def TightBinding(kpoints, model="Model1", return_vectors=True, **model_params):
actual_model_params = dict(DEFAULT_MODEL_PARAMETERS)
actual_model_params.update(model_params)
t0 = actual_model_params["t0"]
t1 = actual_model_params["t1"]
cell = Lattice(points=POINTS, vectors=VECTORS)
ids = [[0, 0], [1, -1], [0, -1], [-1, 0], [-1, 1], [0, 1], [1, 0]]
points_collection = np.concatenate(
[np.dot([i, j], VECTORS) + POINTS for i, j in ids])
inter_hopping_indices = [
[5, 83],
[5, 82],
[6, 82],
[6, 81],
[6, 63],
[7, 63],
[7, 71],
[8, 71],
[8, 53],
[8, 52],
[9, 52],
[9, 51],
]
if model == "Model1":
intra_hopping_indices0 = [[0, 4], [4, 5], [1, 7], [7, 8], [2, 10],
[10, 11]]
intra_hopping_indices1 = [
[0, 1],
[1, 2],
[2, 0],
[0, 3],
[0, 10],
[0, 11],
[1, 4],
[1, 5],
[1, 6],
[2, 7],
[2, 8],
[2, 9],
[3, 4],
[3, 11],
[6, 5],
[6, 7],
[9, 8],
[9, 10],
]
else:
intra_hopping_indices0 = [
[0, 4],
[4, 5],
[5, 6],
[1, 7],
[7, 8],
[8, 9],
[2, 10],
[10, 11],
[11, 3],
]
intra_hopping_indices1 = [
[0, 1],
[1, 2],
[2, 0],
[0, 3],
[0, 10],
[0, 11],
[1, 4],
[1, 5],
[1, 6],
[2, 7],
[2, 8],
[2, 9],
[3, 4],
[6, 7],
[9, 10],
]
terms = []
for coeff, hopping_indices in [
(t0, intra_hopping_indices0),
(t1, intra_hopping_indices1),
(t1, inter_hopping_indices),
]:
for ij in hopping_indices:
p0, p1 = points_collection[ij]
p0_eqv, dR0 = cell.decompose(p0)
p1_eqv, dR1 = cell.decompose(p1)
index0 = cell.getIndex(p0_eqv, fold=False)
index1 = cell.getIndex(p1_eqv, fold=False)
terms.append((index0, index1, coeff, dR0 - dR1))
site_num = cell.point_num
kpoint_num = kpoints.shape[0]
HMs = np.zeros((kpoint_num, site_num, site_num), dtype=np.complex128)
for i, j, coeff, dR in terms:
HMs[:, i, j] += coeff * np.exp(1j * np.matmul(kpoints, dR))
HMs += np.transpose(HMs, (0, 2, 1)).conj()
if return_vectors:
return np.linalg.eigh(HMs)
else:
return np.linalg.eigvalsh(HMs)
dtype=np.float64)
VECTORS = np.array([[4.0, 0.0], [2.0, 2 * SQRT3]], dtype=np.float64)
t0 = 0.0
t1 = -1.00
t2 = -0.25
id = "t0={0:.2f}_t1={1:.2f}_t2={2:.2f}".format(t0, t1, t2)
cluster = Lattice(POINTS, VECTORS)
intra_bonds_1st, inter_bonds_1st = cluster.bonds(nth=1)
intra_bonds_2nd, inter_bonds_2nd = cluster.bonds(nth=2)
intra_bonds_6th, inter_bonds_6th = cluster.bonds(nth=6)
HTerms = []
for bond in intra_bonds_1st + inter_bonds_1st:
p0, p1 = bond.endpoints
p0_eqv, dR0 = cluster.decompose(p0)
p1_eqv, dR1 = cluster.decompose(p1)
index0 = cluster.getIndex(p0_eqv)
index1 = cluster.getIndex(p1_eqv)
HTerms.append((index0, index1, t0, dR0 - dR1))
for bond in intra_bonds_2nd + inter_bonds_2nd:
p0, p1 = bond.endpoints
p0_eqv, dR0 = cluster.decompose(p0)
p1_eqv, dR1 = cluster.decompose(p1)
index0 = cluster.getIndex(p0_eqv)
index1 = cluster.getIndex(p1_eqv)
HTerms.append((index0, index1, t1, dR0 - dR1))
for bond in intra_bonds_6th + inter_bonds_6th:
p0, p1 = bond.endpoints
def TightBinding(kpoints, model="Model1", return_vectors=True, **model_params):
actual_model_params = dict(DEFAULT_MODEL_PARAMETERS)
actual_model_params.update(model_params)
t0 = actual_model_params["t0"]
t1 = actual_model_params["t1"]
mu0 = actual_model_params["mu0"] / 2
mu1 = actual_model_params["mu1"] / 2
ids = ([0, 0], [-1, 0], [1, 0], [0, -1], [0, 1])
cell = Lattice(ALL_POINTS[0:20], TRANSLATION_VECTORS)
points_collection = np.concatenate(
[np.dot(ij, TRANSLATION_VECTORS) + ALL_POINTS[0:20] for ij in ids]
)
intra_hopping_indices0 = [
[0, 1], [0, 2], [0, 3], [0, 4],
[0, 5], [0, 6], [0, 7], [0, 8], [0, 9],
[10, 11], [10, 12], [10, 13], [10, 14],
[10, 15], [10, 16], [10, 17], [10, 18], [10, 19],
]
intra_hopping_indices1 = [
[1, 2], [2, 3], [3, 4], [4, 5],
[5, 6], [6, 7], [7, 8], [8, 9], [9, 1],
[11, 12], [12, 13], [13, 14], [14, 15],
[15, 16], [16, 17], [17, 18], [18, 19], [19, 11],
[6, 11]
]
inter_hopping_indices = [[3, 37], [9, 74]]
if model == "Model2":
intra_hopping_indices1 += [[6, 12], [6, 19], [5, 11], [7, 11]]
inter_hopping_indices += [
[3, 36], [3, 38], [2, 37], [4, 37],
[9, 73], [9, 75], [1, 74], [8, 74],
]
terms = []
zero_dr = np.array([0.0, 0.0], dtype=np.float64)
for point in cell.points:
index = cell.getIndex(point, fold=False)
mu = mu0 if index in (0, 10) else mu1
terms.append((2 * index, 2 * index, mu, zero_dr))
terms.append((2 * index + 1, 2 * index + 1, mu, zero_dr))
for t, ijs in [(t0, intra_hopping_indices0), (t1, intra_hopping_indices1)]:
for ij in ijs:
p0, p1 = points_collection[ij]
coeff = t / np.dot(p1 - p0, p1 - p0)
index0 = cell.getIndex(p0, fold=False)
index1 = cell.getIndex(p1, fold=False)
terms.append((2 * index0, 2 * index1, coeff, zero_dr))
terms.append((2 * index0 + 1, 2 * index1 + 1, coeff, zero_dr))
for ij in inter_hopping_indices:
p0, p1 = points_collection[ij]
coeff = t1 / np.dot(p1 - p0, p1 - p0)
p0_eqv, dR0 = cell.decompose(p0)
p1_eqv, dR1 = cell.decompose(p1)
index0 = cell.getIndex(p0_eqv, fold=False)
index1 = cell.getIndex(p1_eqv, fold=False)
terms.append((2 * index0, 2 * index1, coeff, dR1 - dR0))
terms.append((2 * index0 + 1, 2 * index1 + 1, coeff, dR1 - dR0))
# msg = "({0:2d},{1:2d}), t={2:.8f}, dR=({3:.8f}, {4:.8f})"
# print("Hamiltonian Terms:")
# for i, j, coeff, dR in terms:
# print(msg.format(i, j, coeff, dR[0], dR[1]))
point_num = cell.point_num
shape = (kpoints.shape[0], 2 * point_num, 2 * point_num)
HMs = np.zeros(shape, dtype=np.complex128)
for i, j, coeff, dR in terms:
HMs[:, i, j] += coeff * np.exp(1j * np.matmul(kpoints, dR))
HMs += np.transpose(HMs, (0, 2, 1)).conj()
if return_vectors:
return np.linalg.eigh(HMs)
else:
return np.linalg.eigvalsh(HMs)