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kwant
kwant
Commits
6413c295
Commit
6413c295
authored
Jul 05, 2019
by
Joseph Weston
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whitespace fix
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Pipeline
#19065
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in 38 minutes and 47 seconds
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doc/source/tutorial/magnetic_field.rst
doc/source/tutorial/magnetic_field.rst
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doc/source/tutorial/magnetic_field.rst
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6413c295
...
...
@@ 37,7 +37,7 @@ The eigenstates satisfy :math:`a^\dagger a  n \rangle = n  n \rangle` with the
Landau
level
index
:
math
:`
n
\
geq
0
`,
and
in
coordinate
representation
are
proportional
to
..
math
::
\
psi_n
(
x
,
y
)
=
\
left
(
\
frac
{\
partial
}{
\
partial
w
}

\
frac
{
w
^*}{
4
l_B
^
2
}
\
right
)
w
^
n
e
^{
w
^
2
/
4l
_B
^
2
},
...
...
@@ 47,7 +47,7 @@ with :math:`w = x + i y`. The matrix elements of the ladder operators are
\
langle
n

a

m
\
rangle
=
\
sqrt
{
m
}~\
delta_
{
n
,
m

1
},
\
quad
\
quad
\
langle
n

a
^\
dagger

m
\
rangle
=
\
sqrt
{
m
+
1
}~\
delta_
{
n
,
m
+
1
}.
Truncating
the
basis
to
the
first
:
math
:`
N
`
Landau
levels
allows
us
to
approximate
the
Hamiltonian
as
a
discrete
,
finite
matrix
.
...
...
@@ 56,16 +56,16 @@ We can now formulate the algorithm that Kwant uses to compute Landau levels.
1.
We
take
a
generic
continuum
Hamiltonian
,
written
in
terms
of
the
kinetic
momentum
:
math
:`\
vec
{
k
}`.
The
Hamiltonian
must
be
translationally
invariant
along
the
directions
perpendicular
to
the
field
direction
.
2.
We
substitute
the
momenta
perpendicular
to
the
magnetic
field
with
the
ladder
operators
:
math
:`
a
`
and
:
math
:`
a
^\
dagger
`.
3.
We
construct
a
`
kwant
.
builder
.
Builder
`
using
a
special
lattice
which
includes
the
Landau
level
index
:
math
:`
n
`
as
a
degree
of
freedom
on
each
site
.
The
directions
normal
to
the
field
direction
are
not
included
in
the
builder
,
because
they
are
encoded
in
the
Landau
level
index
.
This
procedure
is
automated
with
`
kwant
.
continuum
.
discretize_landau
`.
This
procedure
is
automated
with
`
kwant
.
continuum
.
discretize_landau
`.
As
an
example
,
let
us
take
the
Bernevig

Hughes

Zhang
model
that
we
first
considered
in
the
discretizer
tutorial
":ref:`discretizebhzmodel`"
:
...
...
@@ 155,7 +155,7 @@ with the Landau levels shown as dashed lines.
h
=
landau_syst
.
hamiltonian_submatrix
(
params
=
params
)
for
ev
in
scipy
.
linalg
.
eigvals
(
h
):
ax
.
axhline
(
ev
,
linestyle
=
''
)
The
dispersion
and
the
Landau
levels
diverge
with
increasing
energy
,
because
the
real
space
discretization
of
the
ribbon
gives
a
worse
approximation
to
the
dispersion
at
higher
energies
.
...
...
@@ 229,4 +229,4 @@ to construct our heterostructure:
.. rubric:: Footnotes
.. [#] `Wikipedia <https://en.wikipedia.org/wiki/Landau_quantization>`_ has
a nice introduction to Landau quantization.
\ No newline at end of file
a nice introduction to Landau quantization.
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