There is an important model which can be used to describe quantum spin Hall insulators, known as the Bernevig-Hughes-Zhang model or, in short, BHZ model. In essence, this model is equivalent to two copies of the Chern insulator Hamiltonian on the square lattice that we studied in the fourth week.

The BHZ Hamiltonian takes the form

\[ H_\textrm{BHZ}(\mathbf{k}) = \begin{pmatrix} h(\mathbf{k}) & 0 \\ 0 & h^*(-\mathbf{k}) \end{pmatrix}\,, \]

with

\[ h(\mathbf{k}) = \epsilon(\mathbf{k}) + \mathbf{d}(\mathbf{k})\cdot \pmb{\sigma}\,. \]

Here \(\pmb\sigma = (\sigma_x, \sigma_y, \sigma_z)\) is a vector of Pauli matrices acting on the electron/hole degree of freedom (the original two bands of the Chern insulator), \(\epsilon(\mathbf{k}) = C - D(k_x^2+k_y^2)\), the vector \(\mathbf{d} = [A k_x, -A k_y, M(\mathbf{k})]\), and \(M(\mathbf{k}) = M - B(k_x^2+k_y^2)\).

You can see that it is basically two copies of the massive Dirac Hamiltonian we used to study Chern insulators. In particular, there is a linear coupling in momentum between the holes and the electrons. The gap in the Hamiltonian is given by the term \(M(\mathbf{k})\), a momentum-dependent effective mass.

By changing the sign of \(M\) from negative to positive, you get a gap closing at \(\mathbf{k}=\pmb{0}\):

This gap closing turns your trivial insulator into a topologically non-trivial quantum spin Hall insulator.

In the rest of this lecture, we will use the BHZ model as a toy-model to illustrate the behavior of a quantum spin Hall insulator using numerical examples. The BHZ model, however, is more than a toy-model, and it can be used to capture the behavior of some real semiconducting materials. For this reason, the BHZ model will be a main protagonist in the next chapter, where we will discuss real materials and the experimental evidence for the quantum spin Hall effect.