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Commit 6d0e0aab authored by Joseph Weston's avatar Joseph Weston
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remove old tutorial script machinery

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doc/source/code/README deleted 100644 → 0
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This directory contains the code examples from the documentation.
Most scripts are present in three related but different versions that
correspond to three different usages.
* Subdirectory 'figure': scripts used for figure generation. Figures
are not displayed but saved to disk.
* Subdirectory 'include': scripts that display figures on screen.
They contain commented marks for including snippets in the
documentation.
* Subdirectory 'download': complete scripts to be offered for download
by readers. Like 'include' but with the include marks removed.
Most scripts are extracted from corresponding '*.py.diff' files inside
'figure/'. These are patches from the 'include' version to the
'figure' version. The patches include complete context and as such
can be used to recreate both files. It's these patches that are kept
under version control.
running 'make html' or 'make latex' inside '/doc' will automatically
update all these scripts according to the following scheme:
---->------------->------
/ \
/ download/x.py \
figure/x.py.diff ^ \
^ \ | \
| -> include/x.py ---(patch)---> figure/x.py
| | |
| | |
\ v /
----<----------(diff)--------------<--------
Thus, it is possible to update figure/x.py.diff, include/x.py or
figure/x.py and any changes will be propagated automatically when
'make' is run. (Only download/x.py is a dead end.) The user will be
informed about any conflicts. The makefile will only update files
that are older than their sources and is careful to propagate time
stamps in order to avoid infinite loops.
Editing only figure/x.py.diff is a sure way to avoid any conflicts.
doc/source/code/figure/_defs.py deleted 100644 → 0
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################################################################
# Make matplotlib work without X11
################################################################
import matplotlib
matplotlib.use('Agg')
################################################################
# Prepend Kwant's build directory to sys.path
################################################################
import sys
from distutils.util import get_platform
sys.path.insert(0, "../../../../build/lib.{0}-{1}.{2}".format(
get_platform(), *sys.version_info[:2]))
################################################################
# Define constants for plotting
################################################################
pt_to_in = 1. / 72.
# Default width of figures in pts
figwidth_pt = 600
figwidth_in = figwidth_pt * pt_to_in
# Width for smaller figures
figwidth_small_pt = 400
figwidth_small_in = figwidth_small_pt * pt_to_in
# Sizes for matplotlib figures
mpl_width_in = figwidth_pt * pt_to_in
mpl_label_size = None # font sizes in points
mpl_tick_size = None
# dpi for conversion from inches
dpi = 90
doc/source/code/figure/ab_ring.py.diff deleted 100644 → 0
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@@ -1,127 +1,196 @@
# Tutorial 2.3.3. Nontrivial shapes
# =================================
#
# Physics background
# ------------------
# Flux-dependent transmission through a quantum ring
#
# Kwant features highlighted
# --------------------------
# - More complex shapes with lattices
# - Allows for discussion of subtleties of `attach_lead` (not in the
# example, but in the tutorial main text)
# - Modifcations of hoppings/sites after they have been added
+import _defs
from cmath import exp
from math import pi
import kwant
# For plotting
from matplotlib import pyplot
#HIDDEN_BEGIN_eusz
def make_system(a=1, t=1.0, W=10, r1=10, r2=20):
# Start with an empty tight-binding system and a single square lattice.
# `a` is the lattice constant (by default set to 1 for simplicity).
lat = kwant.lattice.square(a)
syst = kwant.Builder()
#### Define the scattering region. ####
# Now, we aim for a more complex shape, namely a ring (or annulus)
def ring(pos):
(x, y) = pos
rsq = x ** 2 + y ** 2
return (r1 ** 2 < rsq < r2 ** 2)
#HIDDEN_END_eusz
# and add the corresponding lattice points using the `shape`-function
#HIDDEN_BEGIN_lcak
syst[lat.shape(ring, (0, r1 + 1))] = 4 * t
syst[lat.neighbors()] = -t
#HIDDEN_END_lcak
# In order to introduce a flux through the ring, we introduce a phase on
# the hoppings on the line cut through one of the arms. Since we want to
# change the flux without modifying the Builder instance repeatedly, we
# define the modified hoppings as a function that takes the flux as its
# parameter phi.
#HIDDEN_BEGIN_lvkt
def hopping_phase(site1, site2, phi):
return -t * exp(1j * phi)
def crosses_branchcut(hop):
ix0, iy0 = hop[0].tag
# builder.HoppingKind with the argument (1, 0) below
# returns hoppings ordered as ((i+1, j), (i, j))
return iy0 < 0 and ix0 == 1 # ix1 == 0 then implied
# Modify only those hopings in x-direction that cross the branch cut
def hops_across_cut(syst):
for hop in kwant.builder.HoppingKind((1, 0), lat, lat)(syst):
if crosses_branchcut(hop):
yield hop
syst[hops_across_cut] = hopping_phase
#HIDDEN_END_lvkt
#### Define the leads. ####
# left lead
#HIDDEN_BEGIN_qwgr
sym_lead = kwant.TranslationalSymmetry((-a, 0))
lead = kwant.Builder(sym_lead)
def lead_shape(pos):
(x, y) = pos
return (-W / 2 < y < W / 2)
lead[lat.shape(lead_shape, (0, 0))] = 4 * t
lead[lat.neighbors()] = -t
#HIDDEN_END_qwgr
#### Attach the leads and return the system. ####
#HIDDEN_BEGIN_skbz
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
#HIDDEN_END_skbz
return syst
+def make_system_note1(a=1, t=1.0, W=10, r1=10, r2=20):
+ lat = kwant.lattice.square(a)
+ syst = kwant.Builder()
+ def ring(pos):
+ (x, y) = pos
+ rsq = x**2 + y**2
+ return ( r1**2 < rsq < r2**2)
+ syst[lat.shape(ring, (0, 11))] = 4 * t
+ syst[lat.neighbors()] = -t
+ sym_lead0 = kwant.TranslationalSymmetry((-a, 0))
+ lead0 = kwant.Builder(sym_lead0)
+ def lead_shape(pos):
+ (x, y) = pos
+ return (-1 < x < 1) and ( 0.5 * W < y < 1.5 * W )
+ lead0[lat.shape(lead_shape, (0, W))] = 4 * t
+ lead0[lat.neighbors()] = -t
+ lead1 = lead0.reversed()
+ syst.attach_lead(lead0)
+ syst.attach_lead(lead1)
+ return syst
+
+
+def make_system_note2(a=1, t=1.0, W=10, r1=10, r2=20):
+ lat = kwant.lattice.square(a)
+ syst = kwant.Builder()
+ def ring(pos):
+ (x, y) = pos
+ rsq = x**2 + y**2
+ return ( r1**2 < rsq < r2**2)
+ syst[lat.shape(ring, (0, 11))] = 4 * t
+ syst[lat.neighbors()] = -t
+ sym_lead0 = kwant.TranslationalSymmetry((-a, 0))
+ lead0 = kwant.Builder(sym_lead0)
+ def lead_shape(pos):
+ (x, y) = pos
+ return (-1 < x < 1) and ( -W/2 < y < W/2 )
+ lead0[lat.shape(lead_shape, (0, 0))] = 4 * t
+ lead0[lat.neighbors()] = -t
+ lead1 = lead0.reversed()
+ syst.attach_lead(lead0)
+ syst.attach_lead(lead1, lat(0, 0))
+ return syst
+
+
def plot_conductance(syst, energy, fluxes):
# compute conductance
normalized_fluxes = [flux / (2 * pi) for flux in fluxes]
data = []
for flux in fluxes:
smatrix = kwant.smatrix(syst, energy, params=dict(phi=flux))
data.append(smatrix.transmission(1, 0))
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(normalized_fluxes, data)
- pyplot.xlabel("flux [flux quantum]")
- pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.xlabel("flux [flux quantum]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("conductance [e^2/h]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ fig.savefig("ab_ring_result.pdf")
+ fig.savefig("ab_ring_result.png", dpi=_defs.dpi)
def main():
syst = make_system()
# Check that the system looks as intended.
- kwant.plot(syst)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="ab_ring_syst." + extension,
+ fig_size=size, dpi=_defs.dpi)
+
# Finalize the system.
syst = syst.finalized()
# We should see a conductance that is periodic with the flux quantum
plot_conductance(syst, energy=0.15, fluxes=[0.01 * i * 3 * 2 * pi
for i in range(100)])
+ # Finally, some plots needed for the notes
+ syst = make_system_note1()
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="ab_ring_note1." + extension,
+ fig_size=size, dpi=_defs.dpi)
+ syst = make_system_note2()
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="ab_ring_note2." + extension,
+ fig_size=size, dpi=_defs.dpi)
+
+
# 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()
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doc/source/code/figure/band_structure.py.diff deleted 100644 → 0
+ 0
− 67
View file @ f1929ee7
@@ -1,52 +1,62 @@
# Tutorial 2.4.1. Band structure calculations
# ===========================================
#
# Physics background
# ------------------
# band structure of a simple quantum wire in tight-binding approximation
#
# Kwant features highlighted
# --------------------------
# - Computing the band structure of a finalized lead.
+import _defs
import kwant
# For plotting.
from matplotlib import pyplot
#HIDDEN_BEGIN_zxip
def make_lead(a=1, t=1.0, W=10):
# Start with an empty lead with a single square lattice
lat = kwant.lattice.square(a)
sym_lead = kwant.TranslationalSymmetry((-a, 0))
lead = kwant.Builder(sym_lead)
# build up one unit cell of the lead, and add the hoppings
# to the next unit cell
for j in range(W):
lead[lat(0, j)] = 4 * t
if j > 0:
lead[lat(0, j), lat(0, j - 1)] = -t
lead[lat(1, j), lat(0, j)] = -t
return lead
#HIDDEN_END_zxip
#HIDDEN_BEGIN_pejz
def main():
lead = make_lead().finalized()
- kwant.plotter.bands(lead, show=False)
- pyplot.xlabel("momentum [(lattice constant)^-1]")
- pyplot.ylabel("energy [t]")
- pyplot.show()
+ fig = kwant.plotter.bands(lead, show=False)
+ pyplot.xlabel("momentum [(lattice constant)^-1]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("energy [t]", fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("band_structure_result." + extension, dpi=_defs.dpi)
+
#HIDDEN_END_pejz
# 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()
doc/source/code/figure/closed_system.py.diff deleted 100644 → 0
+ 0
− 168
View file @ f1929ee7
@@ -1,144 +1,161 @@
# Tutorial 2.4.2. Closed systems
# ==============================
#
# Physics background
# ------------------
# Fock-darwin spectrum of a quantum dot (energy spectrum in
# as a function of a magnetic field)
#
# Kwant features highlighted
# --------------------------
# - Use of `hamiltonian_submatrix` in order to obtain a Hamiltonian
# matrix.
+import _defs
from cmath import exp
import numpy as np
import kwant
# For eigenvalue computation
#HIDDEN_BEGIN_tibv
import scipy.sparse.linalg as sla
#HIDDEN_END_tibv
# For plotting
from matplotlib import pyplot
def make_system(a=1, t=1.0, r=10):
# Start with an empty tight-binding system and a single square lattice.
# `a` is the lattice constant (by default set to 1 for simplicity).
#HIDDEN_BEGIN_qlyd
lat = kwant.lattice.square(a, norbs=1)
syst = kwant.Builder()
# Define the quantum dot
def circle(pos):
(x, y) = pos
rsq = x ** 2 + y ** 2
return rsq < r ** 2
def hopx(site1, site2, B):
# The magnetic field is controlled by the parameter B
y = site1.pos[1]
return -t * exp(-1j * B * y)
syst[lat.shape(circle, (0, 0))] = 4 * t
# hoppings in x-direction
syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx
# hoppings in y-directions
syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t
# It's a closed system for a change, so no leads
return syst
#HIDDEN_END_qlyd
#HIDDEN_BEGIN_yvri
def plot_spectrum(syst, Bfields):
energies = []
for B in Bfields:
# Obtain the Hamiltonian as a sparse matrix
ham_mat = syst.hamiltonian_submatrix(params=dict(B=B), sparse=True)
# we only calculate the 15 lowest eigenvalues
ev = sla.eigsh(ham_mat.tocsc(), k=15, sigma=0,
return_eigenvectors=False)
energies.append(ev)
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(Bfields, energies)
- pyplot.xlabel("magnetic field [arbitrary units]")
- pyplot.ylabel("energy [t]")
- pyplot.show()
+ pyplot.xlabel("magnetic field [arbitrary units]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("energy [t]", fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("closed_system_result." + extension, dpi=_defs.dpi)
#HIDDEN_END_yvri
def sorted_eigs(ev):
evals, evecs = ev
evals, evecs = map(np.array, zip(*sorted(zip(evals, evecs.transpose()))))
return evals, evecs.transpose()
#HIDDEN_BEGIN_wave
def plot_wave_function(syst, B=0.001):
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+
# Calculate the wave functions in the system.
ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))
# Plot the probability density of the 10th eigenmode.
- kwant.plotter.map(syst, np.abs(evecs[:, 9])**2,
- colorbar=False, oversampling=1)
+ for extension in ('pdf', 'png'):
+ kwant.plotter.map(
+ syst, np.abs(evecs[:, 9])**2, colorbar=False, oversampling=1,
+ file="closed_system_eigenvector." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_wave
#HIDDEN_BEGIN_current
def plot_current(syst, B=0.001):
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+
# Calculate the wave functions in the system.
ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))
# Calculate and plot the local current of the 10th eigenmode.
J = kwant.operator.Current(syst)
current = J(evecs[:, 9], params=dict(B=B))
- kwant.plotter.current(syst, current, colorbar=False)
+ for extension in ('pdf', 'png'):
+ kwant.plotter.current(
+ syst, current, colorbar=False,
+ file="closed_system_current." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_current
def main():
syst = make_system()
- # Check that the system looks as intended.
- kwant.plot(syst)
-
# Finalize the system.
syst = syst.finalized()
# The following try-clause can be removed once SciPy 0.9 becomes uncommon.
try:
# We should observe energy levels that flow towards Landau
# level energies with increasing magnetic field.
plot_spectrum(syst, [iB * 0.002 for iB in range(100)])
# Plot an eigenmode of a circular dot. Here we create a larger system for
# better spatial resolution.
syst = make_system(r=30).finalized()
plot_wave_function(syst)
plot_current(syst)
except ValueError as e:
if e.message == "Input matrix is not real-valued.":
print("The calculation of eigenvalues failed because of a bug in SciPy 0.9.")
print("Please upgrade to a newer version of SciPy.")
else:
raise
# 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()
doc/source/code/figure/discretize.py.diff deleted 100644 → 0
+ 0
− 248
View file @ f1929ee7
@@ -1,225 +1,239 @@
# Tutorial 2.9. Processing continuum Hamiltonians with discretize
# ===============================================================
#
# Physics background
# ------------------
# - tight-binding approximation of continuous Hamiltonians
#
# Kwant features highlighted
# --------------------------
# - kwant.continuum.discretize
+import _defs
import kwant
#HIDDEN_BEGIN_import
import kwant.continuum
#HIDDEN_END_import
import scipy.sparse.linalg
import scipy.linalg
import numpy as np
# For plotting
import matplotlib as mpl
from matplotlib import pyplot as plt
+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 stadium_system(L=20, W=20):
#HIDDEN_BEGIN_template
hamiltonian = "k_x**2 + k_y**2 + V(x, y)"
template = kwant.continuum.discretize(hamiltonian)
- print(template)
+ with open('discretizer_verbose.txt', 'w') as f:
+ print(template, file=f)
#HIDDEN_END_template
#HIDDEN_BEGIN_fill
def stadium(site):
(x, y) = site.pos
x = max(abs(x) - 20, 0)
return x**2 + y**2 < 30**2
syst = kwant.Builder()
syst.fill(template, stadium, (0, 0));
syst = syst.finalized()
#HIDDEN_END_fill
return syst
#HIDDEN_BEGIN_plot_eigenstate
def plot_eigenstate(syst, n=2, Vx=.0003, Vy=.0005):
def potential(x, y):
return Vx * x + Vy * y
ham = syst.hamiltonian_submatrix(params=dict(V=potential), sparse=True)
evecs = scipy.sparse.linalg.eigsh(ham, k=10, which='SM')[1]
kwant.plotter.map(syst, abs(evecs[:, n])**2, show=False)
#HIDDEN_END_plot_eigenstate
- plt.show()
+ save_figure('discretizer_gs')
def qsh_system(a=20, L=2000, W=1000):
#HIDDEN_BEGIN_define_qsh
hamiltonian = """
+ C * identity(4) + M * kron(sigma_0, sigma_z)
- B * (k_x**2 + k_y**2) * kron(sigma_0, sigma_z)
- D * (k_x**2 + k_y**2) * kron(sigma_0, sigma_0)
+ A * k_x * kron(sigma_z, sigma_x)
- A * k_y * kron(sigma_0, sigma_y)
"""
template = kwant.continuum.discretize(hamiltonian, grid=a)
#HIDDEN_END_define_qsh
#HIDDEN_BEGIN_define_qsh_build
def shape(site):
(x, y) = site.pos
return (0 <= y < W and 0 <= x < L)
def lead_shape(site):
(x, y) = site.pos
return (0 <= y < W)
syst = kwant.Builder()
syst.fill(template, shape, (0, 0))
lead = kwant.Builder(kwant.TranslationalSymmetry([-a, 0]))
lead.fill(template, lead_shape, (0, 0))
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
syst = syst.finalized()
#HIDDEN_END_define_qsh_build
return syst
def analyze_qsh():
params = dict(A=3.65, B=-68.6, D=-51.1, M=-0.01, C=0)
syst = qsh_system()
#HIDDEN_BEGIN_plot_qsh_band
kwant.plotter.bands(syst.leads[0], params=params,
momenta=np.linspace(-0.3, 0.3, 201), show=False)
#HIDDEN_END_plot_qsh_band
plt.grid()
plt.xlim(-.3, 0.3)
plt.ylim(-0.05, 0.05)
plt.xlabel('momentum [1/A]')
plt.ylabel('energy [eV]')
- plt.show()
+ save_figure('discretizer_qsh_band')
#HIDDEN_BEGIN_plot_qsh_wf
+
# get scattering wave functions at E=0
wf = kwant.wave_function(syst, energy=0, params=params)
# prepare density operators
sigma_z = np.array([[1, 0], [0, -1]])
prob_density = kwant.operator.Density(syst, np.eye(4))
spin_density = kwant.operator.Density(syst, np.kron(sigma_z, np.eye(2)))
# calculate expectation values and plot them
wf_sqr = sum(prob_density(psi) for psi in wf(0)) # states from left lead
rho_sz = sum(spin_density(psi) for psi in wf(0)) # states from left lead
fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(16, 4))
kwant.plotter.map(syst, wf_sqr, ax=ax1)
kwant.plotter.map(syst, rho_sz, ax=ax2)
#HIDDEN_END_plot_qsh_wf
ax = ax1
im = [obj for obj in ax.get_children()
if isinstance(obj, mpl.image.AxesImage)][0]
fig.colorbar(im, ax=ax)
ax = ax2
im = [obj for obj in ax.get_children()
if isinstance(obj, mpl.image.AxesImage)][0]
fig.colorbar(im, ax=ax)
ax1.set_title('Probability density')
ax2.set_title('Spin density')
- plt.show()
+ save_figure('discretizer_qsh_wf')
def lattice_spacing():
#HIDDEN_BEGIN_ls_def
hamiltonian = "k_x**2 * identity(2) + alpha * k_x * sigma_y"
params = dict(alpha=.5)
#HIDDEN_END_ls_def
def plot(ax, a=1):
#HIDDEN_BEGIN_ls_hk_cont
h_k = kwant.continuum.lambdify(hamiltonian, locals=params)
k_cont = np.linspace(-4, 4, 201)
e_cont = [scipy.linalg.eigvalsh(h_k(k_x=ki)) for ki in k_cont]
#HIDDEN_END_ls_hk_cont
#HIDDEN_BEGIN_ls_hk_tb
template = kwant.continuum.discretize(hamiltonian, grid=a)
syst = kwant.wraparound.wraparound(template).finalized()
def h_k(k_x):
p = dict(k_x=k_x, **params)
return syst.hamiltonian_submatrix(params=p)
k_tb = np.linspace(-np.pi/a, np.pi/a, 201)
e_tb = [scipy.linalg.eigvalsh(h_k(k_x=a*ki)) for ki in k_tb]
#HIDDEN_END_ls_hk_tb
ax.plot(k_cont, e_cont, 'r-')
ax.plot(k_tb, e_tb, 'k-')
ax.plot([], [], 'r-', label='continuum')
ax.plot([], [], 'k-', label='tight-binding')
ax.set_xlim(-4, 4)
ax.set_ylim(-1, 14)
ax.set_title('a={}'.format(a))
ax.set_xlabel('momentum [a.u.]')
ax.set_ylabel('energy [a.u.]')
ax.grid()
ax.legend()
_, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(12, 4))
plot(ax1, a=1)
plot(ax2, a=.25)
- plt.show()
+ save_figure('discretizer_lattice_spacing')
def substitutions():
#HIDDEN_BEGIN_subs_1
sympify = kwant.continuum.sympify
subs = {'sx': [[0, 1], [1, 0]], 'sz': [[1, 0], [0, -1]]}
e = (
sympify('[[k_x**2, alpha * k_x], [k_x * alpha, -k_x**2]]'),
sympify('k_x**2 * sigma_z + alpha * k_x * sigma_x'),
sympify('k_x**2 * sz + alpha * k_x * sx', locals=subs),
)
- print(e[0] == e[1] == e[2])
+ with open('discretizer_subs_1.txt', 'w') as f:
+ print(e[0] == e[1] == e[2], file=f)
#HIDDEN_END_subs_1
#HIDDEN_BEGIN_subs_2
subs = {'A': 'A(x) + B', 'V': 'V(x) + V_0', 'C': 5}
- print(sympify('k_x * A * k_x + V + C', locals=subs))
+ with open('discretizer_subs_2.txt', 'w') as f:
+ print(sympify('k_x * A * k_x + V + C', locals=subs), file=f)
#HIDDEN_END_subs_2
def main():
#HIDDEN_BEGIN_symbolic_discretization
template = kwant.continuum.discretize('k_x * A(x) * k_x')
- print(template)
+ with open('discretizer_intro_verbose.txt', 'w') as f:
+ print(template, file=f)
#HIDDEN_END_symbolic_discretization
syst = stadium_system()
plot_eigenstate(syst)
analyze_qsh()
lattice_spacing()
substitutions()
# 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()
doc/source/code/figure/faq.py.diff deleted 100644 → 0
+ 0
− 483
View file @ f1929ee7
@@ -1,450 +1,479 @@
# Frequently asked questions
#
# This script is a disorganized collection of code snippets. As a whole, it is
# not meant as an example of good programming practice.
+import _defs
import kwant
import numpy as np
import tinyarray
import matplotlib
from matplotlib import pyplot as plt
matplotlib.rcParams['figure.figsize'] = (3.5, 3.5)
+def save_figure(file_name):
+ for extension in ('pdf', 'png'):
+ plt.savefig('.'.join((file_name, extension)),
+ dpi=_defs.dpi, bbox_inches='tight')
+
+
#### What is a Site?
#HIDDEN_BEGIN_site
a = 1
lat = kwant.lattice.square(a)
syst = kwant.Builder()
syst[lat(1, 0)] = 4
syst[lat(1, 1)] = 4
kwant.plot(syst)
+save_figure("faq_site")
#HIDDEN_END_site
#### What is a hopping?
a = 1
lat = kwant.lattice.square(a)
syst = kwant.Builder()
syst[lat(1, 0)] = 4
syst[lat(1, 1)] = 4
#HIDDEN_BEGIN_hopping
syst[(lat(1, 0), lat(1, 1))] = 1j
#HIDDEN_END_hopping
kwant.plot(syst)
+save_figure("faq_hopping")
#### What is a lattice?
#HIDDEN_BEGIN_lattice_monatomic
# Two monatomic lattices
primitive_vectors = [(1, 0), (0, 1)]
lat_a = kwant.lattice.Monatomic(primitive_vectors, offset=(0, 0))
lat_b = kwant.lattice.Monatomic(primitive_vectors, offset=(0.5, 0.5))
# lat1 is equivalent to kwant.lattice.square()
syst = kwant.Builder()
syst[lat_a(0, 0)] = 4
syst[lat_b(0, 0)] = 4
kwant.plot(syst)
+save_figure("faq_lattice")
#HIDDEN_END_lattice_monatomic
#HIDDEN_BEGIN_lattice_polyatomic
# One polyatomic lattice containing two sublattices
lat = kwant.lattice.Polyatomic([(1, 0), (0, 1)], [(0, 0), (0.5, 0.5)])
sub_a, sub_b = lat.sublattices
#HIDDEN_END_lattice_polyatomic
#### How to make a hole in a system?
#HIDDEN_BEGIN_hole
# Define the lattice and the (empty) system
a = 2
lat = kwant.lattice.cubic(a)
syst = kwant.Builder()
L = 10
W = 10
H = 2
# Add sites to the system in a cuboid
syst[(lat(i, j, k) for i in range(L) for j in range(W) for k in range(H))] = 4
kwant.plot(syst)
+save_figure("faq_hole1")
# Delete sites to create a hole
def in_hole(site):
x, y, z = site.pos / a - (L/2, W/2, H/2) # position relative to centre
return abs(x) < L / 4 and abs(y) < W / 4
for site in filter(in_hole, list(syst.sites())):
del syst[site]
kwant.plot(syst)
+save_figure("faq_hole2")
#HIDDEN_END_hole
#### How can we get access to the sites of our system?
builder = kwant.Builder()
lat = kwant.lattice.square()
builder[(lat(i, j) for i in range(3) for j in range(3))] = 4
#HIDDEN_BEGIN_sites1
# Before finalizing the system
sites = list(builder.sites()) # sites() doe *not* return a list
#HIDDEN_END_sites1
#HIDDEN_BEGIN_sites2
# After finalizing the system
syst = builder.finalized()
sites = syst.sites # syst.sites is an actual list
#HIDDEN_END_sites2
#HIDDEN_BEGIN_sites3
i = syst.id_by_site[lat(0, 2)] # we want the id of the site lat(0, 2)
#HIDDEN_END_sites3
#### How to plot a polyatomic lattice with different colors?
#HIDDEN_BEGIN_colors1
lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
#HIDDEN_END_colors1
#HIDDEN_BEGIN_colors2
# Plot sites from different families in different colors
def family_color(site):
if site.family == a:
return 'red'
if site.family == b:
return 'green'
else:
return 'blue'
def plot_system(syst):
kwant.plot(syst, site_lw=0.1, site_color=family_color)
## Add sites and hoppings.
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4
syst[b(i, j)] = 4
syst[c(i, j)] = 4
syst[lat.neighbors()] = -1
## Plot the system.
plot_system(syst)
+save_figure("faq_colors")
#HIDDEN_END_colors2
-
#### How to create all hoppings in a given direction using Hoppingkind?
# Monatomic lattice
#HIDDEN_BEGIN_direction1
# Create hopping between neighbors with HoppingKind
a = 1
syst = kwant.Builder()
lat = kwant.lattice.square(a)
syst[ (lat(i, j) for i in range(5) for j in range(5)) ] = 4
syst[kwant.builder.HoppingKind((1, 0), lat)] = -1
kwant.plot(syst)
+save_figure("faq_direction1")
#HIDDEN_END_direction1
# Polyatomic lattice
lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
def family_color(site):
if site.family == a:
return 'red'
if site.family == b:
return 'green'
else:
return 'blue'
def plot_system(syst):
kwant.plot(syst, site_size=0.15, site_lw=0.05, site_color=family_color)
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4
syst[b(i, j)] = 4
syst[c(i, j)] = 4
#HIDDEN_BEGIN_direction2
# equivalent to syst[kwant.builder.HoppingKind((0, 1), b)] = -1
syst[kwant.builder.HoppingKind((0, 1), b, b)] = -1
#HIDDEN_END_direction2
plot_system(syst)
+save_figure("faq_direction2")
# Delete the hoppings previously created
del syst[kwant.builder.HoppingKind((0, 1), b, b)]
#HIDDEN_BEGIN_direction3
syst[kwant.builder.HoppingKind((0, 0), a, b)] = -1
syst[kwant.builder.HoppingKind((0, 0), a, c)] = -1
syst[kwant.builder.HoppingKind((0, 0), c, b)] = -1
#HIDDEN_END_direction3
plot_system(syst)
+save_figure("faq_direction3")
#### How to create the hoppings between adjacent sites?
# Monatomic lattice
#HIDDEN_BEGIN_adjacent1
# Create hoppings with lat.neighbors()
syst = kwant.Builder()
lat = kwant.lattice.square()
syst[(lat(i, j) for i in range(3) for j in range(3))] = 4
syst[lat.neighbors()] = -1 # Equivalent to lat.neighbors(1)
kwant.plot(syst)
+save_figure("faq_adjacent1")
del syst[lat.neighbors()] # Delete all nearest-neighbor hoppings
syst[lat.neighbors(2)] = -1
kwant.plot(syst)
+save_figure("faq_adjacent2")
#HIDDEN_END_adjacent1
# Polyatomic lattice
#HIDDEN_BEGIN_FAQ6
# Hoppings using .neighbors()
#HIDDEN_BEGIN_adjacent2
# Create the system
lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4 # red
syst[b(i, j)] = 4 # green
syst[c(i, j)] = 4 # blue
syst[lat.neighbors()] = -1
#HIDDEN_END_adjacent2
plot_system(syst)
+save_figure("faq_adjacent3")
del syst[lat.neighbors()] # Delete the hoppings previously created
#HIDDEN_BEGIN_adjacent3
syst[a.neighbors()] = -1
#HIDDEN_END_adjacent3
plot_system(syst)
+save_figure("faq_adjacent4")
del syst[a.neighbors()] # Delete the hoppings previously created
syst[lat.neighbors(2)] = -1
plot_system(syst)
del syst[lat.neighbors(2)]
#### How to create a lead with a lattice different from the scattering region?
# Plot sites from different families in different colors
def plot_system(syst):
def family_color(site):
if site.family == subA:
return 'blue'
if site.family == subB:
return 'yellow'
else:
return 'green'
kwant.plot(syst, site_lw=0.1, site_color=family_color)
#HIDDEN_BEGIN_different_lattice1
# Define the scattering Region
L = 5
W = 5
lat = kwant.lattice.honeycomb()
subA, subB = lat.sublattices
syst = kwant.Builder()
syst[(subA(i, j) for i in range(L) for j in range(W))] = 4
syst[(subB(i, j) for i in range(L) for j in range(W))] = 4
syst[lat.neighbors()] = -1
#HIDDEN_END_different_lattice1
plot_system(syst)
+save_figure("faq_different_lattice1")
#HIDDEN_BEGIN_different_lattice2
# Create a lead
lat_lead = kwant.lattice.square()
sym_lead1 = kwant.TranslationalSymmetry((0, 1))
lead1 = kwant.Builder(sym_lead1)
lead1[(lat_lead(i, 0) for i in range(2, 7))] = 4
lead1[lat_lead.neighbors()] = -1
#HIDDEN_END_different_lattice2
plot_system(lead1)
+save_figure("faq_different_lattice2")
#HIDDEN_BEGIN_different_lattice3
syst[(lat_lead(i, 5) for i in range(2, 7))] = 4
syst[lat_lead.neighbors()] = -1
# Manually attach sites from graphene to square lattice
syst[((lat_lead(i+2, 5), subB(i, 4)) for i in range(5))] = -1
#HIDDEN_END_different_lattice3
plot_system(syst)
+save_figure("faq_different_lattice3")
#HIDDEN_BEGIN_different_lattice4
syst.attach_lead(lead1)
#HIDDEN_END_different_lattice4
plot_system(syst)
+save_figure("faq_different_lattice4")
#### How to cut a finite system out of a system with translationnal symmetries?
#HIDDEN_BEGIN_fill1
# Create 3d model.
cubic = kwant.lattice.cubic()
sym_3d = kwant.TranslationalSymmetry([1, 0, 0], [0, 1, 0], [0, 0, 1])
model = kwant.Builder(sym_3d)
model[cubic(0, 0, 0)] = 4
model[cubic.neighbors()] = -1
#HIDDEN_END_fill1
#HIDDEN_BEGIN_fill2
# Build scattering region (white).
def cuboid_shape(site):
x, y, z = abs(site.pos)
return x < 4 and y < 10 and z < 3
cuboid = kwant.Builder()
cuboid.fill(model, cuboid_shape, (0, 0, 0));
#HIDDEN_END_fill2
kwant.plot(cuboid);
+save_figure("faq_fill2")
#HIDDEN_BEGIN_fill3
# Build electrode (black).
def electrode_shape(site):
x, y, z = site.pos - (0, 5, 2)
return y**2 + z**2 < 2.3**2
electrode = kwant.Builder(kwant.TranslationalSymmetry([1, 0, 0]))
electrode.fill(model, electrode_shape, (0, 5, 2)) # lead
# Scattering region
cuboid.fill(electrode, lambda s: abs(s.pos[0]) < 7, (0, 5, 4))
cuboid.attach_lead(electrode)
#HIDDEN_END_fill3
kwant.plot(cuboid);
+save_figure("faq_fill3")
#### How does Kwant order the propagating modes of a lead?
#HIDDEN_BEGIN_pm
lat = kwant.lattice.square()
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead[(lat(0, i) for i in range(3))] = 4
lead[lat.neighbors()] = -1
flead = lead.finalized()
E = 2.5
prop_modes, _ = flead.modes(energy=E)
#HIDDEN_END_pm
def plot_and_label_modes(lead, E):
# Plot the different modes
pmodes, _ = lead.modes(energy=E)
kwant.plotter.bands(lead, show=False)
for i, k in enumerate(pmodes.momenta):
plt.plot(k, E, 'ko')
plt.annotate(str(i), xy=(k, E), xytext=(-5, 8),
textcoords='offset points',
bbox=dict(boxstyle='round,pad=0.1',fc='white', alpha=0.7))
plt.plot([-3, 3], [E, E], 'r--')
plt.ylim(E-1, E+1)
plt.xlim(-2, 2)
plt.xlabel("momentum")
plt.ylabel("energy")
plt.show()
plot_and_label_modes(flead, E)
+save_figure('faq_pm1')
# More involved example
s0 = np.eye(2)
sz = np.array([[1, 0], [0, -1]])
lead2 = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead2[(lat(0, i) for i in range(2))] = np.diag([1.8, -1])
lead2[lat.neighbors()] = -1 * sz
flead2 = lead2.finalized()
plot_and_label_modes(flead2, 1)
+save_figure('faq_pm2')
#### How does Kwant order components of an individual wavefunction?
def circle(R):
return lambda r: np.linalg.norm(r) < R
def make_system(lat):
norbs = lat.norbs
syst = kwant.Builder()
syst[lat.shape(circle(3), (0, 0))] = 4 * np.eye(norbs)
syst[lat.neighbors()] = -1 * np.eye(norbs)
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead[(lat(0, i) for i in range(-1, 2))] = 4 * np.eye(norbs)
lead[lat.neighbors()] = -1 * np.eye(norbs)
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst.finalized()
#HIDDEN_BEGIN_ord1
lat = kwant.lattice.square(norbs=1)
syst = make_system(lat)
scattering_states = kwant.wave_function(syst, energy=1)
wf = scattering_states(0)[0] # scattering state from lead 0 incoming in mode 0
idx = syst.id_by_site[lat(0, 0)] # look up index of site
-print('wavefunction on lat(0, 0): ', wf[idx])
+with open('faq_ord1.txt', 'w') as f:
+ print('wavefunction on lat(0, 0): ', wf[idx], file=f)
#HIDDEN_END_ord1
#HIDDEN_BEGIN_ord2
lat = kwant.lattice.square(norbs=2)
syst = make_system(lat)
scattering_states = kwant.wave_function(syst, energy=1)
wf = scattering_states(0)[0] # scattering state from lead 0 incoming in mode 0
idx = syst.id_by_site[lat(0, 0)] # look up index of site
# Group consecutive degrees of freedom from 'wf' together; these correspond
# to degrees of freedom on the same site.
wf = wf.reshape(-1, 2)
-print('wavefunction on lat(0, 0): ', wf[idx])
+with open('faq_ord2.txt', 'w') as f:
+ print('wavefunction on lat(0, 0): ', wf[idx], file=f)
#HIDDEN_END_ord2
doc/source/code/figure/graphene.py.diff deleted 100644 → 0
+ 0
− 217
View file @ f1929ee7
@@ -1,179 +1,203 @@
# Tutorial 2.5. Beyond square lattices: graphene
# ==============================================
#
# Physics background
# ------------------
# Transport through a graphene quantum dot with a pn-junction
#
# Kwant features highlighted
# --------------------------
# - Application of all the aspects of tutorials 1-3 to a more complicated
# lattice, namely graphene
+import _defs
from math import pi, sqrt, tanh
import kwant
# For computing eigenvalues
import scipy.sparse.linalg as sla
# For plotting
from matplotlib import pyplot
# Define the graphene lattice
sin_30, cos_30 = (1 / 2, sqrt(3) / 2)
#HIDDEN_BEGIN_hnla
graphene = kwant.lattice.general([(1, 0), (sin_30, cos_30)],
[(0, 0), (0, 1 / sqrt(3))])
a, b = graphene.sublattices
#HIDDEN_END_hnla
#HIDDEN_BEGIN_shzy
def make_system(r=10, w=2.0, pot=0.1):
#### Define the scattering region. ####
# circular scattering region
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst = kwant.Builder()
# w: width and pot: potential maximum of the p-n junction
def potential(site):
(x, y) = site.pos
d = y * cos_30 + x * sin_30
return pot * tanh(d / w)
syst[graphene.shape(circle, (0, 0))] = potential
#HIDDEN_END_shzy
# specify the hoppings of the graphene lattice in the
# format expected by builder.HoppingKind
#HIDDEN_BEGIN_hsmc
hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))
#HIDDEN_END_hsmc
#HIDDEN_BEGIN_bfwb
syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
#HIDDEN_END_bfwb
# Modify the scattering region
#HIDDEN_BEGIN_efut
del syst[a(0, 0)]
syst[a(-2, 1), b(2, 2)] = -1
#HIDDEN_END_efut
#### Define the leads. ####
#HIDDEN_BEGIN_aakh
# left lead
sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0)))
def lead0_shape(pos):
x, y = pos
return (-0.4 * r < y < 0.4 * r)
lead0 = kwant.Builder(sym0)
lead0[graphene.shape(lead0_shape, (0, 0))] = -pot
lead0[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
# The second lead, going to the top right
sym1 = kwant.TranslationalSymmetry(graphene.vec((0, 1)))
def lead1_shape(pos):
v = pos[1] * sin_30 - pos[0] * cos_30
return (-0.4 * r < v < 0.4 * r)
lead1 = kwant.Builder(sym1)
lead1[graphene.shape(lead1_shape, (0, 0))] = pot
lead1[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
#HIDDEN_END_aakh
#HIDDEN_BEGIN_kmmw
return syst, [lead0, lead1]
#HIDDEN_END_kmmw
#HIDDEN_BEGIN_zydk
def compute_evs(syst):
# Compute some eigenvalues of the closed system
sparse_mat = syst.hamiltonian_submatrix(sparse=True)
evs = sla.eigs(sparse_mat, 2)[0]
print(evs.real)
#HIDDEN_END_zydk
def plot_conductance(syst, energies):
# Compute transmission as a function of energy
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
data.append(smatrix.transmission(0, 1))
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(energies, data)
- pyplot.xlabel("energy [t]")
- pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.xlabel("energy [t]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("conductance [e^2/h]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("graphene_result." + extension, dpi=_defs.dpi)
def plot_bandstructure(flead, momenta):
bands = kwant.physics.Bands(flead)
energies = [bands(k) for k in momenta]
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(momenta, energies)
- pyplot.xlabel("momentum [(lattice constant)^-1]")
- pyplot.ylabel("energy [t]")
- pyplot.show()
+ pyplot.xlabel("momentum [(lattice constant)^-1]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("energy [t]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("graphene_bs." + extension, dpi=_defs.dpi)
#HIDDEN The part of the following code block which begins with family_colors
#HIDDEN is included verbatim in the tutorial text because nested code examples
#HIDDEN are not supported. Remember to update the tutorial text when you
#HIDDEN modify this block.
#HIDDEN_BEGIN_itkk
def main():
pot = 0.1
syst, leads = make_system(pot=pot)
# To highlight the two sublattices of graphene, we plot one with
# a filled, and the other one with an open circle:
def family_colors(site):
return 0 if site.family == a else 1
- # Plot the closed system without leads.
- kwant.plot(syst, site_color=family_colors, site_lw=0.1, colorbar=False)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_color=family_colors, site_lw=0.1, colorbar=False,
+ file="graphene_syst1." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_itkk
# Compute some eigenvalues.
#HIDDEN_BEGIN_jmbi
compute_evs(syst.finalized())
#HIDDEN_END_jmbi
# Attach the leads to the system.
for lead in leads:
syst.attach_lead(lead)
- # Then, plot the system with leads.
- kwant.plot(syst, site_color=family_colors, site_lw=0.1,
- lead_site_lw=0, colorbar=False)
+ size = (_defs.figwidth_in, 0.9 * _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_color=family_colors, colorbar=False, site_lw=0.1,
+ file="graphene_syst2." + extension,
+ fig_size=size, dpi=_defs.dpi, lead_site_lw=0)
# Finalize the system.
syst = syst.finalized()
# Compute the band structure of lead 0.
momenta = [-pi + 0.02 * pi * i for i in range(101)]
plot_bandstructure(syst.leads[0], momenta)
# Plot conductance.
energies = [-2 * pot + 4. / 50. * pot * i for i in range(51)]
plot_conductance(syst, energies)
# 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()
doc/source/code/figure/kernel_polynomial_method.py.diff deleted 100644 → 0
+ 0
− 374
View file @ f1929ee7
@@ -1,219 +1,242 @@
# 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)
#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)
#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)
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)
#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)
#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)
#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)
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)
#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())
#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)
rho_alt_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=rho_alt)
#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)
# 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)
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)
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()
doc/source/code/figure/magnetic_texture.py.diff deleted 100644 → 0
+ 0
− 286
View file @ f1929ee7
@@ -1,245 +1,268 @@
# Tutorial 2.7. Spin textures
# ===========================
#
# Physics background
# ------------------
# - Spin textures
# - Skyrmions
#
# Kwant features highlighted
# --------------------------
# - operators
# - plotting vector fields
+import _defs
+from contextlib import redirect_stdout
+
from math import sin, cos, tanh, pi
import itertools
import numpy as np
import tinyarray as ta
import matplotlib.pyplot as plt
import kwant
sigma_0 = ta.array([[1, 0], [0, 1]])
sigma_x = ta.array([[0, 1], [1, 0]])
sigma_y = ta.array([[0, -1j], [1j, 0]])
sigma_z = ta.array([[1, 0], [0, -1]])
# vector of Pauli matrices σ_αiβ where greek
# letters denote spinor indices
sigma = np.rollaxis(np.array([sigma_x, sigma_y, sigma_z]), 1)
#HIDDEN_BEGIN_model
def field_direction(pos, r0, delta):
x, y = pos
r = np.linalg.norm(pos)
r_tilde = (r - r0) / delta
theta = (tanh(r_tilde) - 1) * (pi / 2)
if r == 0:
m_i = [0, 0, -1]
else:
m_i = [
(x / r) * sin(theta),
(y / r) * sin(theta),
cos(theta),
]
return np.array(m_i)
def scattering_onsite(site, r0, delta, J):
m_i = field_direction(site.pos, r0, delta)
return J * np.dot(m_i, sigma)
def lead_onsite(site, J):
return J * sigma_z
#HIDDEN_END_model
#HIDDEN_BEGIN_syst
lat = kwant.lattice.square(norbs=2)
def make_system(L=80):
syst = kwant.Builder()
def square(pos):
return all(-L/2 < p < L/2 for p in pos)
syst[lat.shape(square, (0, 0))] = scattering_onsite
syst[lat.neighbors()] = -sigma_0
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)),
conservation_law=-sigma_z)
lead[lat.shape(square, (0, 0))] = lead_onsite
lead[lat.neighbors()] = -sigma_0
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst
#HIDDEN_END_syst
-def plot_vector_field(syst, params):
+def save_plot(fname):
+ for extension in ('.pdf', '.png'):
+ plt.savefig(fname + extension,
+ dpi=_defs.dpi, bbox_inches='tight')
+
+
+def plot_vector_field(syst, params, fname=None):
xmin, ymin = min(s.tag for s in syst.sites)
xmax, ymax = max(s.tag for s in syst.sites)
x, y = np.meshgrid(np.arange(xmin, xmax+1), np.arange(ymin, ymax+1))
m_i = [field_direction(p, **params) for p in zip(x.flat, y.flat)]
m_i = np.reshape(m_i, x.shape + (3,))
m_i = np.rollaxis(m_i, 2, 0)
- fig, ax = plt.subplots(1, 1)
+ fig, ax = plt.subplots(1, 1, figsize=(9, 7))
im = ax.quiver(x, y, *m_i, pivot='mid', scale=75)
fig.colorbar(im)
- plt.show()
+ if fname:
+ save_plot(fname)
+ else:
+ plt.show()
-def plot_densities(syst, densities):
- fig, axes = plt.subplots(1, len(densities))
+def plot_densities(syst, densities, fname=None):
+ fig, axes = plt.subplots(1, len(densities), figsize=(7*len(densities), 7))
for ax, (title, rho) in zip(axes, densities):
kwant.plotter.map(syst, rho, ax=ax, a=4)
ax.set_title(title)
- plt.show()
+ if fname:
+ save_plot(fname)
+ else:
+ plt.show()
-def plot_currents(syst, currents):
- fig, axes = plt.subplots(1, len(currents))
+
+def plot_currents(syst, currents, fname=None):
+ fig, axes = plt.subplots(1, len(currents), figsize=(7*len(currents), 7))
if not hasattr(axes, '__len__'):
axes = (axes,)
for ax, (title, current) in zip(axes, currents):
kwant.plotter.current(syst, current, ax=ax, colorbar=False)
ax.set_title(title)
- plt.show()
+ if fname:
+ save_plot(fname)
+ else:
+ plt.show()
def main():
syst = make_system().finalized()
#HIDDEN_BEGIN_wavefunction
params = dict(r0=20, delta=10, J=1)
wf = kwant.wave_function(syst, energy=-1, params=params)
psi = wf(0)[0]
#HIDDEN_END_wavefunction
- plot_vector_field(syst, dict(r0=20, delta=10))
+ plot_vector_field(syst, dict(r0=20, delta=10), fname='mag_field_direction')
#HIDDEN_BEGIN_ldos
# even (odd) indices correspond to spin up (down)
up, down = psi[::2], psi[1::2]
density = np.abs(up)**2 + np.abs(down)**2
#HIDDEN_END_ldos
#HIDDEN_BEGIN_lsdz
# spin down components have a minus sign
spin_z = np.abs(up)**2 - np.abs(down)**2
#HIDDEN_END_lsdz
#HIDDEN_BEGIN_lsdy
# spin down components have a minus sign
spin_y = 1j * (down.conjugate() * up - up.conjugate() * down)
#HIDDEN_END_lsdy
#HIDDEN_BEGIN_lden
rho = kwant.operator.Density(syst)
rho_sz = kwant.operator.Density(syst, sigma_z)
rho_sy = kwant.operator.Density(syst, sigma_y)
# calculate the expectation values of the operators with 'psi'
density = rho(psi)
spin_z = rho_sz(psi)
spin_y = rho_sy(psi)
#HIDDEN_END_lden
plot_densities(syst, [
('$σ_0$', density),
('$σ_z$', spin_z),
('$σ_y$', spin_y),
- ])
+ ], fname='spin_densities')
#HIDDEN_BEGIN_current
J_0 = kwant.operator.Current(syst)
J_z = kwant.operator.Current(syst, sigma_z)
J_y = kwant.operator.Current(syst, sigma_y)
# calculate the expectation values of the operators with 'psi'
current = J_0(psi)
spin_z_current = J_z(psi)
spin_y_current = J_y(psi)
#HIDDEN_END_current
plot_currents(syst, [
('$J_{σ_0}$', current),
('$J_{σ_z}$', spin_z_current),
('$J_{σ_y}$', spin_y_current),
- ])
+ ], fname='spin_currents')
#HIDDEN_BEGIN_following
def following_m_i(site, r0, delta):
m_i = field_direction(site.pos, r0, delta)
return np.dot(m_i, sigma)
J_m = kwant.operator.Current(syst, following_m_i)
# evaluate the operator
m_current = J_m(psi, params=dict(r0=25, delta=10))
#HIDDEN_END_following
plot_currents(syst, [
(r'$J_{\mathbf{m}_i}$', m_current),
('$J_{σ_z}$', spin_z_current),
- ])
+ ], fname='spin_current_comparison')
#HIDDEN_BEGIN_density_cut
def circle(site):
return np.linalg.norm(site.pos) < 20
rho_circle = kwant.operator.Density(syst, where=circle, sum=True)
all_states = np.vstack((wf(0), wf(1)))
dos_in_circle = sum(rho_circle(p) for p in all_states) / (2 * pi)
- print('density of states in circle:', dos_in_circle)
+ with open('circle_dos.txt', 'w') as f:
+ with redirect_stdout(f):
+ print('density of states in circle:', dos_in_circle)
#HIDDEN_END_density_cut
#HIDDEN_BEGIN_current_cut
def left_cut(site_to, site_from):
return site_from.pos[0] <= -39 and site_to.pos[0] > -39
def right_cut(site_to, site_from):
return site_from.pos[0] < 39 and site_to.pos[0] >= 39
J_left = kwant.operator.Current(syst, where=left_cut, sum=True)
J_right = kwant.operator.Current(syst, where=right_cut, sum=True)
Jz_left = kwant.operator.Current(syst, sigma_z, where=left_cut, sum=True)
Jz_right = kwant.operator.Current(syst, sigma_z, where=right_cut, sum=True)
- print('J_left:', J_left(psi), ' J_right:', J_right(psi))
- print('Jz_left:', Jz_left(psi), ' Jz_right:', Jz_right(psi))
+ with open('current_cut.txt', 'w') as f:
+ with redirect_stdout(f):
+ print('J_left:', J_left(psi), ' J_right:', J_right(psi))
+ print('Jz_left:', Jz_left(psi), ' Jz_right:', Jz_right(psi))
#HIDDEN_END_current_cut
#HIDDEN_BEGIN_bind
J_m = kwant.operator.Current(syst, following_m_i)
J_z = kwant.operator.Current(syst, sigma_z)
J_m_bound = J_m.bind(params=dict(r0=25, delta=10, J=1))
J_z_bound = J_z.bind(params=dict(r0=25, delta=10, J=1))
# Sum current local from all scattering states on the left at energy=-1
wf_left = wf(0)
J_m_left = sum(J_m_bound(p) for p in wf_left)
J_z_left = sum(J_z_bound(p) for p in wf_left)
#HIDDEN_END_bind
plot_currents(syst, [
(r'$J_{\mathbf{m}_i}$ (from left)', J_m_left),
(r'$J_{σ_z}$ (from left)', J_z_left),
- ])
+ ], fname='bound_current')
if __name__ == '__main__':
main()
doc/source/code/figure/plot_graphene.py.diff deleted 100644 → 0
+ 0
− 141
View file @ f1929ee7
@@ -1,112 +1,130 @@
# Tutorial 2.8.1. 2D example: graphene quantum dot
# ================================================
#
# Physics background
# ------------------
# - graphene edge states
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting
+import _defs
import kwant
from matplotlib import pyplot
#HIDDEN_BEGIN_makesyst
lat = kwant.lattice.honeycomb()
a, b = lat.sublattices
def make_system(r=8, t=-1, tp=-0.1):
def circle(pos):
x, y = pos
return x**2 + y**2 < r**2
syst = kwant.Builder()
- syst[lat.shape(circle, (0, 0))] = 0
+ syst[lat.shape(circle, (0,0))] = 0
syst[lat.neighbors()] = t
syst.eradicate_dangling()
if tp:
syst[lat.neighbors(2)] = tp
return syst
#HIDDEN_END_makesyst
#HIDDEN_BEGIN_plotsyst1
def plot_system(syst):
- kwant.plot(syst)
#HIDDEN_END_plotsyst1
- # the standard plot is ok, but not very intelligible. One can do
- # better by playing wioth colors and linewidths
+ # standard plot - not very intelligible for this particular situation
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="plot_graphene_syst1." + extension,
+ fig_size=size, dpi=_defs.dpi)
# use color and linewidths to get a better plot
#HIDDEN_BEGIN_plotsyst2
def family_color(site):
return 'black' if site.family == a else 'white'
def hopping_lw(site1, site2):
return 0.04 if site1.family == site2.family else 0.1
- kwant.plot(syst, site_lw=0.1, site_color=family_color, hop_lw=hopping_lw)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_lw=0.1, site_color=family_color,
+ hop_lw=hopping_lw, file="plot_graphene_syst2." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plotsyst2
#HIDDEN_BEGIN_plotdata1
def plot_data(syst, n):
import scipy.linalg as la
syst = syst.finalized()
ham = syst.hamiltonian_submatrix()
evecs = la.eigh(ham)[1]
wf = abs(evecs[:, n])**2
#HIDDEN_END_plotdata1
# the usual - works great in general, looks just a bit crufty for
# small systems
#HIDDEN_BEGIN_plotdata2
- kwant.plotter.map(syst, wf, oversampling=10, cmap='gist_heat_r')
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plotter.map(syst, wf, oversampling=10, cmap='gist_heat_r',
+ file="plot_graphene_data1." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plotdata2
# use two different sort of triangles to cleanly fill the space
#HIDDEN_BEGIN_plotdata3
def family_shape(i):
site = syst.sites[i]
return ('p', 3, 180) if site.family == a else ('p', 3, 0)
def family_color(i):
return 'black' if syst.sites[i].family == a else 'white'
- kwant.plot(syst, site_color=wf, site_symbol=family_shape,
- site_size=0.5, hop_lw=0, cmap='gist_heat_r')
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_color=wf, site_symbol=family_shape,
+ site_size=0.5, hop_lw=0, cmap='gist_heat_r',
+ file="plot_graphene_data2." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plotdata3
# plot by changing the symbols itself
#HIDDEN_BEGIN_plotdata4
def site_size(i):
return 3 * wf[i] / wf.max()
- kwant.plot(syst, site_size=site_size, site_color=(0, 0, 1, 0.3),
- hop_lw=0.1)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_size=site_size, site_color=(0,0,1,0.3),
+ hop_lw=0.1, file="plot_graphene_data3." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plotdata4
def main():
# plot the graphene system in different styles
syst = make_system()
plot_system(syst)
# compute a wavefunction (number 225) and plot it in different
# styles
syst = make_system(tp=0)
plot_data(syst, 225)
# 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()
doc/source/code/figure/plot_qahe.py.diff deleted 100644 → 0
+ 0
− 78
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# Comprehensive example: quantum anomalous Hall effect
# ====================================================
#
# Physics background
# ------------------
# + Quantum anomalous Hall effect
#
# Features highlighted
# --------------------
# + Use of `kwant.continuum` to discretize a continuum Hamiltonian
# + Use of `kwant.operator` to compute local current
# + Use of `kwant.plotter.current` to plot local current
+import _defs
import math
import matplotlib.pyplot
import kwant
import kwant.continuum
# 2 band model exhibiting quantum anomalous Hall effect
#HIDDEN_BEGIN_model
def make_model(a):
ham = ("alpha * (k_x * sigma_x - k_y * sigma_y)"
"+ (m + beta * kk) * sigma_z"
"+ (gamma * kk + U) * sigma_0")
subs = {"kk": "k_x**2 + k_y**2"}
return kwant.continuum.discretize(ham, locals=subs, grid=a)
#HIDDEN_END_model
def make_system(model, L):
def lead_shape(site):
x, y = site.pos / L
return abs(y) < 0.5
# QPC shape: a rectangle with 2 gaussians
# etched out of the top and bottom edge.
def central_shape(site):
x, y = site.pos / L
return abs(x) < 3/5 and abs(y) < 0.5 - 0.4 * math.exp(-40 * x**2)
lead = kwant.Builder(kwant.TranslationalSymmetry(
model.lattice.vec((-1, 0))))
lead.fill(model, lead_shape, (0, 0))
syst = kwant.Builder()
syst.fill(model, central_shape, (0, 0))
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst.finalized()
def main():
# Set up our model and system, and define the model parameters.
params = dict(alpha=0.365, beta=0.686, gamma=0.512, m=-0.01, U=0)
model = make_model(1)
syst = make_system(model, 70)
kwant.plot(syst)
# Calculate the scattering states at energy 'm' coming from the left
# lead, and the associated particle current.
psi = kwant.wave_function(syst, energy=params['m'], params=params)(0)
#HIDDEN_BEGIN_current
J = kwant.operator.Current(syst).bind(params=params)
current = sum(J(p) for p in psi)
- kwant.plotter.current(syst, current)
+ for extension in ('pdf', 'png'):
+ kwant.plotter.current(syst, current,
+ file="plot_qahe_current." + extension,
+ dpi=_defs.dpi)
#HIDDEN_END_current
if __name__ == '__main__':
main()
doc/source/code/figure/plot_zincblende.py.diff deleted 100644 → 0
+ 0
− 71
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# Tutorial 2.8.2. 3D example: zincblende structure
# ================================================
#
# Physical background
# -------------------
# - 3D Bravais lattices
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting in 3D
+import _defs
import kwant
from matplotlib import pyplot
#HIDDEN_BEGIN_zincblende1
lat = kwant.lattice.general([(0, 0.5, 0.5), (0.5, 0, 0.5), (0.5, 0.5, 0)],
[(0, 0, 0), (0.25, 0.25, 0.25)])
a, b = lat.sublattices
#HIDDEN_END_zincblende1
#HIDDEN_BEGIN_zincblende2
def make_cuboid(a=15, b=10, c=5):
def cuboid_shape(pos):
x, y, z = pos
return 0 <= x < a and 0 <= y < b and 0 <= z < c
syst = kwant.Builder()
syst[lat.shape(cuboid_shape, (0, 0, 0))] = None
syst[lat.neighbors()] = None
return syst
#HIDDEN_END_zincblende2
def main():
# the standard plotting style for 3D is mainly useful for
# checking shapes:
#HIDDEN_BEGIN_plot1
syst = make_cuboid()
- kwant.plot(syst)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="plot_zincblende_syst1." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plot1
# visualize the crystal structure better for a very small system
#HIDDEN_BEGIN_plot2
syst = make_cuboid(a=1.5, b=1.5, c=1.5)
def family_colors(site):
return 'r' if site.family == a else 'g'
- kwant.plot(syst, site_size=0.18, site_lw=0.01, hop_lw=0.05,
- site_color=family_colors)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_size=0.18, site_lw=0.01, hop_lw=0.05,
+ site_color=family_colors,
+ file="plot_zincblende_syst2." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_plot2
# 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()
doc/source/code/figure/quantum_well.py.diff deleted 100644 → 0
+ 0
− 103
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@@ -1,88 +1,95 @@
# Tutorial 2.3.2. Spatially dependent values through functions
# ============================================================
#
# Physics background
# ------------------
# transmission through a quantum well
#
# Kwant features highlighted
# --------------------------
# - Functions as values in Builder
+import _defs
import kwant
# For plotting
from matplotlib import pyplot
#HIDDEN_BEGIN_ehso
def make_system(a=1, t=1.0, W=10, L=30, L_well=10):
# Start with an empty tight-binding system and a single square lattice.
# `a` is the lattice constant (by default set to 1 for simplicity.
lat = kwant.lattice.square(a)
syst = kwant.Builder()
#### Define the scattering region. ####
# Potential profile
def potential(site, pot):
(x, y) = site.pos
if (L - L_well) / 2 < x < (L + L_well) / 2:
return pot
else:
return 0
#HIDDEN_END_ehso
#HIDDEN_BEGIN_coid
def onsite(site, pot):
return 4 * t + potential(site, pot)
syst[(lat(x, y) for x in range(L) for y in range(W))] = onsite
syst[lat.neighbors()] = -t
#HIDDEN_END_coid
#### Define and attach the leads. ####
lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0)))
lead[(lat(0, j) for j in range(W))] = 4 * t
lead[lat.neighbors()] = -t
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst
def plot_conductance(syst, energy, welldepths):
#HIDDEN_BEGIN_sqvr
# Compute conductance
data = []
for welldepth in welldepths:
smatrix = kwant.smatrix(syst, energy, params=dict(pot=-welldepth))
data.append(smatrix.transmission(1, 0))
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(welldepths, data)
- pyplot.xlabel("well depth [t]")
- pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.xlabel("well depth [t]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("conductance [e^2/h]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("quantum_well_result." + extension, dpi=_defs.dpi)
#HIDDEN_END_sqvr
def main():
syst = make_system()
- # Check that the system looks as intended.
- kwant.plot(syst)
-
# Finalize the system.
syst = syst.finalized()
# We should see conductance steps.
plot_conductance(syst, energy=0.2,
welldepths=[0.01 * i for i in range(100)])
# 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()
doc/source/code/figure/quantum_wire.py.diff deleted 100644 → 0
+ 0
− 136
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# Tutorial 2.2.2. Transport through a quantum wire
# ================================================
#
# Physics background
# ------------------
# Conductance of a quantum wire; subbands
#
# Kwant features highlighted
# --------------------------
# - Builder for setting up transport systems easily
# - Making scattering region and leads
# - Using the simple sparse solver for computing Landauer conductance
+import _defs
from matplotlib import pyplot
#HIDDEN_BEGIN_dwhx
import kwant
#HIDDEN_END_dwhx
# First, define the tight-binding system
#HIDDEN_BEGIN_goiq
syst = kwant.Builder()
#HIDDEN_END_goiq
# Here, we are only working with square lattices
#HIDDEN_BEGIN_suwo
a = 1
lat = kwant.lattice.square(a)
#HIDDEN_END_suwo
#HIDDEN_BEGIN_zfvr
t = 1.0
W = 10
L = 30
# Define the scattering region
for i in range(L):
for j in range(W):
# On-site Hamiltonian
syst[lat(i, j)] = 4 * t
# Hopping in y-direction
if j > 0:
syst[lat(i, j), lat(i, j - 1)] = -t
# Hopping in x-direction
if i > 0:
syst[lat(i, j), lat(i - 1, j)] = -t
#HIDDEN_END_zfvr
# Then, define and attach the leads:
# First the lead to the left
# (Note: TranslationalSymmetry takes a real-space vector)
#HIDDEN_BEGIN_xcmc
sym_left_lead = kwant.TranslationalSymmetry((-a, 0))
left_lead = kwant.Builder(sym_left_lead)
#HIDDEN_END_xcmc
#HIDDEN_BEGIN_ndez
for j in range(W):
left_lead[lat(0, j)] = 4 * t
if j > 0:
left_lead[lat(0, j), lat(0, j - 1)] = -t
left_lead[lat(1, j), lat(0, j)] = -t
#HIDDEN_END_ndez
#HIDDEN_BEGIN_fskr
syst.attach_lead(left_lead)
#HIDDEN_END_fskr
# Then the lead to the right
#HIDDEN_BEGIN_xhqc
sym_right_lead = kwant.TranslationalSymmetry((a, 0))
right_lead = kwant.Builder(sym_right_lead)
for j in range(W):
right_lead[lat(0, j)] = 4 * t
if j > 0:
right_lead[lat(0, j), lat(0, j - 1)] = -t
right_lead[lat(1, j), lat(0, j)] = -t
syst.attach_lead(right_lead)
#HIDDEN_END_xhqc
# Plot it, to make sure it's OK
#HIDDEN_BEGIN_wsgh
-kwant.plot(syst)
+size = (_defs.figwidth_in, 0.3 * _defs.figwidth_in)
+for extension in ('pdf', 'png'):
+ kwant.plot(syst, file="quantum_wire_syst." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_wsgh
# Finalize the system
#HIDDEN_BEGIN_dngj
syst = syst.finalized()
#HIDDEN_END_dngj
# Now that we have the system, we can compute conductance
#HIDDEN_BEGIN_buzn
energies = []
data = []
for ie in range(100):
energy = ie * 0.01
# compute the scattering matrix at a given energy
smatrix = kwant.smatrix(syst, energy)
# compute the transmission probability from lead 0 to
# lead 1
energies.append(energy)
data.append(smatrix.transmission(1, 0))
#HIDDEN_END_buzn
# Use matplotlib to write output
# We should see conductance steps
#HIDDEN_BEGIN_lliv
-pyplot.figure()
+fig = pyplot.figure()
pyplot.plot(energies, data)
-pyplot.xlabel("energy [t]")
-pyplot.ylabel("conductance [e^2/h]")
-pyplot.show()
+pyplot.xlabel("energy [t]", fontsize=_defs.mpl_label_size)
+pyplot.ylabel("conductance [e^2/h]", fontsize=_defs.mpl_label_size)
+pyplot.setp(fig.get_axes()[0].get_xticklabels(), fontsize=_defs.mpl_tick_size)
+pyplot.setp(fig.get_axes()[0].get_yticklabels(), fontsize=_defs.mpl_tick_size)
+fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+for extension in ('pdf', 'png'):
+ fig.savefig("quantum_wire_result." + extension, dpi=_defs.dpi)
#HIDDEN_END_lliv
doc/source/code/figure/spin_orbit.py.diff deleted 100644 → 0
+ 0
− 118
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# Tutorial 2.3.1. Matrix structure of on-site and hopping elements
# ================================================================
#
# Physics background
# ------------------
# Gaps in quantum wires with spin-orbit coupling and Zeeman splititng,
# as theoretically predicted in
# http://prl.aps.org/abstract/PRL/v90/i25/e256601
# and (supposedly) experimentally oberved in
# http://www.nature.com/nphys/journal/v6/n5/abs/nphys1626.html
#
# Kwant features highlighted
# --------------------------
# - Numpy matrices as values in Builder
+import _defs
import kwant
# For plotting
from matplotlib import pyplot
# For matrix support
#HIDDEN_BEGIN_xumz
import tinyarray
#HIDDEN_END_xumz
# define Pauli-matrices for convenience
#HIDDEN_BEGIN_hwbt
sigma_0 = tinyarray.array([[1, 0], [0, 1]])
sigma_x = tinyarray.array([[0, 1], [1, 0]])
sigma_y = tinyarray.array([[0, -1j], [1j, 0]])
sigma_z = tinyarray.array([[1, 0], [0, -1]])
#HIDDEN_END_hwbt
def make_system(t=1.0, alpha=0.5, e_z=0.08, W=10, L=30):
# Start with an empty tight-binding system and a single square lattice.
# `a` is the lattice constant (by default set to 1 for simplicity).
lat = kwant.lattice.square()
syst = kwant.Builder()
#### Define the scattering region. ####
#HIDDEN_BEGIN_uxrm
syst[(lat(x, y) for x in range(L) for y in range(W))] = \
4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
-t * sigma_0 + 1j * alpha * sigma_y / 2
# hoppings in y-directions
syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
-t * sigma_0 - 1j * alpha * sigma_x / 2
#HIDDEN_END_uxrm
#### Define the left lead. ####
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
#HIDDEN_BEGIN_yliu
lead[(lat(0, j) for j in range(W))] = 4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
-t * sigma_0 + 1j * alpha * sigma_y / 2
# hoppings in y-directions
lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
-t * sigma_0 - 1j * alpha * sigma_x / 2
#HIDDEN_END_yliu
#### Attach the leads and return the finalized system. ####
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst
def plot_conductance(syst, energies):
# Compute conductance
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
data.append(smatrix.transmission(1, 0))
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(energies, data)
- pyplot.xlabel("energy [t]")
- pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.xlabel("energy [t]", fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("conductance [e^2/h]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("spin_orbit_result." + extension, dpi=_defs.dpi)
def main():
syst = make_system()
- # Check that the system looks as intended.
- kwant.plot(syst)
-
# Finalize the system.
syst = syst.finalized()
# We should see non-monotonic conductance steps.
plot_conductance(syst, energies=[0.01 * i - 0.3 for i in range(100)])
# 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()
doc/source/code/figure/superconductor.py.diff deleted 100644 → 0
+ 0
− 148
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@@ -1,132 +1,141 @@
# Tutorial 2.6. "Superconductors": orbitals, conservation laws and symmetries
# ===========================================================================
#
# Physics background
# ------------------
# - conductance of a NS-junction (Andreev reflection, superconducting gap)
#
# Kwant features highlighted
# --------------------------
# - Implementing electron and hole ("orbital") degrees of freedom
# using conservation laws.
# - Use of discrete symmetries to relate scattering states.
+import _defs
import kwant
import tinyarray
import numpy as np
# For plotting
from matplotlib import pyplot
+from contextlib import redirect_stdout
tau_x = tinyarray.array([[0, 1], [1, 0]])
tau_y = tinyarray.array([[0, -1j], [1j, 0]])
tau_z = tinyarray.array([[1, 0], [0, -1]])
#HIDDEN_BEGIN_nbvn
def make_system(a=1, W=10, L=10, barrier=1.5, barrierpos=(3, 4),
mu=0.4, Delta=0.1, Deltapos=4, t=1.0):
# Start with an empty tight-binding system. On each site, there
# are now electron and hole orbitals, so we must specify the
# number of orbitals per site. The orbital structure is the same
# as in the Hamiltonian.
lat = kwant.lattice.square(norbs=2)
syst = kwant.Builder()
#### Define the scattering region. ####
# The superconducting order parameter couples electron and hole orbitals
# on each site, and hence enters as an onsite potential.
# The pairing is only included beyond the point 'Deltapos' in the scattering region.
syst[(lat(x, y) for x in range(Deltapos) for y in range(W))] = (4 * t - mu) * tau_z
syst[(lat(x, y) for x in range(Deltapos, L) for y in range(W))] = (4 * t - mu) * tau_z + Delta * tau_x
# The tunnel barrier
syst[(lat(x, y) for x in range(barrierpos[0], barrierpos[1])
for y in range(W))] = (4 * t + barrier - mu) * tau_z
# Hoppings
syst[lat.neighbors()] = -t * tau_z
#HIDDEN_END_nbvn
#HIDDEN_BEGIN_ttth
#### Define the leads. ####
# Left lead - normal, so the order parameter is zero.
sym_left = kwant.TranslationalSymmetry((-a, 0))
# Specify the conservation law used to treat electrons and holes separately.
# We only do this in the left lead, where the pairing is zero.
lead0 = kwant.Builder(sym_left, conservation_law=-tau_z, particle_hole=tau_y)
lead0[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z
lead0[lat.neighbors()] = -t * tau_z
#HIDDEN_END_ttth
#HIDDEN_BEGIN_zuuw
# Right lead - superconducting, so the order parameter is included.
sym_right = kwant.TranslationalSymmetry((a, 0))
lead1 = kwant.Builder(sym_right)
lead1[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z + Delta * tau_x
lead1[lat.neighbors()] = -t * tau_z
#### Attach the leads and return the system. ####
syst.attach_lead(lead0)
syst.attach_lead(lead1)
return syst
#HIDDEN_END_zuuw
#HIDDEN_BEGIN_jbjt
def plot_conductance(syst, energies):
# Compute conductance
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
# Conductance is N - R_ee + R_he
data.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] -
smatrix.transmission((0, 0), (0, 0)) +
smatrix.transmission((0, 1), (0, 0)))
#HIDDEN_END_jbjt
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("superconductor_transport_result." + extension,
+ dpi=_defs.dpi)
#HIDDEN_BEGIN_pqmp
def check_PHS(syst):
# Scattering matrix
s = kwant.smatrix(syst, energy=0)
# Electron to electron block
s_ee = s.submatrix((0,0), (0,0))
# Hole to hole block
s_hh = s.submatrix((0,1), (0,1))
print('s_ee: \n', np.round(s_ee, 3))
print('s_hh: \n', np.round(s_hh[::-1, ::-1], 3))
print('s_ee - s_hh^*: \n',
np.round(s_ee - s_hh[::-1, ::-1].conj(), 3), '\n')
# Electron to hole block
s_he = s.submatrix((0,1), (0,0))
# Hole to electron block
s_eh = s.submatrix((0,0), (0,1))
print('s_he: \n', np.round(s_he, 3))
print('s_eh: \n', np.round(s_eh[::-1, ::-1], 3))
print('s_he + s_eh^*: \n',
np.round(s_he + s_eh[::-1, ::-1].conj(), 3))
#HIDDEN_END_pqmp
def main():
syst = make_system(W=10)
- # Check that the system looks as intended.
- kwant.plot(syst)
-
# Finalize the system.
syst = syst.finalized()
# Check particle-hole symmetry of the scattering matrix
- check_PHS(syst)
+ with open('check_PHS_out.txt', 'w') as f:
+ with redirect_stdout(f):
+ check_PHS(syst)
# Compute and plot the conductance
plot_conductance(syst, energies=[0.002 * i for i in range(-10, 100)])
# 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()
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