def test_closest():
rng = ensure_rng(10)
for sym_dim in range(1, 4):
for space_dim in range(sym_dim, 4):
lat = kwant.lattice.general(ta.identity(space_dim))
# Choose random periods.
while True:
periods = rng.randint(-10, 11, (sym_dim, space_dim))
if np.linalg.det(np.dot(periods, periods.T)) > 0.1:
# Periods are reasonably linearly independent.
break
syst = builder.Builder(kwant.TranslationalSymmetry(*periods))
for tag in rng.randint(-30, 31, (4, space_dim)):
# Add site and connect it to the others.
old_sites = list(syst.sites())
new_site = lat(*tag)
syst[new_site] = None
syst[((new_site, os) for os in old_sites)] = None
# Test consistency with fill().
for point in 200 * rng.random_sample((10, space_dim)) - 100:
closest = syst.closest(point)
dist = closest.pos - point
dist = ta.dot(dist, dist)
syst2 = builder.Builder()
syst2.fill(syst, inside_disc(point, 2 * dist), closest)
assert syst2.closest(point) == closest
for site in syst2.sites():
dd = site.pos - point
dd = ta.dot(dd, dd)
assert dd >= 0.999999 * dist
def act(self, element, a, b=None):
m_part = self._get_site_family_data(a.family)[0]
try:
delta = ta.dot(m_part, element)
except ValueError:
msg = 'Expecting a {0}-tuple group element, but got `{1}` instead.'
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta = -delta
if b is None:
return builder.Site(a.family, a.tag + delta, True)
elif b.family == a.family:
return (builder.Site(a.family, a.tag + delta, True),
builder.Site(b.family, b.tag + delta, True))
else:
m_part = self._get_site_family_data(b.family)[0]
try:
delta2 = ta.dot(m_part, element)
except ValueError:
msg = ('Expecting a {0}-tuple group element, '
'but got `{1}` instead.')
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta2 = -delta2
return (builder.Site(a.family, a.tag + delta, True),
builder.Site(b.family, b.tag + delta2, True))
def act(self, element, a, b=None):
element = ta.array(element)
if element.dtype is not int:
raise ValueError("group element must be a tuple of integers")
m_part = self._get_site_family_data(a.family)[0]
try:
delta = ta.dot(m_part, element)
except ValueError:
msg = 'Expecting a {0}-tuple group element, but got `{1}` instead.'
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta = -delta
if b is None:
return builder.Site(a.family, a.tag + delta, True)
elif b.family == a.family:
return (builder.Site(a.family, a.tag + delta, True),
builder.Site(b.family, b.tag + delta, True))
else:
m_part = self._get_site_family_data(b.family)[0]
try:
delta2 = ta.dot(m_part, element)
except ValueError:
msg = ('Expecting a {0}-tuple group element, '
'but got `{1}` instead.')
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta2 = -delta2
return (builder.Site(a.family, a.tag + delta, True),
builder.Site(b.family, b.tag + delta2, True))
def test_dot():
# Check acceptance of non-tinyarray arguments.
assert ta.dot([1, 2], (3, 4)) == 11
for dtype in dtypes:
shape_pairs = [(1, 1), (2, 2), (3, 3),
(0, 0),
(0, (0, 1)), ((0, 1), 1),
(0, (0, 2)), ((0, 2), 2),
(1, (1, 2)), ((2, 1), 1),
(2, (2, 1)), ((1, 2), 2),
(2, (2, 3)), ((3, 2), 2),
((1, 1), (1, 1)), ((2, 2), (2, 2)),
((3, 3), (3, 3)), ((2, 3), (3, 2)), ((2, 1), (1, 2)),
((2, 3, 4), (4, 3)),
((2, 3, 4), 4),
((3, 4), (2, 4, 3)),
(4, (2, 4, 3))]
# We have to use almost_equal here because the result of numpy's dot
# does not always agree to the last bit with a naive implementation.
# (This is probably due to their usage of SSE or parallelization.)
#
# On my machine in summer 2012 with Python 2.7 and 3.2 the program
#
# import numpy as np
# a = np.array([13.2, 14.3, 15.4, 16.5])
# b = np.array([-5.0, -3.9, -2.8, -1.7])
# r = np.dot(a, b)
# rr = sum(x * y for x, y in zip(a, b))
# print(r - rr)
#
# outputs 2.84217094304e-14.
for sa, sb in shape_pairs:
a = make(sa, dtype)
b = make(sb, dtype) - 5
assert_almost_equal(ta.dot(ta.array(a), ta.array(b)), np.dot(a, b),
13)
shape_pairs = [((), 2), (2, ()),
(1, 2),
(1, (2, 2)), ((1, 1), 2),
((2, 2), (3, 2)),
((2, 3, 2), (4, 3)),
((2, 3, 4), 3),
((3, 3), (2, 4, 3)),
(3, (2, 4, 3))]
for sa, sb in shape_pairs:
a = make(sa, dtype)
b = make(sb, dtype) - 5
raises(ValueError, ta.dot, ta.array(a.tolist()),
ta.array(b.tolist()))
raises(ValueError, ta.dot, ta.array(a), ta.array(b))
def test_dot():
# Check acceptance of non-tinyarray arguments.
assert_equal(ta.dot([1, 2], (3, 4)), 11)
for dtype in dtypes:
shape_pairs = [(1, 1), (2, 2), (3, 3),
(0, 0),
(0, (0, 1)), ((0, 1), 1),
(0, (0, 2)), ((0, 2), 2),
(1, (1, 2)), ((2, 1), 1),
(2, (2, 1)), ((1, 2), 2),
(2, (2, 3)), ((3, 2), 2),
((1, 1), (1, 1)), ((2, 2), (2, 2)),
((3, 3), (3, 3)), ((2, 3), (3, 2)), ((2, 1), (1, 2)),
((2, 3, 4), (4, 3)),
((2, 3, 4), 4),
((3, 4), (2, 4, 3)),
(4, (2, 4, 3))]
# We have to use almost_equal here because the result of numpy's dot
# does not always agree to the last bit with a naive implementation.
# (This is probably due to their usage of SSE or parallelization.)
#
# On my machine in summer 2012 with Python 2.7 and 3.2 the program
#
# import numpy as np
# a = np.array([13.2, 14.3, 15.4, 16.5])
# b = np.array([-5.0, -3.9, -2.8, -1.7])
# r = np.dot(a, b)
# rr = sum(x * y for x, y in zip(a, b))
# print(r - rr)
#
# outputs 2.84217094304e-14.
for sa, sb in shape_pairs:
a = make(sa, dtype)
b = make(sb, dtype) - 5
assert_almost_equal(ta.dot(ta.array(a), ta.array(b)), np.dot(a, b),
13)
shape_pairs = [((), 2), (2, ()),
(1, 2),
(1, (2, 2)), ((1, 1), 2),
((2, 2), (3, 2)),
((2, 3, 2), (4, 3)),
((2, 3, 4), 3),
((3, 3), (2, 4, 3)),
(3, (2, 4, 3))]
for sa, sb in shape_pairs:
a = make(sa, dtype)
b = make(sb, dtype) - 5
assert_raises(ValueError, ta.dot, ta.array(a.tolist()),
ta.array(b.tolist()))
assert_raises(ValueError, ta.dot, ta.array(a), ta.array(b))
def __init__(self, prim_vecs, basis, name='', norbs=None):
prim_vecs = ta.array(prim_vecs, float)
if prim_vecs.ndim != 2:
raise ValueError('`prim_vecs` must be a 2d array-like object.')
dim = prim_vecs.shape[1]
if name is None:
name = ''
if isinstance(name, str):
name = [name + str(i) for i in range(len(basis))]
if prim_vecs.shape[0] > dim:
raise ValueError('Number of primitive vectors exceeds '
'the space dimensionality.')
basis = ta.array(basis, float)
if basis.ndim != 2:
raise ValueError('`basis` must be a 2d array-like object.')
if basis.shape[1] != dim:
raise ValueError('Basis dimensionality does not match '
'the space dimensionality.')
try:
norbs = list(norbs)
if len(norbs) != len(basis):
raise ValueError('Length of `norbs` is not the same as '
'the number of basis vectors')
except TypeError:
norbs = [norbs] * len(basis)
self.sublattices = [Monatomic(prim_vecs, offset, sname, norb)
for offset, sname, norb in zip(basis, name, norbs)]
# Sequence of primitive vectors of the lattice.
self._prim_vecs = prim_vecs
# Precalculation of auxiliary arrays for real space calculations.
self.reduced_vecs, self.transf = lll.lll(prim_vecs)
self.voronoi = ta.dot(lll.voronoi(self.reduced_vecs), self.transf)
def __init__(self, prim_vecs, basis, name='', norbs=None):
prim_vecs = ta.array(prim_vecs, float)
_check_prim_vecs(prim_vecs)
dim = prim_vecs.shape[1]
if name is None:
name = ''
if isinstance(name, str):
name = [name + str(i) for i in range(len(basis))]
basis = ta.array(basis, float)
if basis.ndim != 2:
raise ValueError('`basis` must be a 2d array-like object.')
if basis.shape[1] != dim:
raise ValueError('Basis dimensionality does not match '
'the space dimensionality.')
try:
norbs = list(norbs)
if len(norbs) != len(basis):
raise ValueError('Length of `norbs` is not the same as '
'the number of basis vectors')
except TypeError:
norbs = [norbs] * len(basis)
self.sublattices = [
Monatomic(prim_vecs, offset, sname, norb)
for offset, sname, norb in zip(basis, name, norbs)
]
# Sequence of primitive vectors of the lattice.
self._prim_vecs = prim_vecs
# Precalculation of auxiliary arrays for real space calculations.
self.reduced_vecs, self.transf = lll.lll(prim_vecs)
self.voronoi = ta.dot(lll.voronoi(self.reduced_vecs), self.transf)
def which(self, site):
det_x_inv_m_part, det_m = self._get_site_family_data(site.family)[-2:]
if isinstance(site, system.Site):
result = ta.dot(det_x_inv_m_part, site.tag) // det_m
elif isinstance(site, system.SiteArray):
result = np.dot(det_x_inv_m_part, site.tags.transpose()) // det_m
else:
raise TypeError("'site' must be a Site or a SiteArray")
return -result if self.is_reversed else result
def subgroup(self, *generators):
"""Return the subgroup generated by a sequence of group elements.
Parameters
----------
*generators: sequence of int
Each generator must have length ``self.num_directions``.
"""
generators = ta.array(generators)
if generators.dtype != int:
raise ValueError('Generators must be sequences of integers.')
return TranslationalSymmetry(*ta.dot(generators, self.periods))
def vec(self, int_vec):
"""
Return the coordinates of a Bravais lattice vector in real space.
Parameters
----------
vec : integer vector
Returns
-------
output : real vector
"""
return ta.dot(int_vec, self._prim_vecs)
def vec(self, int_vec):
"""
Return the coordinates of a Bravais lattice vector in real space.
Parameters
----------
vec : integer vector
Returns
-------
output : real vector
"""
return ta.dot(int_vec, self._prim_vecs)
def act(self, element, a, b=None):
is_site = isinstance(a, system.Site)
# Tinyarray for small arrays (single site) else numpy
array_mod = ta if is_site else np
element = array_mod.array(element)
if not np.issubdtype(element.dtype, np.integer):
raise ValueError("group element must be a tuple of integers")
if (len(element.shape) == 2 and is_site):
raise ValueError("must provide a single group element when "
"acting on single sites.")
if (len(element.shape) == 1 and not is_site):
# We can act on a whole SiteArray with a single group element
# using numpy broadcasting.
element = element.reshape(1, -1)
m_part = self._get_site_family_data(a.family)[0]
try:
delta = array_mod.dot(m_part, element)
except ValueError:
msg = 'Expecting a {0}-tuple group element, but got `{1}` instead.'
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta = -delta
if b is None:
if is_site:
return system.Site(a.family, a.tag + delta, True)
else:
return system.SiteArray(a.family, a.tags + delta.transpose())
elif b.family == a.family:
if is_site:
return (system.Site(a.family, a.tag + delta, True),
system.Site(b.family, b.tag + delta, True))
else:
return (system.SiteArray(a.family, a.tags + delta.transpose()),
system.SiteArray(b.family, b.tags + delta.transpose()))
else:
m_part = self._get_site_family_data(b.family)[0]
try:
delta2 = ta.dot(m_part, element)
except ValueError:
msg = ('Expecting a {0}-tuple group element, '
'but got `{1}` instead.')
raise ValueError(msg.format(self.num_directions, element))
if self.is_reversed:
delta2 = -delta2
if is_site:
return (system.Site(a.family, a.tag + delta, True),
system.Site(b.family, b.tag + delta2, True))
else:
return (system.SiteArray(a.family, a.tags + delta.transpose()),
system.SiteArray(b.family,
b.tags + delta2.transpose()))
def has_subgroup(self, other):
if isinstance(other, system.NoSymmetry):
return True
elif not isinstance(other, TranslationalSymmetry):
raise ValueError("Unknown symmetry type.")
if other.periods.shape[1] != self.periods.shape[1]:
return False # Mismatch of spatial dimensionalities.
inv = np.linalg.pinv(self.periods)
factors = np.dot(other.periods, inv)
# Absolute tolerance is correct in the following since we want an error
# relative to the closest integer.
return (np.allclose(factors, np.round(factors), rtol=0, atol=1e-8)
and np.allclose(ta.dot(factors, self.periods), other.periods))
def f(*args, **kwargs):
a, b, *args = args
assert not (args and kwargs)
sym_b = sym.act(elem, b)
k = [kwargs[k] for k in momenta] if kwargs else args[mnp:]
phase = cmath.exp(1j * ta.dot(elem, k))
if not callable(val):
v = val
elif kwargs:
if not takes_kwargs:
kwargs = {p: kwargs[p] for p in extra_params}
v = val(a, sym_b, **kwargs)
else:
v = val(a, sym_b, *args[:mnp])
return phase * v
def __init__(self, prim_vecs, offset=None, name='', norbs=None):
prim_vecs = ta.array(prim_vecs, float)
if prim_vecs.ndim != 2:
raise ValueError('``prim_vecs`` must be a 2d array-like object.')
dim = prim_vecs.shape[1]
if name is None:
name = ''
if prim_vecs.shape[0] > dim:
raise ValueError('Number of primitive vectors exceeds '
'the space dimensionality.')
if offset is None:
offset = ta.zeros(dim)
else:
offset = ta.array(offset, float)
if offset.shape != (dim, ):
raise ValueError('Dimensionality of offset does not match '
'that of the space.')
msg = '{0}({1}, {2}, {3}, {4})'
cl = self.__module__ + '.' + self.__class__.__name__
canonical_repr = msg.format(cl, short_array_repr(prim_vecs),
short_array_repr(offset), repr(name),
repr(norbs))
super().__init__(canonical_repr, name, norbs)
self.sublattices = [self]
self._prim_vecs = prim_vecs
self.inv_pv = ta.array(np.linalg.pinv(prim_vecs))
self.offset = offset
# Precalculation of auxiliary arrays for real space calculations.
self.reduced_vecs, self.transf = lll.lll(prim_vecs)
self.voronoi = ta.dot(lll.voronoi(self.reduced_vecs), self.transf)
self.dim = dim
self.lattice_dim = len(prim_vecs)
if name != '':
msg = "<Monatomic lattice {0}{1}>"
orbs = ' with {0} orbitals'.format(
self.norbs) if self.norbs else ''
self.cached_str = msg.format(name, orbs)
else:
msg = "<unnamed Monatomic lattice, vectors {0}, origin [{1}]{2}>"
orbs = ', with {0} orbitals'.format(norbs) if norbs else ''
self.cached_str = msg.format(short_array_str(self._prim_vecs),
short_array_str(self.offset), orbs)
def __init__(self, prim_vecs, offset=None, name='', norbs=None):
prim_vecs = ta.array(prim_vecs, float)
if prim_vecs.ndim != 2:
raise ValueError('``prim_vecs`` must be a 2d array-like object.')
dim = prim_vecs.shape[1]
if name is None:
name = ''
if prim_vecs.shape[0] > dim:
raise ValueError('Number of primitive vectors exceeds '
'the space dimensionality.')
if offset is None:
offset = ta.zeros(dim)
else:
offset = ta.array(offset, float)
if offset.shape != (dim,):
raise ValueError('Dimensionality of offset does not match '
'that of the space.')
msg = '{0}({1}, {2}, {3}, {4})'
cl = self.__module__ + '.' + self.__class__.__name__
canonical_repr = msg.format(cl, short_array_repr(prim_vecs),
short_array_repr(offset),
repr(name), repr(norbs))
super().__init__(canonical_repr, name, norbs)
self.sublattices = [self]
self._prim_vecs = prim_vecs
self.inv_pv = ta.array(np.linalg.pinv(prim_vecs))
self.offset = offset
# Precalculation of auxiliary arrays for real space calculations.
self.reduced_vecs, self.transf = lll.lll(prim_vecs)
self.voronoi = ta.dot(lll.voronoi(self.reduced_vecs), self.transf)
self.dim = dim
self.lattice_dim = len(prim_vecs)
if name != '':
msg = "<Monatomic lattice {0}{1}>"
orbs = ' with {0} orbitals'.format(self.norbs) if self.norbs else ''
self.cached_str = msg.format(name, orbs)
else:
msg = "<unnamed Monatomic lattice, vectors {0}, origin [{1}]{2}>"
orbs = ', with {0} orbitals'.format(norbs) if norbs else ''
self.cached_str = msg.format(short_array_str(self._prim_vecs),
short_array_str(self.offset), orbs)
def _mul(R1, R2):
# Cached multiplication of spatial parts.
if is_sympy_matrix(R1) and is_sympy_matrix(R2):
# If spatial parts are sympy matrices, use cached multiplication.
R = R1 * R2
elif not (is_sympy_matrix(R1) or is_sympy_matrix(R2)):
# If arrays, use dot
R = ta.dot(R1, R2)
elif ((is_sympy_matrix(R1) or is_sympy_matrix(R2)) and
(isinstance(R1, ta.ndarray_int) or isinstance(R2, ta.ndarray_int))):
# Multiplying sympy and integer tinyarray is ok, should result in sympy
R = sympy.ImmutableMatrix(R1) * sympy.ImmutableMatrix(R2)
else:
raise ValueError("Mixing of sympy and floating point in the spatial part R is not allowed. "
"To avoid this error, make sure that all PointGroupElements are initialized "
"with either floating point arrays or sympy matrices as rotations. "
"Integer arrays are allowed in both cases.")
R = _make_int(R)
return R
def disk(site):
pos = site.pos
return ta.dot(pos, pos) < 13
def halfplane(site):
return ta.dot(site.pos - (-1, 1), (-0.9, 0.63)) > 0
def vec_norm(vec):
return np.sqrt(ta.dot(vec, vec))
def test_fill():
g = kwant.lattice.square()
sym_x = kwant.TranslationalSymmetry((-1, 0))
sym_xy = kwant.TranslationalSymmetry((-1, 0), (0, 1))
template_1d = builder.Builder(sym_x)
template_1d[g(0, 0)] = None
template_1d[g.neighbors()] = None
def line_200(site):
return -100 <= site.pos[0] < 100
## Test that copying a builder by "fill" preserves everything.
for sym, func in [
(kwant.TranslationalSymmetry(*np.diag([3, 4, 5])), lambda pos: True),
(builder.NoSymmetry(), lambda pos: ta.dot(pos, pos) < 17)
]:
cubic = kwant.lattice.general(ta.identity(3))
# Make a weird system.
orig = kwant.Builder(sym)
sites = cubic.shape(func, (0, 0, 0))
for i, site in enumerate(orig.expand(sites)):
if i % 7 == 0:
continue
orig[site] = i
for i, hopp in enumerate(orig.expand(cubic.neighbors(1))):
if i % 11 == 0:
continue
orig[hopp] = i * 1.2345
for i, hopp in enumerate(orig.expand(cubic.neighbors(2))):
if i % 13 == 0:
continue
orig[hopp] = i * 1j
# Clone the original using fill.
clone = kwant.Builder(sym)
clone.fill(orig, lambda s: True, (0, 0, 0))
# Verify that both are identical.
assert set(clone.site_value_pairs()) == set(orig.site_value_pairs())
assert (set(clone.hopping_value_pairs()) == set(
orig.hopping_value_pairs()))
## Test for warning when "start" is out.
target = builder.Builder()
for start in [(-101, 0), (101, 0)]:
with warns(RuntimeWarning):
target.fill(template_1d, line_200, start)
## Test filling of infinite builder.
for n in [1, 2, 4]:
sym_n = kwant.TranslationalSymmetry((n, 0))
for start in [g(0, 0), g(20, 0)]:
target = builder.Builder(sym_n)
sites = target.fill(template_1d,
lambda s: True,
start,
max_sites=10)
assert len(sites) == n
assert len(list(target.hoppings())) == n
assert set(sym_n.to_fd(s) for s in sites) == set(target.sites())
## test max_sites
target = builder.Builder()
for max_sites in (-1, 0):
with raises(ValueError):
target.fill(template_1d,
lambda site: True,
g(0, 0),
max_sites=max_sites)
assert len(list(target.sites())) == 0
target = builder.Builder()
with raises(RuntimeError):
target.fill(template_1d, line_200, g(0, 0), max_sites=10)
## test filling
target = builder.Builder()
added_sites = target.fill(template_1d, line_200, g(0, 0))
assert len(added_sites) == 200
# raise warning if target already contains all starting sites
with warns(RuntimeWarning):
target.fill(template_1d, line_200, g(0, 0))
## test multiplying unit cell size in 1D
n_cells = 10
sym_nx = kwant.TranslationalSymmetry(*(sym_x.periods * n_cells))
target = builder.Builder(sym_nx)
target.fill(template_1d, lambda site: True, g(0, 0))
should_be_syst = builder.Builder(sym_nx)
should_be_syst[(g(i, 0) for i in range(n_cells))] = None
should_be_syst[g.neighbors()] = None
assert sorted(target.sites()) == sorted(should_be_syst.sites())
assert sorted(target.hoppings()) == sorted(should_be_syst.hoppings())
## test multiplying unit cell size in 2D
template_2d = builder.Builder(sym_xy)
template_2d[g(0, 0)] = None
template_2d[g.neighbors()] = None
template_2d[builder.HoppingKind((2, 2), g)] = None
nm_cells = (3, 5)
sym_nmxy = kwant.TranslationalSymmetry(*(sym_xy.periods * nm_cells))
target = builder.Builder(sym_nmxy)
target.fill(template_2d, lambda site: True, g(0, 0))
should_be_syst = builder.Builder(sym_nmxy)
should_be_syst[(g(i, j) for i in range(10) for j in range(10))] = None
should_be_syst[g.neighbors()] = None
should_be_syst[builder.HoppingKind((2, 2), g)] = None
assert sorted(target.sites()) == sorted(should_be_syst.sites())
assert sorted(target.hoppings()) == sorted(should_be_syst.hoppings())
## test filling 0D builder with 2D builder
def square_shape(site):
x, y = site.tag
return 0 <= x < 10 and 0 <= y < 10
target = builder.Builder()
target.fill(template_2d, square_shape, g(0, 0))
should_be_syst = builder.Builder()
should_be_syst[(g(i, j) for i in range(10) for j in range(10))] = None
should_be_syst[g.neighbors()] = None
should_be_syst[builder.HoppingKind((2, 2), g)] = None
assert sorted(target.sites()) == sorted(should_be_syst.sites())
assert sorted(target.hoppings()) == sorted(should_be_syst.hoppings())
## test that 'fill' respects the symmetry of the target builder
lat = kwant.lattice.chain(a=1)
template = builder.Builder(kwant.TranslationalSymmetry((-1, )))
template[lat(0)] = 2
template[lat.neighbors()] = -1
target = builder.Builder(kwant.TranslationalSymmetry((-2, )))
target[lat(0)] = None
to_target_fd = target.symmetry.to_fd
# Refuses to fill the target because target already contains the starting
# site.
with warns(RuntimeWarning):
target.fill(template, lambda x: True, lat(0))
# should only add a single site (and hopping)
new_sites = target.fill(template, lambda x: True, lat(1))
assert target[lat(0)] is None # should not be overwritten by template
assert target[lat(-1)] == template[lat(0)]
assert len(new_sites) == 1
assert to_target_fd(new_sites[0]) == to_target_fd(lat(-1))
def f(a, *args):
phase = cmath.exp(1j * ta.dot(elem, args[mnp:]))
v = val(a, sym.act(elem, a), *args[:mnp]) if callable(val) else val
pv = phase * v
return pv + herm_conj(pv)
def f(a, b, *args):
phase = cmath.exp(1j * ta.dot(elem, args[mnp:]))
v = val(a, sym.act(elem, b), *args[:mnp]) if callable(val) else val
return phase * v
def make_lattice(a, theta):
x = ta.dot(R(theta), (a, 0))
y = ta.dot(R(theta), (0, a))
return kwant.lattice.general([x, y], norbs=1)
def pos(self, tag):
"""Return the real-space position of the site with a given tag."""
return ta.dot(tag, self._prim_vecs) + self.offset
def f(*args):
a, b, *args = args
phase = cmath.exp(1j * ta.dot(elem, args[mnp:]))
v = val(a, sym.act(elem, b), *args[:mnp]) if callable(val) else val
return phase * v
def which(self, site):
det_x_inv_m_part, det_m = self._get_site_family_data(site.family)[-2:]
result = ta.dot(det_x_inv_m_part, site.tag) // det_m
return -result if self.is_reversed else result
def f(*args):
a, *args = args
phase = cmath.exp(1j * ta.dot(elem, args[mnp:]))
v = val(a, sym.act(elem, a), *args[:mnp]) if callable(val) else val
pv = phase * v
return pv + herm_conj(pv)
def pos(self, tag):
"""Return the real-space position of the site with a given tag."""
return ta.dot(tag, self._prim_vecs) + self.offset
def wire_shape(pos):
rel_pos = pos - center
projection = rel_pos - direction * ta.dot(direction, rel_pos)
return sum(projection * projection) <= r_squared
def shape(site):
d = site.pos - center
dd = ta.dot(d, d)
return dd <= rr
def make_lattice(a, salt='0'):
theta_x = kwant.digest.uniform('x', salt=salt) * np.pi / 6
theta_y = kwant.digest.uniform('y', salt=salt) * np.pi / 6
x = ta.dot(R(theta_x), (a, 0))
y = ta.dot(R(theta_y), (0, a))
return kwant.lattice.general([x, y], norbs=1)
def plot_2d_bands(syst, k_x=31, k_y=31, params=None,
mask_brillouin_zone=False, extend_bbox=0, file=None,
show=True, dpi=None, fig_size=None, ax=None):
"""Plot 2D band structure of a wrapped around system.
This function is primarily useful for systems that have translational
symmetry vectors that are non-orthogonal (e.g. graphene). This function
will properly plot the band structure in an orthonormal basis in k-space,
as opposed to in the basis of reciprocal lattice vectors (which would
produce a "skewed" Brillouin zone).
If your system has orthogonal lattice vectors, you are probably better
off using `kwant.plotter.spectrum`.
Parameters
----------
syst : `kwant.system.FiniteSystem`
A 2D system that was finalized from a Builder produced by
`kwant.wraparound.wraparound`. Note that this *must* be a finite
system; so `kwant.wraparound.wraparound` should have been called with
``keep=None``.
k_x, k_y : int or sequence of float, default: 31
Either a number of sampling points, or a sequence of points at which
the band structure is to be evaluated, in units of inverse length.
params : dict, optional
Dictionary of parameter names and their values, not including the
momentum parameters.
mask_brillouin_zone : bool, default: False
If True, then the band structure will only be plotted over the first
Brillouin zone. By default the band structure is plotted over a
rectangular bounding box that contains the Brillouin zone.
extend_bbox : float, default: 0
Amount by which to extend the region over which the band structure is
plotted, expressed as a proportion of the Brillouin zone bounding box
length. i.e. ``extend_bbox=0.1`` will extend the region by 10% (in all
directions).
file : string or file object, optional
The output file. If None, output will be shown instead.
show : bool, default: False
Whether ``matplotlib.pyplot.show()`` is to be called, and the output is
to be shown immediately. Defaults to `True`.
dpi : float, optional
Number of pixels per inch. If not set the ``matplotlib`` default is
used.
fig_size : tuple, optional
Figure size `(width, height)` in inches. If not set, the default
``matplotlib`` value is used.
ax : ``matplotlib.axes.Axes`` instance, optional
If `ax` is not `None`, no new figure is created, but the plot is done
within the existing Axes `ax`. in this case, `file`, `show`, `dpi`
and `fig_size` are ignored.
Returns
-------
fig : matplotlib figure
A figure with the output if `ax` is not set, else None.
Notes
-----
This function produces plots where the units of momentum are inverse
length. This is contrary to `kwant.plotter.bands`, where the units
of momentum are inverse lattice constant.
If the lattice vectors for the symmetry of ``syst`` are not orthogonal,
then part of the plotted band structure will be outside the first Brillouin
zone (inside the bounding box of the brillouin zone). Setting
``mask_brillouin_zone=True`` will cause the plot to be truncated outside of
the first Brillouin zone.
See Also
--------
kwant.plotter.spectrum
"""
if not hasattr(syst, '_wrapped_symmetry'):
raise TypeError("Expecting a system that was produced by "
"'kwant.wraparound.wraparound'.")
if not isinstance(syst, system.FiniteSystem):
msg = ("All symmetry directions must be wrapped around: specify "
"'keep=None' when calling 'kwant.wraparound.wraparound'.")
raise TypeError(msg)
params = params or {}
lat_ndim, space_ndim = syst._wrapped_symmetry.periods.shape
if lat_ndim != 2:
raise ValueError("Expected a system with a 2D translational symmetry.")
if space_ndim != lat_ndim:
raise ValueError("Lattice dimension must equal realspace dimension.")
# columns of B are lattice vectors
B = np.array(syst._wrapped_symmetry.periods).T
# columns of A are reciprocal lattice vectors
A = np.linalg.pinv(B).T
## calculate the bounding box for the 1st Brillouin zone
# Get lattice points that neighbor the origin, in basis of lattice vectors
reduced_vecs, transf = lll.lll(A.T)
neighbors = ta.dot(lll.voronoi(reduced_vecs), transf)
# Add the origin to these points.
klat_points = np.concatenate(([[0] * lat_ndim], neighbors))
# Transform to cartesian coordinates and rescale.
# Will be used in 'outside_bz' function, later on.
klat_points = 2 * np.pi * np.dot(klat_points, A.T)
# Calculate the Voronoi cell vertices
vor = scipy.spatial.Voronoi(klat_points)
around_origin = vor.point_region[0]
bz_vertices = vor.vertices[vor.regions[around_origin]]
# extract bounding box
k_max = np.max(np.abs(bz_vertices), axis=0)
## build grid along each axis, if needed
ks = []
for k, km in zip((k_x, k_y), k_max):
k = np.array(k)
if not k.shape:
if extend_bbox:
km += km * extend_bbox
k = np.linspace(-km, km, k)
ks.append(k)
# TODO: It is very inefficient to call 'momentum_to_lattice' once for
# each point (for trivial Hamiltonians 60% of the time is spent
# doing this). We should instead transform the whole grid in one call.
def momentum_to_lattice(k):
k, residuals = scipy.linalg.lstsq(A, k)[:2]
if np.any(abs(residuals) > 1e-7):
raise RuntimeError("Requested momentum doesn't correspond"
" to any lattice momentum.")
return k
def ham(k_x, k_y=None, **params):
# transform into the basis of reciprocal lattice vectors
k = momentum_to_lattice([k_x] if k_y is None else [k_x, k_y])
p = dict(zip(syst._momentum_names, k), **params)
return syst.hamiltonian_submatrix(params=p, sparse=False)
def outside_bz(k_x, k_y, **_):
dm = scipy.spatial.distance_matrix(klat_points, [[k_x, k_y]])
return np.argmin(dm) != 0 # is origin no closest 'klat_point' to 'k'?
fig = plotter.spectrum(ham,
x=('k_x', ks[0]),
y=('k_y', ks[1]) if lat_ndim == 2 else None,
params=params,
mask=(outside_bz if mask_brillouin_zone else None),
file=file, show=show, dpi=dpi,
fig_size=fig_size, ax=ax)
return fig
def which(self, site):
det_x_inv_m_part, det_m = self._get_site_family_data(site.family)[-2:]
result = ta.dot(det_x_inv_m_part, site.tag) // det_m
return -result if self.is_reversed else result
lat = kwant.lattice.triangular()
sys = kwant.Builder()
sys[lat.shape(circle, (0, 0))] = 0
sys[lat.neighbors()] = 1
lead_dirs = [lat.vec((-3, 1)), lat.vec((0, -1)), lat.vec((2, -1))]
for d in lead_dirs:
lead = kwant.Builder(kwant.TranslationalSymmetry(d))
lead[lat.wire((0, 0), 2.1)] = 0
lead[lat.neighbors()] = 1
sys.attach_lead(lead)
fig = kwant.plot(sys, show=False)
ax = pyplot.gca()
pyplot.text(-2, 5.5, 'scattering region', size=15)
pyplot.text(-10, -1, 'lead 0', color='red', size=15)
pyplot.text(-3, -7.7, 'lead 1', color='red', size=15)
pyplot.text(5.5, 0, 'lead 2', color='red', size=15)
for dir, offset in zip(lead_dirs, [10.5, 8, 8.6]):
dir = dir / math.sqrt(ta.dot(dir, dir))
for i in [0, 0.4, 0.8]:
ax.add_artist(Circle(dir * (offset + i), 0.06, fc='k'))
pyplot.axis('off')
pyplot.xlim((-11, 9))
pyplot.ylim((-8, 6))
fig.tight_layout()
fig.set_size_inches(*(5, 3.75))
fig.savefig("tbsys.pdf")
def wire_shape(pos):
rel_pos = pos - center
projection = rel_pos - direction * ta.dot(direction, rel_pos)
return sum(projection * projection) <= r_squared
