2.11. Adding magnetic field

Computing Landau levels in a harmonic oscillator basis

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The complete source code of this example can be found in landau-levels.py

When electrons move in an external magnetic field, their motion perpendicular to the field direction is quantized into discrete Landau levels. Kwant implements an efficient scheme for computing the Landau levels of arbitrary continuum Hamiltonians. The general scheme revolves around rewriting the Hamiltonian in terms of a basis of harmonic oscillator states [1], and is commonly illustrated in textbooks for quadratic Hamiltonians.

To demonstrate the general scheme, let us consider a magnetic field oriented along the \(z\) direction \(\vec{B} = (0, 0, B)\), such that electron motion in the \(xy\) plane is Landau quantized. The magnetic field enters the Hamiltonian through the kinetic momentum

\[\hbar \vec{k} = - i \hbar \nabla + e\vec{A}(x, y).\]

In the symmetric gauge \(\vec{A}(x, y) = (B/2)[-y, x, 0]\), we introduce ladder operators with the substitution

\[k_x = \frac{1}{\sqrt{2} l_B} (a + a^\dagger), \quad \quad k_y = \frac{i}{\sqrt{2} l_B} (a - a^\dagger),\]

with the magnetic length \(l_B = \sqrt{\hbar/eB}\). The ladder operators obey the commutation relation

\[[a, a^\dagger] = 1,\]

and define a quantum harmonic oscillator. We can thus write any electron continuum Hamiltonian in terms of \(a\) and \(a^\dagger\). Such a Hamiltonian has a simple matrix representation in the eigenbasis of the number operator \(a^\dagger a\). The eigenstates satisfy \(a^\dagger a | n \rangle = n | n \rangle\) with the integer Landau level index \(n \geq 0\), and in coordinate representation are proportional to

\[\psi_n (x, y) = \left( \frac{\partial}{ \partial w} - \frac{w^*}{4 l_B^2} \right) w^n e^{-|w|^2/4l_B^2},\]

with \(w = x + i y\). 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 \(N\) Landau levels allows us to approximate the Hamiltonian as a discrete, finite matrix.

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 \(\vec{k}\). 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 \(a\) and \(a^\dagger\).

3. We construct a kwant.builder.Builder using a special lattice which includes the Landau level index \(n\) 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.

As an example, let us take the Bernevig-Hughes-Zhang model that we first considered in the discretizer tutorial “Models with more structure: Bernevig-Hughes-Zhang”:

\[C + M σ_0 \otimes σ_z + F(k_x^2 + k_y^2) σ_0 \otimes σ_z + D(k_x^2 + k_y^2) σ_0 \otimes σ_0 + A k_x σ_z \otimes σ_x + A k_y σ_0 \otimes σ_y.\]

We can discretize this Hamiltonian in a basis of Landau levels as follows

import numpy as np
import scipy.linalg
from matplotlib import pyplot

import kwant
import kwant.continuum

import matplotlib
import matplotlib.pyplot
from matplotlib_inline.backend_inline import set_matplotlib_formats

matplotlib.rcParams['figure.figsize'] = matplotlib.pyplot.figaspect(1) * 2
set_matplotlib_formats('svg')

hamiltonian = """
   + C * identity(4) + M * kron(sigma_0, sigma_z)
   - F * (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)
"""

syst = kwant.continuum.discretize_landau(hamiltonian, N=10)
syst = syst.finalized()

We can then plot the spectrum of the system as a function of magnetic field, and observe a crossing of Landau levels at finite magnetic field near zero energy, characteristic of a quantum spin Hall insulator with band inversion.

params = dict(A=3.645, F =-68.6, D=-51.2, M=-0.01, C=0)
b_values = np.linspace(0.0001, 0.0004, 200)

fig = kwant.plotter.spectrum(syst, ('B', b_values), params=params, show=False)
pyplot.ylim(-0.1, 0.2);

Comparing with tight-binding

In the limit where fewer than one quantum of flux is threaded through a plaquette of the discretization lattice we can compare the discretization in Landau levels with a discretization in realspace.

lat = kwant.lattice.square(norbs=1)
syst = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))

def peierls(to_site, from_site, B):
    y = from_site.tag[1]
    return -1 * np.exp(-1j * B * y)

syst[(lat(0, j) for j in range(-19, 20))] = 4
syst[lat.neighbors()] = -1
syst[kwant.HoppingKind((1, 0), lat)] = peierls
syst = syst.finalized()

landau_syst = kwant.continuum.discretize_landau("k_x**2 + k_y**2", N=5)
landau_syst = landau_syst.finalized()

Here we plot the dispersion relation for the discretized ribbon and compare it with the Landau levels shown as dashed lines.

fig, ax = pyplot.subplots(1, 1)
ax.set_xlabel("momentum")
ax.set_ylabel("energy")
ax.set_ylim(0, 1)

params = dict(B=0.1)

kwant.plotter.bands(syst, ax=ax, params=params)

h = landau_syst.hamiltonian_submatrix(params=params)
for ev in scipy.linalg.eigvalsh(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.

Discretizing 3D models

Although the preceding examples have only included the plane perpendicular to the magnetic field, the Landau level quantization also works if the direction parallel to the field is included. In fact, although the system must be translationally invariant in the plane perpendicular to the field, the system may be arbitrary along the parallel direction. For example, it is therefore possible to model a heterostructure and/or set up a scattering problem along the field direction.

Let’s say that we wish to to model a heterostructure with a varying potential \(V\) along the direction of a magnetic field, \(z\), that includes Zeeman splitting and Rashba spin-orbit coupling:

\[\frac{\hbar^2}{2m}\sigma_0(k_x^2 + k_y^2 + k_z^2) + V(z)\sigma_0 + \frac{\mu_B B}{2}\sigma_z + \hbar\alpha(\sigma_x k_y - \sigma_y k_x).\]

We can discretize this Hamiltonian in a basis of Landau levels as before:

continuum_hamiltonian = """
    (k_x**2 + k_y**2 + k_z**2) * sigma_0
    + V(z) * sigma_0
    + mu * B * sigma_z / 2
    + alpha * (sigma_x * k_y - sigma_y * k_x)
"""

template = kwant.continuum.discretize_landau(continuum_hamiltonian, N=10)

This creates a system with a single translational symmetry, along the \(z\) direction, which we can use as a template to construct our heterostructure:

def hetero_structure(site):
    z, = site.pos
    return 0 <= z < 10

def hetero_potential(z):
    if z < 2:
      return 0
    elif z < 7:
      return 0.5
    else:
      return 0.7

syst = kwant.Builder()
syst.fill(template, hetero_structure, (0,))

syst = syst.finalized()

params = dict(
    B=0.5,
    mu=0.2,
    alpha=0.4,
    V=hetero_potential,
)

syst.hamiltonian_submatrix(params=params);

Footnotes

[1]

Wikipedia has a nice introduction to Landau quantization.