svwfrep_irreps(permgroup, matrix_irreps_dict, sphrep_full).items():
irrepd = W.shape[0]
if irrepd == 1:
mat = np.reshape(W, W.shape[-4:])
else:
mat = np.zeros(W.shape[-4:], dtype=complex) # TODO the result should be real — check!
for d in range(irrepd):
mat += W[d,d]
if not np.allclose(mat.imag, 0):
raise ValueError("The imaginary part of the resulting projector should be zero, damn!")
else:
summedprojs[repi] = mat.real
return summedprojs
def gen_point_group_svwfrep_projectors2_w_bases(permgroup, matrix_irreps_dict, sphrep_full):
return gen_point_group_svwfrep_projectors2(permgroup, matrix_irreps_dict, sphrep_full, do_bases = True)
def gen_point_group_svwfrep_projectors2(permgroup, matrix_irreps_dict, sphrep_full, do_bases = False):
'''
an approach as in gen_hexlattice_Kpoint_svwf_rep_projectors; for comparison and testing
'''
if (do_bases):
bases = dict()
projectors = dict()
for repi, W in gen_point_group_svwfrep_irreps(permgroup, matrix_irreps_dict, sphrep_full).items():
nelem = W.shape[-1] # however, this should change between iterations
totalvecs = 0
tmplist = list()
for t in (0,1):
for y in range(nelem):
for ai in range(W.shape[0]):
for bi in range(W.shape[1]):
v = np.zeros((2, nelem))
v[t,y] = 1
v1 = np.tensordot(W[ai,bi], v, axes = ([-2,-1],[0,1]))
if not np.allclose(v1,0):
v1 = normalize(v1)
for v2 in tmplist:
dot = np.tensordot(v1.conjugate(),v2, axes=([-2,-1],[0,1]))
if not (np.allclose(dot,0)):
if not np.allclose(np.abs(dot),1):
raise ValueError('You have to fix this piece of code.')
break
else:
totalvecs += 1
tmplist.append(v1)
theprojector = np.zeros((2,nelem, 2, nelem), dtype = float)
if do_bases:
thebasis = np.zeros((len(tmplist), 2, nelem), dtype=complex)
for i, v in enumerate(tmplist):
thebasis[i] = v
bases[repi] = thebasis
for v in tmplist:
theprojector += (v[:,:,ň,ň] * v.conjugate()[ň,ň,:,:]).real
for x in [0, 1, -1, sqrt(.5), -sqrt(.5), .5, -.5]:
theprojector[np.isclose(theprojector,x)] = x
projectors[repi] = theprojector
if do_bases:
return projectors, bases
else:
return projectors
# Group D3h; mostly legacy code (kept because of the the honeycomb lattice K-point code, whose generalised version not yet implemented)
# Note that the size argument of permutations is necessary, otherwise e.g. c*c and b*b would not be evaluated equal
# N.B. the weird elements as Permutation(N) it means identity permutation of size N+1.
rot3_perm = Permutation(0,1,2, size=5) # C3 rotation
xflip_perm = Permutation(0,2, size=5) # vertical mirror
zflip_perm = Permutation(3,4, size=5) # horizontal mirror
D3h_srcgens = [rot3_perm,xflip_perm,zflip_perm]
D3h_permgroup = PermutationGroup(*D3h_srcgens) # D3h
D3h_irreps = {
# Bradley, Cracknell p. 61
"E'" : generate_grouprep(D3h_permgroup, epsilon, D3h_srcgens, [alif, lam, epsilon], immultop = np.dot, imcmp = np.allclose),
"E''" : generate_grouprep(D3h_permgroup, epsilon, D3h_srcgens, [alif, lam, -epsilon], immultop = np.dot, imcmp = np.allclose),
# Bradley, Cracknell p. 59, or Dresselhaus, Table A.14 (p. 482)
"A1'" : generate_grouprep(D3h_permgroup, 1, D3h_srcgens, [1,1,1]),
"A2'" : generate_grouprep(D3h_permgroup, 1, D3h_srcgens, [1,-1,1]),
"A1''" : generate_grouprep(D3h_permgroup, 1, D3h_srcgens, [1,-1,-1]),
"A2''" : generate_grouprep(D3h_permgroup, 1, D3h_srcgens, [1,1,-1]),
}
#TODO lepší název fce; legacy, use group_info['D3h'].generate_grouprep() instead
def gen_point_D3h_svwf_rep(lMax, vflip = 'x'):
'''
Gives the projection operators $P_kl('\Gamma')$ from Dresselhaus (4.28)
for all irreps $\Gamma$ of D3h.;
as an array with indices [k,l,t,y,t,y]
'''
my, ny = get_mn_y(lMax)
nelem = len(my)
C3_yy = WignerD_yy_fromvector(lMax, np.array([0,0,2*pi/3]))
C3_tyty = np.moveaxis(np.eye(2)[:,:,ň,ň] * C3_yy, 2,1)
zfl_tyty = zflip_tyty(lMax)
#yfl_tyty = yflip_tyty(lMax)
#xfl_tyty = xflip_tyty(lMax)
vfl_tyty = yflip_tyty(lMax) if vflip == 'y' else xflip_tyty(lMax)
I_tyty = np.moveaxis(np.eye(2)[:,:,ň,ň] * np.eye(nelem), 2,1)
order = D3h_permgroup.order()
sphrep_full = generate_grouprep(D3h_permgroup, I_tyty, D3h_srcgens, [C3_tyty, vfl_tyty, zfl_tyty],
immultop = mmult_tyty, imcmp = np.allclose)
sphreps = dict()
for repkey, matrixrep in D3h_irreps.items():
arepmatrix = matrixrep[rot3_perm] # just one of the matrices to get the shape etc
if isinstance(arepmatrix, numbers.Number):
dim = 1 # repre dimension
preprocess = lambda x: np.array([[x]])
elif isinstance(arepmatrix, np.ndarray):
if(len(arepmatrix.shape)) != 2 or arepmatrix.shape[0] != arepmatrix.shape[1]:
raise ValueError("Arrays representing irrep matrices must be of square shape")
dim = arepmatrix.shape[0]
preprocess = lambda x: x
else:
raise ValueError("Irrep is not a square array or number")
sphrep = np.zeros((dim,dim,2,nelem,2,nelem), dtype=complex)
for i in D3h_permgroup.elements:
sphrep += preprocess(matrixrep[i]).conj().transpose()[:,:,ň,ň,ň,ň] * sphrep_full[i]
sphrep *= dim / order
# clean the nonexact values here
for x in [0, 0.5, -0.5, 0.5j, -0.5j]:
sphrep[np.isclose(sphrep,x)]=x
sphreps[repkey] = sphrep
return sphreps
def gen_hexlattice_Kpoint_svwf_rep(lMax, psi, vflip = 'x'):
my, ny = get_mn_y(lMax)
nelem = len(my)
C3_yy = WignerD_yy_fromvector(lMax, np.array([0,0,2*pi/3]))
C3_tyty = np.moveaxis(np.eye(2)[:,:,ň,ň] * C3_yy, 2,1)
zfl_tyty = zflip_tyty(lMax)
#yfl_tyty = yflip_tyty(lMax)
#xfl_tyty = xflip_tyty(lMax)
vfl_tyty = yflip_tyty(lMax) if vflip == 'y' else xflip_tyty(lMax)
I_tyty = np.moveaxis(np.eye(2)[:,:,ň,ň] * np.eye(nelem), 2,1)
hex_C3_K_ptypty = np.diag([exp(-psi*1j*2*pi/3),exp(+psi*1j*2*pi/3)])[:,ň,ň,:,ň,ň] * C3_tyty[ň,:,:,ň,:,:]
hex_zfl_ptypty = np.eye(2)[:,ň,ň,:,ň,ň] * zfl_tyty[ň,:,:,ň,:,:]
#hex_xfl_ptypty = np.array([[0,1],[1,0]])[:,ň,ň,:,ň,ň] * xfl_tyty[ň,:,:,ň,:,:]
hex_vfl_ptypty = np.array([[0,1],[1,0]])[:,ň,ň,:,ň,ň] * vfl_tyty[ň,:,:,ň,:,:]
hex_I_ptypty = np.eye((2*2*nelem)).reshape((2,2,nelem,2,2,nelem))
order = D3h_permgroup.order()
hex_K_sphrep_full = generate_grouprep(D3h_permgroup, hex_I_ptypty, D3h_srcgens, [hex_C3_K_ptypty, hex_vfl_ptypty, hex_zfl_ptypty],
immultop = mmult_ptypty, imcmp = np.allclose)
hex_K_sphreps = dict()
for repkey, matrixrep in D3h_irreps.items():
arepmatrix = matrixrep[rot3_perm] # just one of the matrices to get the shape etc
if isinstance(arepmatrix, numbers.Number):
dim = 1 # repre dimension
preprocess = lambda x: np.array([[x]])
elif isinstance(arepmatrix, np.ndarray):
if(len(arepmatrix.shape)) != 2 or arepmatrix.shape[0] != arepmatrix.shape[1]:
raise ValueError("Arrays representing irrep matrices must be of square shape")
dim = arepmatrix.shape[0]
preprocess = lambda x: x
else:
raise ValueError("Irrep is not a square array or number")
sphrep = np.zeros((dim,dim,2,2,nelem,2,2,nelem), dtype=complex)
for i in D3h_permgroup.elements:
sphrep += preprocess(matrixrep[i]).conj().transpose()[:,:,ň,ň,ň,ň,ň,ň] * hex_K_sphrep_full[i]
sphrep *= dim / order
# clean the nonexact values here
for x in [0, 0.5, -0.5, 0.5j, -0.5j]:
sphrep[np.isclose(sphrep,x)]=x
hex_K_sphreps[repkey] = sphrep
return hex_K_sphreps
def normalize(v):
norm = np.linalg.norm(v.reshape((np.prod(v.shape),)), ord=2)
if norm == 0:
return v*np.nan
return v / norm
def gen_hexlattice_Kpoint_svwf_rep_projectors(lMax, psi, vflip='x', do_bases=False):
nelem = lMax * (lMax+2)
projectors = dict()
if do_bases:
bases = dict()
for repi, W in gen_hexlattice_Kpoint_svwf_rep(lMax,psi,vflip=vflip).items():
totalvecs = 0
tmplist = list()
for p in (0,1):
for t in (0,1):
for y in range(nelem):
for ai in range(W.shape[0]):
for bi in range(W.shape[1]):
v = np.zeros((2,2,nelem))
v[p,t,y] = 1
#v = np.ones((2,2,nelem))
v1 = np.tensordot(W[ai,bi],v, axes = ([-3,-2,-1],[0,1,2]))
if not np.allclose(v1,0):
v1 = normalize(v1)
for v2 in tmplist:
dot = np.tensordot(v1.conjugate(),v2,axes = ([-3,-2,-1],[0,1,2]))
if not np.allclose(dot,0):
if not np.allclose(np.abs(dot),1):
raise ValueError('You have to fix this piece of code.')# TODO maybe I should make sure that the absolute value is around 1
break
else:
totalvecs += 1
tmplist.append(v1)
#for index, x in np.ndenumerate(v1):
# if x!=0:
# print(index, x)
#print('----------')
theprojector = np.zeros((2,2,nelem,2,2,nelem), dtype = float)
if do_bases:
thebasis = np.zeros((len(tmplist), 2,2,nelem), dtype=complex)
for i, v in enumerate(tmplist):
thebasis[i] = v
bases[repi] = thebasis
for v in tmplist:
theprojector += (v[:,:,:,ň,ň,ň] * v.conjugate()[ň,ň,ň,:,:,:]).real # TODO check is it possible to have imaginary elements?
for x in [0, 1, -1,sqrt(0.5),-sqrt(0.5),0.5,-0.5]:
theprojector[np.isclose(theprojector,x)]=x
projectors[repi] = theprojector
if do_bases:
return projectors, bases
else:
return projectors
point_group_info = { # representation info of some useful point groups
# TODO real trivial without generators
'trivial_g' : SVWFPointGroupInfo('trivial_g',
# permutation group generators
( # I put here the at least the identity for now (it is reduntant, but some functions are not robust enough to have an empty set of generators
Permutation(),
),
# dictionary with irrep generators
{
"A" : (1,),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (identity_tyty(lMax),),
# quaternion rep generators
rep3d_gens = (
IRot3.identity(),
)
),
'C2' : SVWFPointGroupInfo('C2',
# permutation group generators
(Permutation(0,1), # 180 deg rotation around z axis
),
# dictionary with irrep generators
{
# Bradley, Cracknell p. 57;
'A': (1,),
'B': (-1,),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (zrotN_tyty(2, lMax),),
# quaternion rep generators
rep3d_gens = (
IRot3.zrotN(2),
)
),
'C2v' : SVWFPointGroupInfo('C2v',
# permutation group generators
(Permutation(0,1, size=4)(2,3), # x -> - x mirror operation (i.e. yz mirror plane)
Permutation(0,3, size=4)(1,2), # y -> - y mirror operation (i.e. xz mirror plane)
),
# dictionary with irrep generators
{
# Bradley, Cracknell p. 58; not sure about the labels / axes here
'A1': (1,1),
'B2': (-1,1),
'A2': (-1,-1),
'B1': (1,-1),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (xflip_tyty(lMax), yflip_tyty(lMax)),
# quaternion rep generators
rep3d_gens = (
IRot3.xflip(),
IRot3.yflip(),
)
),
'D2h' : SVWFPointGroupInfo('D2h',
# permutation group generators
(Permutation(0,1, size=6)(2,3), # x -> - x mirror operation (i.e. yz mirror plane)
Permutation(0,3, size=6)(1,2), # y -> - y mirror operation (i.e. xz mirror plane)
# ^^^ btw, I guess that Permutation(0,1, size=6) and Permutation(2,3, size=6) would
# do exactly the same job (they should; CHECK)
Permutation(4,5, size=6) # z -> - z mirror operation (i.e. xy mirror plane)
),
# dictionary with irrep generators
{
# Product of C2v and zflip; not sure about the labels / axes here
"A1'": (1,1,1),
"B2'": (-1,1,1),
"A2'": (-1,-1,1),
"B1'": (1,-1,1),
"A1''": (-1,-1,-1),
"B2''": (1,-1,-1),
"A2''": (1,1,-1),
"B1''": (-1,1,-1),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (xflip_tyty(lMax), yflip_tyty(lMax), zflip_tyty(lMax)),
# quaternion rep generators
rep3d_gens = (
IRot3.xflip(),
IRot3.yflip(),
IRot3.zflip(),
)
),
'C4' : SVWFPointGroupInfo('C4',
# permutation group generators
(Permutation(0,1,2,3, size=4),), #C4 rotation
# dictionary with irrep generators
{
# Bradley, Cracknell p. 58
'A': (1,),
'B': (-1,),
'1E': (-1j,),
'2E': (1j,),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (zrotN_tyty(4, lMax), ),
# quaternion rep generators
rep3d_gens = (
IRot3.zrotN(4),
)
),
'C4v' : SVWFPointGroupInfo('C4v',
# permutation group generators
(Permutation(0,1,2,3, size=4), #C4 rotation
Permutation(0,1, size=4)(2,3)), # x -> - x mirror operation (i.e. yz mirror plane)
# dictionary with irrep generators
{
# Bradley, Cracknell p. 62
'E': (ra, -lam),
# Bradley, Cracknell p. 59, or Dresselhaus, Table A.18
'A1': (1,1),
'A2': (1,-1),
'B1': (-1,1),
'B2': (-1,-1),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (zrotN_tyty(4, lMax), xflip_tyty(lMax)),
# quaternion rep generators
rep3d_gens = (
IRot3.zrotN(4),
IRot3.xflip(),
)
),
'D4h' : SVWFPointGroupInfo('D4h',
# permutation group generators
(Permutation(0,1,2,3, size=6), # C4 rotation
Permutation(0,1, size=6)(2,3), # x -> - x mirror operation (i.e. yz mirror plane)
Permutation(4,5, size=6), # horizontal mirror operation z -> -z (i.e. xy mirror plane)
),
# dictionary with irrep generators
{ # product of C4v and zflip
"E'": (ra, -lam, epsilon),
"E''":(ra, -lam, -epsilon),
"A1'": (1,1,1),
"A2'": (1,-1,1),
"A1''": (1,-1,-1),
"A2''": (1,1,-1),
"B1'": (-1,1,1),
"B2'": (-1,-1,1),
"B1''": (-1,-1,-1),
"B2''": (-1,1,-1),
},
# function that generates a tuple with svwf representation generators
lambda lMax : (zrotN_tyty(4, lMax), xflip_tyty(lMax), zflip_tyty(lMax)),
# quaternion rep generators
rep3d_gens = (
IRot3.zrotN(4),
IRot3.xflip(),
IRot3.zflip(),
)
),
'D3h' : SVWFPointGroupInfo('D3h',
# permutation group generators
( Permutation(0,1,2, size=5), # C3 rotation
Permutation(0,2, size=5), # vertical mirror
Permutation(3,4, size=5), # horizontal mirror z -> -z (i.e. xy mirror plane)
),
# dictionary with irrep generators
{ # Bradley, Cracknell p. 61
"E'" : (alif, lam, epsilon),
"E''" : (alif, lam, -epsilon),
# Bradley, Cracknell p. 59, or Dresselhaus, Table A.14 (p. 482)
"A1'" : (1,1,1),
"A2'" : (1,-1,1),
"A1''" : (1,-1,-1),
"A2''" : (1,1,-1),
},
# function that generates a tuple with svwf representation generators
lambda lMax, vflip: (zrotN_tyty(3, lMax), yflip_tyty(lMax) if vflip == 'y' else xflip_tyty(lMax), zflip_tyty(lMax)),
# quaternion rep generators
rep3d_gens = (
IRot3.zrotN(3),
IRot3.xflip(), # if vflip == 'y' else IRot3.xflip(), # FIXME enable to choose
IRot3.zflip(),
)
),
'x_and_z_flip': SVWFPointGroupInfo(
'x_and_z_flip',
(
Permutation(0,1, size=4), # x -> -x mirror op
Permutation(2,3, size=4), # z -> -z mirror op
),
{
"P'": (1, 1),
"R'": (-1, 1),
"P''": (-1,-1),
"R''": (1, -1),
},
lambda lMax : (xflip_tyty(lMax), zflip_tyty(lMax)),
rep3d_gens = (
IRot3.xflip(),
IRot3.zflip(),
)
),
'y_and_z_flip': SVWFPointGroupInfo(
'y_and_z_flip',
(
Permutation(0,1, size=4), # y -> -y mirror op
Permutation(2,3, size=4), # z -> -z mirror op
),
{
"P'": (1, 1),
"R'": (-1, 1),
"P''": (-1,-1),
"R''": (1, -1),
},
lambda lMax : (yflip_tyty(lMax), zflip_tyty(lMax)),
rep3d_gens = (
IRot3.yflip(),
IRot3.zflip(),
)
),
}