def get_clusters_rel_by_symmetry(cluster, symmetry='d4'):
"""
Return all distinct clusters related to the initial cluster by symmetry.
:param cluster: geometry object representing a cluster
:param symmetry: TODO implement others besides D4
:return cluster_symm_partners: a list of geometry objects, including the initial cluster which are related by the
specified symmetry
"""
rot_fn = symm.getRotFn(4)
refl_fn = symm.getReflFn([0, 1])
cluster_symm_partners = [cluster]
# list coordinates of all D4 symmetries
places_round = 14
xys_list = []
xys_list.append(np.round(rot_fn(cluster.xlocs, cluster.ylocs), places_round))
xys_list.append(np.round(rot_fn(xys_list[0][0, :], xys_list[0][1, :]), places_round))
xys_list.append(np.round(rot_fn(xys_list[1][0, :], xys_list[1][1, :]), places_round))
xys_list.append(np.round(refl_fn(cluster.xlocs, cluster.ylocs), places_round))
xys_list.append(np.round(refl_fn(xys_list[0][0, :], xys_list[0][1, :]), places_round))
xys_list.append(np.round(refl_fn(xys_list[1][0, :], xys_list[1][1, :]), places_round))
xys_list.append(np.round(refl_fn(xys_list[2][0, :], xys_list[2][1, :]), places_round))
# add distinct clusters to a list
for xys in xys_list:
c = geom.Geometry.createNonPeriodicGeometry(xys[0, :], xys[1, :])
c.permute_sites(c.get_sorting_permutation())
if not [h for h in cluster_symm_partners if h.isequal_adjacency(c)]:
cluster_symm_partners.append(c)
return cluster_symm_partners
def test_full_symm_3byb3(self):
U = 20 * (np.random.rand() - 0.5)
t = np.random.rand()
gm = geom.Geometry.createSquareGeometry(3,
3,
0,
0,
bc1_open=False,
bc2_open=False)
model = ed_fermions.fermions(gm, U, t, ns=np.array([2, 2]))
# no symmetry
hamiltonian_full = model.createH(projector=model.n_projector)
eig_vals_full, eig_vects_full = model.diagH(hamiltonian_full)
# rotational symmetry
cx, cy = model.geometry.get_center_of_mass()
rot_fn = symm.getRotFn(4, cx=cx, cy=cy)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, model.geometry)
rot_op = model.n_projector.dot(
model.get_xform_op(rot_cycles).dot(
model.n_projector.conj().transpose()))
# reflection symmetry
refl_fn = symm.getReflFn(np.array([[0], [1]]), cx=cx, cy=cy)
refl_cycles, max_cycle_len_refl = symm.findSiteCycles(
refl_fn, model.geometry)
refl_op = model.n_projector.dot(
model.get_xform_op(refl_cycles).dot(
model.n_projector.conj().transpose()))
# translational symmetry
tx_fn = symm.getTranslFn(np.array([[1], [0]]))
tx_cycles, tx_max = symm.findSiteCycles(tx_fn, model.geometry)
tx_op_full = model.get_xform_op(tx_cycles)
tx_op = model.n_projector.dot(
tx_op_full.dot(model.n_projector.conj().transpose()))
ty_fn = symm.getTranslFn((np.array([[0], [1]])))
ty_cycles, ty_max = symm.findSiteCycles(ty_fn, model.geometry)
ty_op_full = model.get_xform_op(ty_cycles)
ty_op = model.n_projector.dot(
ty_op_full.dot(model.n_projector.conj().transpose()))
symm_projs = symm.getC4V_and_3byb3(rot_op, refl_op, tx_op, ty_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = model.createH(projector=proj.dot(model.n_projector))
eig_vals_sector, eig_vects_sector = model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
# why only accurate to 10 decimal places?
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 10)
def test_d4_symm(self):
"""
Test d4 symmetry by diagonalizing random spin system with and without it
:return:
"""
cluster = geom.Geometry.createSquareGeometry(3,
3,
0,
0,
bc1_open=False,
bc2_open=False)
jx = np.random.rand()
jy = np.random.rand()
jz = np.random.rand()
hx = np.random.rand()
hy = np.random.rand()
hz = np.random.rand()
# diagonalize full hamiltonian
spin_model = tvi.spinSystem(cluster,
jx,
jy,
jz,
hx,
hy,
hz,
use_ryd_detunes=0)
hamiltonian_full = spin_model.createH()
eig_vals_full, eig_vects_full = spin_model.diagH(hamiltonian_full)
# use rotationl symmetry
cx, cy = spin_model.geometry.get_center_of_mass()
rot_fn = symm.getRotFn(4, cx=cx, cy=cy)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, spin_model.geometry)
rot_op = spin_model.get_xform_op(rot_cycles)
refl_fn = symm.getReflFn(np.array([0, 1]), cx=cx, cy=cy)
refl_cycles, max_cycle_len_ref = symm.findSiteCycles(
refl_fn, spin_model.geometry)
refl_op = spin_model.get_xform_op(refl_cycles)
symm_projs = symm.getD4Projectors(rot_op, refl_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = spin_model.createH(projector=proj)
eig_vals_sector, eig_vects_sector = spin_model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
# why only accurate to 10 decimal places?
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 10)
def test_d2_symmetry_3by3(self):
"""
Test D2 symmetry (generated by 180 degree rotation and a reflection) on a 3x3 Hubbard cluster
with open boundary conditions.
:return:
"""
# todo: do all number sectors
U = 20 * (np.random.rand() - 0.5)
t = np.random.rand()
gm = geom.Geometry.createSquareGeometry(3,
3,
0,
0,
bc1_open=False,
bc2_open=False)
model = ed_fermions.fermions(gm, U, t, ns=np.array([3, 4]), nspecies=2)
# no symmetry
hamiltonian_full = model.createH(projector=model.n_projector)
eig_vals_full, eig_vects_full = model.diagH(hamiltonian_full)
# rotational symmetry
cx, cy = model.geometry.get_center_of_mass()
rot_fn = symm.getRotFn(2, cx=cx, cy=cy)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, model.geometry)
rot_op_full = model.get_xform_op(rot_cycles)
rot_op = model.n_projector.dot(
rot_op_full.dot(model.n_projector.conj().transpose()))
# reflection symmetry
refl_fn = symm.getReflFn(np.array([[0], [1]]), cx=cx, cy=cy)
refl_cycles, max_cycle_len_refl = symm.findSiteCycles(
refl_fn, model.geometry)
refl_op_full = model.get_xform_op(refl_cycles)
refl_op = model.n_projector.dot(
refl_op_full.dot(model.n_projector.conj().transpose()))
symm_projs = symm.getD2Projectors(rot_op, refl_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = model.createH(projector=proj.dot(model.n_projector))
eig_vals_sector, eig_vects_sector = model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
# why only accurate to 10 decimal places?
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 10)
def test_d3_symm_6sites(self):
adjacency_mat = np.array([[0, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 0],
[0, 1, 0, 0, 1, 0], [1, 1, 0, 0, 1, 1],
[0, 1, 1, 1, 0, 1], [0, 0, 0, 1, 1, 0]])
cluster = geom.Geometry.createNonPeriodicGeometry(
xlocs=(0, 1, 2, 0.5, 1.5, 1),
ylocs=(0, 0, 0, np.sqrt(3) / 2, np.sqrt(3) / 2, np.sqrt(3)),
adjacency_mat=adjacency_mat)
jx = np.random.rand()
jy = np.random.rand()
jz = np.random.rand()
hx = np.random.rand()
hy = np.random.rand()
hz = np.random.rand()
# diagonalize full hamiltonian
spin_model = tvi.spinSystem(cluster,
jx,
jy,
jz,
hx,
hy,
hz,
use_ryd_detunes=0)
hamiltonian_full = spin_model.createH()
eig_vals_full, eig_vects_full = spin_model.diagH(hamiltonian_full)
# use rotationl symmetry
# triangle center
cx, cy = 1, 1 / np.sqrt(3)
rot_fn = symm.getRotFn(3, cx=cx, cy=cy)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, spin_model.geometry)
rot_op = spin_model.get_xform_op(rot_cycles)
refl_fn = symm.getReflFn(np.array([0, 1]), cx=cx, cy=cy)
refl_cycles, max_cycle_len_ref = symm.findSiteCycles(
refl_fn, spin_model.geometry)
refl_op = spin_model.get_xform_op(refl_cycles)
symm_projs = symm.getD3Projectors(rot_op, refl_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = spin_model.createH(projector=proj)
eig_vals_sector, eig_vects_sector = spin_model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 12)
def test_d3_symm_3sites(self):
"""
Test d3 symmetry by diagonalizing random spin system with and without it
:return:
"""
# adjacency_mat = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]])
# cluster = geom.Geometry.createNonPeriodicGeometry(xlocs=(0,0.5,1), ylocs=(0,np.sqrt(3)/2,0), adjacency_mat=adjacency_mat)
cluster = geom.Geometry.createRegularPolygonGeometry(3)
jx = np.random.rand()
jy = np.random.rand()
jz = np.random.rand()
hx = np.random.rand()
hy = np.random.rand()
hz = np.random.rand()
# diagonalize full hamiltonian
spin_model = tvi.spinSystem(cluster,
jx,
jy,
jz,
hx,
hy,
hz,
use_ryd_detunes=0)
hamiltonian_full = spin_model.createH()
eig_vals_full, eig_vects_full = spin_model.diagH(hamiltonian_full)
# use rotationl symmetry
rot_fn = symm.getRotFn(3)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, spin_model.geometry)
rot_op = spin_model.get_xform_op(rot_cycles)
refl_fn = symm.getReflFn(np.array([0, 1]))
refl_cycles, max_cycle_len_ref = symm.findSiteCycles(
refl_fn, spin_model.geometry)
refl_op = spin_model.get_xform_op(refl_cycles)
symm_projs = symm.getD3Projectors(rot_op, refl_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = spin_model.createH(projector=proj)
eig_vals_sector, eig_vects_sector = spin_model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
# why only accurate to 10 decimal places?
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 12)
def test_get_refl_fn(self):
"""
Test coordinate reflection function
:return:
"""
refl_vect = np.array([[1.0], [1.0]])
refl_fn = ed_symmetry.getReflFn(refl_vect, cx=-1.0, cy=2.0)
x = np.array([3.0])
y = np.array([5.0])
x_refl, y_refl = refl_fn(x, y)
x_refl_expected = np.array([2.0])
y_refl_expected = np.array([6.0])
self.assertAlmostEqual(np.linalg.norm(x_refl - x_refl_expected), 0, 14)
self.assertAlmostEqual(np.linalg.norm(y_refl - y_refl_expected), 0, 14)
def test_d7_symm_7sites(self):
cluster = geom.Geometry.createRegularPolygonGeometry(7)
jx = np.random.rand()
jy = np.random.rand()
jz = np.random.rand()
hx = np.random.rand()
hy = np.random.rand()
hz = np.random.rand()
# diagonalize full hamiltonian
spin_model = tvi.spinSystem(cluster,
jx,
jy,
jz,
hx,
hy,
hz,
use_ryd_detunes=0)
hamiltonian_full = spin_model.createH()
eig_vals_full, eig_vects_full = spin_model.diagH(hamiltonian_full)
# use rotationl symmetry
rot_fn = symm.getRotFn(7)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, spin_model.geometry)
rot_op = spin_model.get_xform_op(rot_cycles)
refl_fn = symm.getReflFn(np.array([0, 1]))
refl_cycles, max_cycle_len_ref = symm.findSiteCycles(
refl_fn, spin_model.geometry)
refl_op = spin_model.get_xform_op(refl_cycles)
symm_projs = symm.getD6Projectors(rot_op, refl_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = spin_model.createH(projector=proj)
eig_vals_sector, eig_vects_sector = spin_model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 14)
def test_full_symm_3x3(self):
cluster = geom.Geometry.createSquareGeometry(3,
3,
0,
0,
bc1_open=False,
bc2_open=False)
jx = np.random.rand()
jy = np.random.rand()
jz = np.random.rand()
hx = np.random.rand()
hy = np.random.rand()
hz = np.random.rand()
# diagonalize full hamiltonian
spin_model = tvi.spinSystem(cluster,
jx,
jy,
jz,
hx,
hy,
hz,
use_ryd_detunes=0)
hamiltonian_full = spin_model.createH()
eig_vals_full, eig_vects_full = spin_model.diagH(hamiltonian_full)
xtransl_fn = symm.getTranslFn(np.array([[1], [0]]))
xtransl_cycles, max_cycle_len_translx = symm.findSiteCycles(
xtransl_fn, spin_model.geometry)
tx_op = spin_model.get_xform_op(xtransl_cycles)
# y-translations
ytransl_fn = symm.getTranslFn(np.array([[0], [1]]))
ytransl_cycles, max_cycle_len_transly = symm.findSiteCycles(
ytransl_fn, spin_model.geometry)
ty_op = spin_model.get_xform_op(ytransl_cycles)
# rotation
cx, cy = spin_model.geometry.get_center_of_mass()
rot_fn = symm.getRotFn(4, cx=cx, cy=cy)
rot_cycles, max_cycle_len_rot = symm.findSiteCycles(
rot_fn, spin_model.geometry)
rot_op = spin_model.get_xform_op(rot_cycles)
# reflection
# reflection about y-axis
refl_fn = symm.getReflFn(np.array([0, 1]), cx=cx, cy=cy)
refl_cycles, max_cycle_len_refl = symm.findSiteCycles(
refl_fn, spin_model.geometry)
refl_op = spin_model.get_xform_op(refl_cycles)
symm_projs = symm.getC4V_and_3byb3(rot_op, refl_op, tx_op, ty_op)
eig_vals_sectors = []
for ii, proj in enumerate(symm_projs):
h_sector = spin_model.createH(projector=proj)
eig_vals_sector, eig_vects_sector = spin_model.diagH(h_sector)
eig_vals_sectors.append(eig_vals_sector)
# why only accurate to 10 decimal places?
eigs_all_sectors = np.sort(np.concatenate(eig_vals_sectors))
max_diff = np.abs(eig_vals_full - eigs_all_sectors).max()
self.assertAlmostEqual(max_diff, 0, 10)