def term_hopping(hopping_dict, hop_mat, atoms,
sublattices, coords_dict):
"""Find Kwant hoppings in a qsymm.BlochModel term that has a lattice
translation in the Bloch factor.
"""
# Iterate over combinations of atoms, set hoppings between each
for atom1, atom2 in it.product(atoms, atoms):
# Take the block from atom1 to atom2
hop = hop_mat[ranges[atom1], ranges[atom2]]
# Only include nonzero hoppings
if allclose(hop, 0):
continue
# Adjust hopping vector to Bloch form basis
r_lattice = (
r_vec
+ np.array(coords_dict[atom1])
- np.array(coords_dict[atom2])
)
# Bring vector to basis of lattice vectors
lat_basis = np.linalg.solve(np.vstack(lat_vecs).T, r_lattice)
lat_basis = make_int(lat_basis)
# Should only have hoppings that are integer multiples of
# lattice vectors
if lat_basis is not None:
hop_dir = builder.HoppingKind(-lat_basis,
sublattices[atom1],
sublattices[atom2])
# Set the hopping as the matrix times the hopping amplitude
hopping_dict[hop_dir] += Model({coeff: hop}, momenta=momenta)
else:
raise RuntimeError('A nonzero hopping not matching a '
'lattice vector was found.')
return hopping_dict
def term_onsite(onsites_dict, hopping_dict, hop_mat, atoms,
sublattices, coords_dict):
"""Find the Kwant onsites and hoppings in a qsymm.BlochModel term
that has no lattice translation in the Bloch factor.
"""
for atom1, atom2 in it.product(atoms, atoms):
# Subblock within the same sublattice is onsite
hop = hop_mat[ranges[atom1], ranges[atom2]]
if sublattices[atom1] == sublattices[atom2]:
onsites_dict[atom1] += Model({coeff: hop}, momenta=momenta)
# Blocks between sublattices are hoppings between sublattices
# at the same position.
# Only include nonzero hoppings
elif not allclose(hop, 0):
if not allclose(np.array(coords_dict[atom1]),
np.array(coords_dict[atom2])):
raise ValueError(
"Position of sites not compatible with qsymm model.")
lat_basis = np.array(zer)
hop = Model({coeff: hop}, momenta=momenta)
hop_dir = builder.HoppingKind(-lat_basis, sublattices[atom1],
sublattices[atom2])
hopping_dict[hop_dir] += hop
return onsites_dict, hopping_dict
def test_consistency_kwant():
"""Make a random 1D Model, convert it to a builder, and compare
the Bloch representation of the Model with that which Kwant uses
in wraparound and in Bands. Then, convert the builder back to a Model
and compare with the original Model.
For comparison, we also make the system using Kwant only.
"""
orbs = 4
T = np.random.rand(2 * orbs,
2 * orbs) + 1j * np.random.rand(2 * orbs, 2 * orbs)
H = np.random.rand(2 * orbs,
2 * orbs) + 1j * np.random.rand(2 * orbs, 2 * orbs)
H += H.T.conj()
# Make the 1D Model manually using only qsymm features.
c0, c1 = sympy.symbols('c0 c1', real=True)
kx = _commutative_momenta[0]
Ham = Model({c0 * e**(-I * kx): T}, momenta=[0])
Ham += Ham.T().conj()
Ham += Model({c1: H}, momenta=[0])
# Two superimposed atoms, same number of orbitals on each
norbs = OrderedDict([('A', orbs), ('B', orbs)])
atom_coords = [(0.3, ), (0.3, )]
lat_vecs = [(1, )] # Lattice vector
# Make a Kwant builder out of the qsymm Model
model_syst = model_to_builder(Ham, norbs, lat_vecs, atom_coords)
fmodel_syst = model_syst.finalized()
# Make the same system manually using only Kwant features.
lat = kwant.lattice.general(np.array([[1.]]), [(0., )], norbs=2 * orbs)
kwant_syst = kwant.Builder(kwant.TranslationalSymmetry(*lat.prim_vecs))
def onsite(site, c1):
return c1 * H
def hopping(site1, site2, c0):
return c0 * T
sublat = lat.sublattices[0]
kwant_syst[sublat(0, )] = onsite
hopp = kwant.builder.HoppingKind((1, ), sublat)
kwant_syst[hopp] = hopping
fkwant_syst = kwant_syst.finalized()
# Make sure we are consistent with bands calculations in kwant
# The Bloch Hamiltonian used in Kwant for the bands computation
# is h(k) = exp(-i*k)*hop + onsite + exp(i*k)*hop.T.conj.
# We also check that all is consistent with wraparound
coeffs = (0.7, 1.2)
params = dict(c0=coeffs[0], c1=coeffs[1])
kwant_hop = fkwant_syst.inter_cell_hopping(params=params)
kwant_onsite = fkwant_syst.cell_hamiltonian(params=params)
model_kwant_hop = fmodel_syst.inter_cell_hopping(params=params)
model_kwant_onsite = fmodel_syst.cell_hamiltonian(params=params)
assert allclose(model_kwant_hop, coeffs[0] * T)
assert allclose(model_kwant_hop, kwant_hop)
assert allclose(model_kwant_onsite, kwant_onsite)
h_model_kwant = (
lambda k: np.exp(-1j * k) * model_kwant_hop + model_kwant_onsite + np.
exp(1j * k) * model_kwant_hop.T.conj()) # As in kwant.Bands
h_model = Ham.lambdify()
wsyst = kwant.wraparound.wraparound(model_syst).finalized()
for _ in range(20):
k = (np.random.rand() - 0.5) * 2 * np.pi
assert allclose(h_model_kwant(k), h_model(coeffs[0], coeffs[1], k))
params['k_x'] = k
h_wrap = wsyst.hamiltonian_submatrix(params=params)
assert allclose(h_model(coeffs[0], coeffs[1], k), h_wrap)
# Get the model back from the builder
# From the Kwant builder based on original Model
Ham1 = builder_to_model(model_syst,
momenta=Ham.momenta).tomodel(nsimplify=True)
# From the pure Kwant builder
Ham2 = builder_to_model(kwant_syst,
momenta=Ham.momenta).tomodel(nsimplify=True)
assert Ham == Ham1
assert Ham == Ham2
def test_graphene_to_kwant():
norbs = OrderedDict([('A', 1), ('B', 1)
]) # A and B atom per unit cell, one orbital each
hopping_vectors = [('A', 'B', [1, 0])
] # Hopping between neighbouring A and B atoms
# Atomic coordinates within the unit cell
atom_coords = [(0, 0), (1, 0)]
# We set the interatom distance to 1, so the lattice vectors have length sqrt(3)
lat_vecs = [(3 / 2, np.sqrt(3) / 2), (3 / 2, -np.sqrt(3) / 2)]
# Time reversal
TR = PointGroupElement(sympy.eye(2), True, False, np.eye(2))
# Chiral symmetry
C = PointGroupElement(sympy.eye(2), False, True, np.array([[1, 0], [0,
-1]]))
# Atom A rotates into A, B into B.
sphi = 2 * sympy.pi / 3
RC3 = sympy.Matrix([[sympy.cos(sphi), -sympy.sin(sphi)],
[sympy.sin(sphi), sympy.cos(sphi)]])
C3 = PointGroupElement(RC3, False, False, np.eye(2))
# Generate graphene Hamiltonian in Kwant from qsymm
symmetries = [C, TR, C3]
# Generate using a family
family = bloch_family(hopping_vectors, symmetries, norbs)
syst_from_family = model_to_builder(family,
norbs,
lat_vecs,
atom_coords,
coeffs=None)
# Generate using a single Model object
g = sympy.Symbol('g', real=True)
ham = hamiltonian_from_family(family, coeffs=[g])
ham = Model(hamiltonian=ham, momenta=family[0].momenta)
syst_from_model = model_to_builder(ham, norbs, lat_vecs, atom_coords)
# Make the graphene Hamiltonian using kwant only
atoms, orbs = zip(*[(atom, norb) for atom, norb in norbs.items()])
# Make the kwant lattice
lat = kwant.lattice.general(lat_vecs, atom_coords, norbs=orbs)
# Store sublattices by name
sublattices = {
atom: sublat
for atom, sublat in zip(atoms, lat.sublattices)
}
sym = kwant.TranslationalSymmetry(*lat_vecs)
bulk = kwant.Builder(sym)
bulk[[sublattices['A'](0, 0), sublattices['B'](0, 0)]] = 0
def hop(site1, site2, c0):
return c0
bulk[lat.neighbors()] = hop
fsyst_family = kwant.wraparound.wraparound(syst_from_family).finalized()
fsyst_model = kwant.wraparound.wraparound(syst_from_model).finalized()
fsyst_kwant = kwant.wraparound.wraparound(bulk).finalized()
# Check that the energies are identical at random points in the Brillouin zone
coeff = 0.5 + np.random.rand()
for _ in range(20):
kx, ky = 3 * np.pi * (np.random.rand(2) - 0.5)
params = dict(c0=coeff, k_x=kx, k_y=ky)
hamiltonian1 = fsyst_kwant.hamiltonian_submatrix(params=params,
sparse=False)
hamiltonian2 = fsyst_family.hamiltonian_submatrix(params=params,
sparse=False)
assert allclose(hamiltonian1, hamiltonian2)
params = dict(g=coeff, k_x=kx, k_y=ky)
hamiltonian3 = fsyst_model.hamiltonian_submatrix(params=params,
sparse=False)
assert allclose(hamiltonian2, hamiltonian3)
# Include random onsites as well
one = sympy.numbers.One()
onsites = [
Model({one: np.array([[1, 0], [0, 0]])}, momenta=family[0].momenta),
Model({one: np.array([[0, 0], [0, 1]])}, momenta=family[0].momenta)
]
family = family + onsites
syst_from_family = model_to_builder(family,
norbs,
lat_vecs,
atom_coords,
coeffs=None)
gs = list(sympy.symbols('g0:%d' % 3, real=True))
ham = hamiltonian_from_family(family, coeffs=gs)
ham = Model(hamiltonian=ham, momenta=family[0].momenta)
syst_from_model = model_to_builder(ham, norbs, lat_vecs, atom_coords)
def onsite_A(site, c1):
return c1
def onsite_B(site, c2):
return c2
bulk[[sublattices['A'](0, 0)]] = onsite_A
bulk[[sublattices['B'](0, 0)]] = onsite_B
fsyst_family = kwant.wraparound.wraparound(syst_from_family).finalized()
fsyst_model = kwant.wraparound.wraparound(syst_from_model).finalized()
fsyst_kwant = kwant.wraparound.wraparound(bulk).finalized()
# Check equivalence of the Hamiltonian at random points in the BZ
coeffs = 0.5 + np.random.rand(3)
for _ in range(20):
kx, ky = 3 * np.pi * (np.random.rand(2) - 0.5)
params = dict(c0=coeffs[0], c1=coeffs[1], c2=coeffs[2], k_x=kx, k_y=ky)
hamiltonian1 = fsyst_kwant.hamiltonian_submatrix(params=params,
sparse=False)
hamiltonian2 = fsyst_family.hamiltonian_submatrix(params=params,
sparse=False)
assert allclose(hamiltonian1, hamiltonian2)
params = dict(g0=coeffs[0], g1=coeffs[1], g2=coeffs[2], k_x=kx, k_y=ky)
hamiltonian3 = fsyst_model.hamiltonian_submatrix(params=params,
sparse=False)
assert allclose(hamiltonian2, hamiltonian3)
def model_to_builder(model, norbs, lat_vecs, atom_coords, *, coeffs=None):
"""Make a `~kwant.builder.Builder` out of qsymm.Models or qsymm.BlochModels.
Parameters
----------
model : qsymm.Model, qsymm.BlochModel, or an iterable thereof
The Hamiltonian (or terms of the Hamiltonian) to convert to a
Builder.
norbs : OrderedDict or sequence of pairs
Maps sites to the number of orbitals per site in a unit cell.
lat_vecs : list of arrays
Lattice vectors of the underlying tight binding lattice.
atom_coords : list of arrays
Positions of the sites (or atoms) within a unit cell.
The ordering of the atoms is the same as in norbs.
coeffs : list of sympy.Symbol, default None.
Constant prefactors for the individual terms in model, if model
is a list of multiple objects. If model is a single Model or BlochModel
object, this argument is ignored. By default assigns the coefficient
c_n to element model[n].
Returns
-------
syst : `~kwant.builder.Builder`
The unfinalized Kwant system representing the qsymm Model(s).
Notes
-----
Onsite terms that are not provided in the input model are set
to zero by default.
The input model(s) representing the tight binding Hamiltonian in
Bloch form should follow the convention where the difference in the real
space atomic positions appear in the Bloch factors.
"""
def make_int(R):
# If close to an integer array convert to integer tinyarray, else
# return None
R_int = ta.array(np.round(R), int)
if qsymm.linalg.allclose(R, R_int):
return R_int
else:
return None
def term_onsite(onsites_dict, hopping_dict, hop_mat, atoms,
sublattices, coords_dict):
"""Find the Kwant onsites and hoppings in a qsymm.BlochModel term
that has no lattice translation in the Bloch factor.
"""
for atom1, atom2 in it.product(atoms, atoms):
# Subblock within the same sublattice is onsite
hop = hop_mat[ranges[atom1], ranges[atom2]]
if sublattices[atom1] == sublattices[atom2]:
onsites_dict[atom1] += Model({coeff: hop}, momenta=momenta)
# Blocks between sublattices are hoppings between sublattices
# at the same position.
# Only include nonzero hoppings
elif not allclose(hop, 0):
if not allclose(np.array(coords_dict[atom1]),
np.array(coords_dict[atom2])):
raise ValueError(
"Position of sites not compatible with qsymm model.")
lat_basis = np.array(zer)
hop = Model({coeff: hop}, momenta=momenta)
hop_dir = builder.HoppingKind(-lat_basis, sublattices[atom1],
sublattices[atom2])
hopping_dict[hop_dir] += hop
return onsites_dict, hopping_dict
def term_hopping(hopping_dict, hop_mat, atoms,
sublattices, coords_dict):
"""Find Kwant hoppings in a qsymm.BlochModel term that has a lattice
translation in the Bloch factor.
"""
# Iterate over combinations of atoms, set hoppings between each
for atom1, atom2 in it.product(atoms, atoms):
# Take the block from atom1 to atom2
hop = hop_mat[ranges[atom1], ranges[atom2]]
# Only include nonzero hoppings
if allclose(hop, 0):
continue
# Adjust hopping vector to Bloch form basis
r_lattice = (
r_vec
+ np.array(coords_dict[atom1])
- np.array(coords_dict[atom2])
)
# Bring vector to basis of lattice vectors
lat_basis = np.linalg.solve(np.vstack(lat_vecs).T, r_lattice)
lat_basis = make_int(lat_basis)
# Should only have hoppings that are integer multiples of
# lattice vectors
if lat_basis is not None:
hop_dir = builder.HoppingKind(-lat_basis,
sublattices[atom1],
sublattices[atom2])
# Set the hopping as the matrix times the hopping amplitude
hopping_dict[hop_dir] += Model({coeff: hop}, momenta=momenta)
else:
raise RuntimeError('A nonzero hopping not matching a '
'lattice vector was found.')
return hopping_dict
# Disambiguate single model instances from iterables thereof. Because
# Model is itself iterable (subclasses dict) this is a bit cumbersome.
if isinstance(model, Model):
# BlochModel can't yet handle getting a Blochmodel as input
if not isinstance(model, BlochModel):
model = BlochModel(model)
else:
model = BlochModel(hamiltonian_from_family(
model, coeffs=coeffs, nsimplify=False, tosympy=False))
# 'momentum' and 'zer' are used in the closures defined above, so don't
# move these declarations down.
momenta = model.momenta
if len(momenta) != len(lat_vecs):
raise ValueError("Dimension of the lattice and number of "
"momenta do not match.")
zer = [0] * len(momenta)
# Subblocks of the Hamiltonian for different atoms.
N = 0
if not any([isinstance(norbs, OrderedDict), isinstance(norbs, list),
isinstance(norbs, tuple)]):
raise ValueError('norbs must be OrderedDict, tuple, or list.')
else:
norbs = OrderedDict(norbs)
ranges = dict()
for a, n in norbs.items():
ranges[a] = slice(N, N + n)
N += n
# Extract atoms and number of orbitals per atom,
# store the position of each atom
atoms, orbs = zip(*norbs.items())
coords_dict = dict(zip(atoms, atom_coords))
# Make the kwant lattice
lat = lattice.general(lat_vecs, atom_coords, norbs=orbs)
# Store sublattices by name
sublattices = dict(zip(atoms, lat.sublattices))
# Keep track of the hoppings and onsites by storing those
# which have already been set.
hopping_dict = defaultdict(dict)
onsites_dict = defaultdict(dict)
# Iterate over all terms in the model.
for key, hop_mat in model.items():
# Determine whether this term is an onsite or a hopping, extract
# overall symbolic coefficient if any, extract the exponential
# part describing the hopping if present.
r_vec, coeff = key
# Onsite term; modifies onsites_dict and hopping_dict in-place
if allclose(r_vec, 0):
term_onsite(
onsites_dict, hopping_dict, hop_mat,
atoms, sublattices, coords_dict)
# Hopping term; modifies hopping_dict in-place
else:
term_hopping(hopping_dict, hop_mat, atoms,
sublattices, coords_dict)
# If some onsite terms are not set, we set them to zero.
for atom in atoms:
if atom not in onsites_dict:
onsites_dict[atom] = Model(
{sympy.numbers.One(): np.zeros((norbs[atom], norbs[atom]))},
momenta=momenta)
# Make the Kwant system, and set all onsites and hoppings.
sym = lattice.TranslationalSymmetry(*lat_vecs)
syst = builder.Builder(sym)
# Iterate over all onsites and set them
for atom, onsite in onsites_dict.items():
syst[sublattices[atom](*zer)] = onsite.lambdify(onsite=True)
# Finally, iterate over all the hoppings and set them
for direction, hopping in hopping_dict.items():
syst[direction] = hopping.lambdify(hopping=True)
return syst