diff --git a/doc/source/tutorial/magnetic_field.rst b/doc/source/tutorial/magnetic_field.rst index 6bb6fdcdc91a10bc36a66b59c13e2cdee6fcc2a6..5bf242e7c6422b446858b4e797764204872ad449 100644 --- a/doc/source/tutorial/magnetic_field.rst +++ b/doc/source/tutorial/magnetic_field.rst @@ -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:`discretize-bhz-model`": @@ -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.