@@ -1,341 +1,367 @@
# Tutorial 2.8. Calculating spectral density with the Kernel Polynomial Method
# ============================================================================
#
# Physics background
# ------------------
# - Chebyshev polynomials, random trace approximation, spectral densities.
#
# Kwant features highlighted
# --------------------------
# - kpm module,kwant operators.
import scipy
+import _defs
+from contextlib import redirect_stdout
+
# For plotting
from matplotlib import pyplot as plt
#HIDDEN_BEGIN_sys1
# necessary imports
import kwant
import numpy as np
# define the system
def make_syst(r=30, t=-1, a=1):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1)
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.shape(circle, (0, 0))] = 0.
syst[lat.neighbors()] = t
syst.eradicate_dangling()
return syst
#HIDDEN_END_sys1
#HIDDEN_BEGIN_sys2
# define a Haldane system
def make_syst_topo(r=30, a=1, t=1, t2=0.5):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1, name=['a', 'b'])
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.shape(circle, (0, 0))] = 0.
syst[lat.neighbors()] = t
# add second neighbours hoppings
syst[lat.a.neighbors()] = 1j * t2
syst[lat.b.neighbors()] = -1j * t2
syst.eradicate_dangling()
return lat, syst.finalized()
#HIDDEN_END_sys2
#HIDDEN_BEGIN_sys3
# define the system
def make_syst_staggered(r=30, t=-1, a=1, m=0.1):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1)
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.a.shape(circle, (0, 0))] = m
syst[lat.b.shape(circle, (0, 0))] = -m
syst[lat.neighbors()] = t
syst.eradicate_dangling()
return syst
#HIDDEN_END_sys3
# Plot several density of states curves on the same axes.
-def plot_dos(labels_to_data):
+def plot_dos(labels_to_data, file_name=None, ylabel="DoS [a.u.]"):
plt.figure(figsize=(5,4))
for label, (x, y) in labels_to_data:
plt.plot(x, y.real, label=label, linewidth=2)
plt.legend(loc=2, framealpha=0.5)
plt.xlabel("energy [t]")
plt.ylabel(ylabel)
- plt.show()
+ save_figure(file_name)
plt.clf()
# Plot fill density of states plus curves on the same axes.
-def plot_dos_and_curves(dos labels_to_data):
+def plot_dos_and_curves(dos, labels_to_data, file_name=None, ylabel="DoS [a.u.]"):
plt.figure(figsize=(5,4))
plt.fill_between(dos[0], dos[1], label="DoS [a.u.]",
alpha=0.5, color='gray')
for label, (x, y) in labels_to_data:
plt.plot(x, y, label=label, linewidth=2)
plt.legend(loc=2, framealpha=0.5)
plt.xlabel("energy [t]")
plt.ylabel(ylabel)
- plt.show()
+ save_figure(file_name)
plt.clf()
def site_size_conversion(densities):
return 3 * np.abs(densities) / max(densities)
# Plot several local density of states maps in different subplots
def plot_ldos(syst, densities, file_name=None):
fig, axes = plt.subplots(1, len(densities), figsize=(7*len(densities), 7))
for ax, (title, rho) in zip(axes, densities):
kwant.plotter.density(syst, rho.real, ax=ax)
ax.set_title(title)
ax.set(adjustable='box', aspect='equal')
- plt.show()
+ save_figure(file_name)
plt.clf()
+def save_figure(file_name):
+ if not file_name:
+ return
+ for extension in ('pdf', 'png'):
+ plt.savefig('.'.join((file_name,extension)),
+ dpi=_defs.dpi, bbox_inches='tight')
+
+
def simple_dos_example():
#HIDDEN_BEGIN_kpm1
fsyst = make_syst().finalized()
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
#HIDDEN_END_kpm1
#HIDDEN_BEGIN_kpm2
energies, densities = spectrum()
#HIDDEN_END_kpm2
#HIDDEN_BEGIN_kpm3
energy_subset = np.linspace(0, 2)
density_subset = spectrum(energy_subset)
#HIDDEN_END_kpm3
plot_dos([
('densities', (energies, densities)),
('density subset', (energy_subset, density_subset)),
- ])
+ ],
+ file_name='kpm_dos'
+ )
def dos_integrating_example(fsyst):
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
#HIDDEN_BEGIN_int1
- print('identity resolution:', spectrum.integrate())
+ with open('kpm_normalization.txt', 'w') as f:
+ with redirect_stdout(f):
+ print('identity resolution:', spectrum.integrate())
#HIDDEN_END_int1
#HIDDEN_BEGIN_int2
# Fermi energy 0.1 and temperature 0.2
fermi = lambda E: 1 / (np.exp((E - 0.1) / 0.2) + 1)
- print('number of filled states:', spectrum.integrate(fermi))
+ with open('kpm_total_states.txt', 'w') as f:
+ with redirect_stdout(f):
+ print('number of filled states:', spectrum.integrate(fermi))
#HIDDEN_END_int2
def increasing_accuracy_example(fsyst):
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
original_dos = spectrum() # get unaltered DoS
#HIDDEN_BEGIN_acc1
spectrum.add_moments(energy_resolution=0.03)
#HIDDEN_END_acc1
increased_resolution_dos = spectrum()
plot_dos([
('density', original_dos),
('higher energy resolution', increased_resolution_dos),
- ])
+ ],
+ file_name='kpm_dos_acc'
+ )
#HIDDEN_BEGIN_acc2
spectrum.add_moments(100)
spectrum.add_vectors(5)
#HIDDEN_END_acc2
increased_moments_dos = spectrum()
plot_dos([
('density', original_dos),
('higher number of moments', increased_moments_dos),
- ])
+ ],
+ file_name='kpm_dos_r'
+ )
def operator_example(fsyst):
#HIDDEN_BEGIN_op1
# identity matrix
matrix_op = scipy.sparse.eye(len(fsyst.sites))
matrix_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=matrix_op, rng=0)
#HIDDEN_END_op1
#HIDDEN_BEGIN_op2
# 'sum=True' means we sum over all the sites
kwant_op = kwant.operator.Density(fsyst, sum=True)
operator_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=kwant_op, rng=0)
#HIDDEN_END_op2
plot_dos([
('identity matrix', matrix_spectrum()),
('kwant.operator.Density', operator_spectrum()),
])
def ldos_example(fsyst):
#HIDDEN_BEGIN_op3
# 'sum=False' is the default, but we include it explicitly here for clarity.
kwant_op = kwant.operator.Density(fsyst, sum=False)
local_dos = kwant.kpm.SpectralDensity(fsyst, operator=kwant_op, rng=0)
#HIDDEN_END_op3
#HIDDEN_BEGIN_op4
zero_energy_ldos = local_dos(energy=0)
finite_energy_ldos = local_dos(energy=1)
plot_ldos(fsyst, [
('energy = 0', zero_energy_ldos),
('energy = 1', finite_energy_ldos)
- ])
+ ],
+ file_name='kpm_ldos'
+ )
#HIDDEN_END_op4
def ldos_sites_example():
fsyst = make_syst_staggered().finalized()
#HIDDEN_BEGIN_op5
# find 'A' and 'B' sites in the unit cell at the center of the disk
center_tag = np.array([0, 0])
where = lambda s: s.tag == center_tag
# make local vectors
vector_factory = kwant.kpm.LocalVectors(fsyst, where)
#HIDDEN_END_op5
#HIDDEN_BEGIN_op6
# 'num_vectors' can be unspecified when using 'LocalVectors'
local_dos = kwant.kpm.SpectralDensity(fsyst, num_vectors=None,
vector_factory=vector_factory,
mean=False,
rng=0)
energies, densities = local_dos()
plot_dos([
('A sublattice', (energies, densities[:, 0])),
('B sublattice', (energies, densities[:, 1])),
- ])
+ ],
+ file_name='kpm_ldos_sites'
+ )
#HIDDEN_END_op6
def vector_factory_example(fsyst):
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
#HIDDEN_BEGIN_fact1
# construct a generator of vectors with n random elements -1 or +1.
n = fsyst.hamiltonian_submatrix(sparse=True).shape[0]
def binary_vectors():
while True:
yield np.rint(np.random.random_sample(n)) * 2 - 1
custom_factory = kwant.kpm.SpectralDensity(fsyst,
vector_factory=binary_vectors(),
rng=0)
#HIDDEN_END_fact1
plot_dos([
('default vector factory', spectrum()),
('binary vector factory', custom_factory()),
])
def bilinear_map_operator_example(fsyst):
#HIDDEN_BEGIN_blm
rho = kwant.operator.Density(fsyst, sum=True)
# sesquilinear map that does the same thing as `rho`
def rho_alt(bra, ket):
return np.vdot(bra, ket)
rho_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=rho, rng=0)
rho_alt_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=rho_alt, rng=0)
#HIDDEN_END_blm
plot_dos([
('kwant.operator.Density', rho_spectrum()),
('bilinear operator', rho_alt_spectrum()),
])
def conductivity_example():
#HIDDEN_BEGIN_cond
# construct the Haldane model
lat, fsyst = make_syst_topo()
# find 'A' and 'B' sites in the unit cell at the center of the disk
where = lambda s: np.linalg.norm(s.pos) < 1
# component 'xx'
s_factory = kwant.kpm.LocalVectors(fsyst, where)
cond_xx = kwant.kpm.conductivity(fsyst, alpha='x', beta='x', mean=True,
num_vectors=None, vector_factory=s_factory,
rng=0)
# component 'xy'
s_factory = kwant.kpm.LocalVectors(fsyst, where)
cond_xy = kwant.kpm.conductivity(fsyst, alpha='x', beta='y', mean=True,
num_vectors=None, vector_factory=s_factory,
rng=0)
energies = cond_xx.energies
cond_array_xx = np.array([cond_xx(e, temperature=0.01) for e in energies])
cond_array_xy = np.array([cond_xy(e, temperature=0.01) for e in energies])
# area of the unit cell per site
area_per_site = np.abs(np.cross(*lat.prim_vecs)) / len(lat.sublattices)
cond_array_xx /= area_per_site
cond_array_xy /= area_per_site
#HIDDEN_END_cond
# ldos
s_factory = kwant.kpm.LocalVectors(fsyst, where)
spectrum = kwant.kpm.SpectralDensity(fsyst, num_vectors=None,
vector_factory=s_factory,
rng=0)
plot_dos_and_curves(
(spectrum.energies, spectrum.densities * 8),
[
(r'Longitudinal conductivity $\sigma_{xx} / 4$',
(energies, cond_array_xx / 4)),
(r'Hall conductivity $\sigma_{xy}$',
(energies, cond_array_xy))],
ylabel=r'$\sigma [e^2/h]$',
file_name='kpm_cond'
)
def main():
simple_dos_example()
fsyst = make_syst().finalized()
dos_integrating_example(fsyst)
increasing_accuracy_example(fsyst)
operator_example(fsyst)
ldos_example(fsyst)
ldos_sites_example()
vector_factory_example(fsyst)
bilinear_map_operator_example(fsyst)
conductivity_example()
# Call the main function if the script gets executed (as opposed to imported).
# See <http://docs.python.org/library/__main__.html>.
if __name__ == '__main__':
main()