What is the dispersion of the surface state of the \(3D\) topological insulator?

We know that if we fix one momentum (say \(k_x\)) to zero, the Hamiltonian of the remaining system is that of a quantum spin Hall insulator. For this system we know that the Hamiltonian of the edge states is just that of a pair of counter-propagating modes, so

\[H = v \sigma_y k_y.\]

Here, the matrix \(\sigma_y\) acts on the degrees of freedom of these two surface modes, and doesn’t correspond to particle spin.

Since time-reversal symmetry changes the sign of \(k_y\), it must also change the sign of \(\sigma_y\), so the time-reversal operator must be \(\mathcal{T} = i \sigma_y K\).

What if we consider a nonzero \(k_x\)? Generically, the two modes are then coupled by an extra term in the Hamiltonian. This term should be proportional to \(k_x\), and since it couples the modes it must also include a Pauli matrix, which we can just choose to be \(\sigma_x\).

So if the surface of the topological insulator is isotropic, its Hamiltonian is merely

\[H=v \mathbf{\sigma} \cdot \mathbf{k}.\]

Let’s have a quick look at it to get a more concrete understanding:

What you see here is the dispersion of the two lowest energy bands of a thin slice of a 3D BHZ model.

The system is topological when \(M<0\). As expected, the lowest energy state then has a Dirac dispersion, and surface states are formed.

The distinguishing feature of the strong topological insulator is that it has an odd number of Dirac cones in total. In fact, the reason why it is called “strong” is also the reason why an odd number of Dirac cones is special.

To see what is unique, let us add an infinitesimal magnetic field and compute the Chern number of the surface state. We know that the number of Dirac cones is odd. From our study of Chern insulators, we know that the change in the Chern number between \(B = -\varepsilon\) and \(B = +\varepsilon\) is just the number of the Dirac cones. This is because we open a gap at each of them.

Since the Chern number is odd under time reversal, we come to a paradoxical conclusion: if we break time reversal, we end up with a half-integer Hall conductance \(\sigma_{xy} = e^2/h (n + 1/2)\) on a surface of a topological insulator.

This is of course not possible in any purely 2D system, since the Hall conductance must be an integer, and therefore the surface state of a strong topological insulator cannot be created without the topological bulk.

The statement that it is impossible to have a tight-binding Hamiltonian with time-reversal symmetry and a single Dirac cone is known as the “fermion doubling theorem”. There are several tricks that one can perform to work around this limitation in a numerical simulation, but we won’t cover them in the course.