In [1]:
import sys

sys.path.append("../../code")
from init_mooc_nb import *

init_notebook()
import scipy

%output size=150 fig='png'
pi_ticks = [(-np.pi, r"$-\pi$"), (0, "0"), (np.pi, r"$\pi$")]

def haldane(w=20, boundary="zigzag"):
def ribbon_shape_zigzag(pos):
return -0.5 / np.sqrt(3) - 0.1 <= pos[1] < np.sqrt(3) * w / 2 + 0.01

def ribbon_shape_armchair(pos):
return -1 <= pos[0] < w

def onsite(site, p):
if site.family == a:
return p.m
else:
return -p.m

def nn_hopping(site1, site2, p):
return p.t

def nnn_hopping(site1, site2, p):
return 1j * p.t_2

lat = kwant.lattice.honeycomb()
a, b = lat.sublattices
nnn_hoppings_a = (((-1, 0), a, a), ((0, 1), a, a), ((1, -1), a, a))
nnn_hoppings_b = (((1, 0), b, b), ((0, -1), b, b), ((-1, 1), b, b))

if boundary == "zigzag":
syst = kwant.Builder(kwant.TranslationalSymmetry((1, 0)))
syst[lat.shape(ribbon_shape_zigzag, (0, 0))] = onsite
elif boundary == "armchair":
syst = kwant.Builder(kwant.TranslationalSymmetry((0, np.sqrt(3))))
syst[lat.shape(ribbon_shape_armchair, (0, 0))] = onsite
else:
syst = kwant.Builder(kwant.TranslationalSymmetry(*lat.prim_vecs))
syst[lat.shape(lambda pos: True, (0, 0))] = onsite

syst[lat.neighbors()] = nn_hopping
syst[
[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]
] = nnn_hopping
syst[
[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]
] = nnn_hopping

return syst

def Qi_Wu_Zhang():
def onsite(site, p):
return -p.mu * pauli.sz

def hopx(site1, site2, p):
return -0.5j * p.delta * pauli.sy - p.t * pauli.sz

def hopy(site1, site2, p):
return -1j * p.gamma * pauli.sx - p.gamma * pauli.sz

lat = kwant.lattice.square()

syst = kwant.Builder(kwant.TranslationalSymmetry(*lat.prim_vecs))
syst[lat.shape(lambda pos: True, (0, 0))] = onsite
syst[kwant.HoppingKind((1, 0), lat)] = hopx
syst[kwant.HoppingKind((0, 1), lat)] = hopy
return syst

def berry_curvature(syst, p, ks, num_filled_bands=1):
"""Berry curvature of a system.

Parameters:
-----------
sys : kwant.Builder
A 2D infinite system.
p : SimpleNamespace
The arguments expected by the system.
ks : 1D array-like
Values of momentum grid to be used for Berry curvature calculation.
num_filled_bands : int
The number of filled bands.

Returns:
--------
bc : 2D array
Berry curvature on each square in a ks x ks grid.
"""
# Calculate an array of eigenvectors.
B = np.array(syst.symmetry.periods).T
A = B @ np.linalg.inv(B.T @ B)

syst = kwant.wraparound.wraparound(syst).finalized()

def energy(kx, ky):
k = np.array([kx, ky])
kx, ky = np.linalg.solve(A, k)
H = syst.hamiltonian_submatrix(params=dict(p=p, k_x=kx, k_y=ky), sparse=False)
return scipy.linalg.eigh(H)[1]

vectors = np.array(
[[energy(kx, ky)[:, :num_filled_bands] for kx in ks] for ky in ks]
)

# The actual Berry curvature calculation
vectors_x = np.roll(vectors, 1, 0)
vectors_xy = np.roll(vectors_x, 1, 1)
vectors_y = np.roll(vectors, 1, 1)

shifted_vecs = [vectors, vectors_x, vectors_xy, vectors_y]

v_shape = vectors.shape

shifted_vecs = [i.reshape(-1, v_shape[-2], v_shape[-1]) for i in shifted_vecs]

dets = np.ones(len(shifted_vecs[0]), dtype=complex)
for vec, shifted in zip(shifted_vecs, np.roll(shifted_vecs, 1, 0)):
dets *= [np.linalg.det(a.T.conj() @ b) for a, b in zip(vec, shifted)]
bc = np.angle(dets).reshape(int(np.sqrt(len(dets))), -1)

bc = (bc + np.pi / 2) % (np.pi) - np.pi / 2

return bc

def plot_berry_curvature(syst, p, ks=None, title=None):
if ks is None:
ks = np.linspace(-np.pi, np.pi, 150, endpoint=False)
bc = berry_curvature(syst, p, ks)[1:-1, 1:-1]
vmax = max(np.abs(bc).min(), np.abs(bc).max())
kwargs = {
"bounds": (ks.min(), ks.min(), ks.max(), ks.max()),
"kdims": [r"$k_x$", r"$k_y$"],
}

if callable(title):
kwargs["label"] = title(p)

plot = {"xticks": pi_ticks, "yticks": pi_ticks}
style = {"clims": [-vmax, vmax]}
return holoviews.Image(bc, **kwargs).opts(plot=plot, style=style)

def title(p):
title = r"$t={:.2}$, $t_2={:.2}$, $M={:.2}$"
return title.format(p.t, p.t_2, p.m)

Populated the namespace with:
np, matplotlib, kwant, holoviews, init_notebook, SimpleNamespace, pprint_matrix, scientific_number, pretty_fmt_complex, plt, pf, display_html
from code/edx_components:
MoocVideo, MoocDiscussion, MoocCheckboxesAssessment, MoocMultipleChoiceAssessment, MoocSelfAssessment
from code/functions:
spectrum, hamiltonian_array, h_k, pauli
Using kwant 1.4.2 and holoviews 1.13.2
Executed on 2021-03-06 at 14:56:38.781078.