Last week we understood the quantum Hall effect in terms of a pumping argument that we attributed to Laughlin.

Our pumping argument involved putting our system on a cylinder and adiabatically pumping a magnetic flux \(\Phi\) through the cylinder so that the Hamiltonian returns to itself. The flux enters the Hamiltonian through minimal substitution as \(H(\mathbf{k})\rightarrow H(\mathbf{k}+e\mathbf{A})\) where \({\bf A}=\hat{\mathbf{y}}\,\Phi/L\).

Thus we can understand the effects of flux pumping on the Hamiltonian in terms of a change in momentum. When the flux is changed by the appropriate number of quanta, the momentum \(\mathbf{k}\) changes by a reciprocal lattice vector and, hence the Bloch Hamiltonian returns to its original value. To simplify the discussion, in the following we will use a square Brillouin zone, with \(k_x\) and \(k_y\) defined in an interval \([0, 2\pi]\), but all our arguments also apply for the hexagonal Brillouin zone of graphene.

Let’s imagine the adiabatic time-evolution of an eigenstate \(\left|\psi(\mathbf{k})\right\rangle\) of this Hamiltonian, with energy \(E(\mathbf{k})\), as \(\mathbf{k}\) is changed slowly. Suppose the Hamiltonian is such that \(\left|\psi(\mathbf{k})\right\rangle\) remains non-degenerate as in the case of the Haldane model. We can then adiabatically explore an energy band by moving \(\mathbf{k}\), without the risk of encountering a level crossing. After a while, let’s say a time \(T\), we bring \(\mathbf{k}\) back to its initial value after going around the entire Brillouin Zone. For instance, we can consider the following closed path \(C\), where \(k_y\) changes by \(2\pi\) at a fixed \(k_x\), starting from \(k_y=0\):

We then ask: what is the final quantum state at the time \(T\)? For a long time people guessed that it would just be given by the initial state \(\left|\psi(k_x, k_y+2\pi)\right\rangle\equiv\left|\psi(k_x, k_y)\right\rangle\) times the usual phase \(\exp\left(-i \int_0^T E[\mathbf{k}(t)]\,d t\right)\), which an eigenstate of the Hamiltonian accumulates with time.

This would be rather boring. Berry instead realized that for a closed loops there is an additional phase \(\gamma\), which in our case may depend on \(k_x\):

\[\gamma(C) = \oint_C\,\mathbf{A}(\mathbf{k})\,\cdot d\mathbf{k}\,.\]

Here, \(\mathbf{A}(\mathbf{k})=i\left\langle\,\psi(\mathbf{k}) \,|\,\nabla_\mathbf{k}\,\psi(\mathbf{k})\right\rangle\) is a vector with two complex entries, which are obtained by taking the derivatives of \(\left|\psi(\mathbf{k})\right\rangle\) with respect to \(k_x\) and \(k_y\) and then taking the inner product with \(\left\langle\psi(\mathbf{k})\right|\). This vector goes by the rather obscure name of Berry connection. In our example, the final quantum state at the end of the cycle is thus

\[\exp\,\left[i\gamma(k_x)\right]\,\exp\,\left(-i \int_0^T E[\mathbf{k}(t)]\,d t\right)\,\left|\psi(\mathbf{k})\right\rangle\,.\]

We have made explicit the fact that \(\gamma\) in our case may depend on \(k_x\). We will not derive the formula for the Berry phase, something which can be done directly from the Schrödinger equation, see for instance here. What is important to know about \(\gamma\) is that it is a geometric phase: its value depends on the path \(C\) but not on how the path is performed in time, so not on the particular expression for \(\mathbf{k}(t)\). We’ll soon see that sometimes it can have an even stronger, topological character.

Flux pumping

The phase \(\gamma(k_x)\) must bear information about the charge pumped during an adiabatic cycle over \(k_y\). Now we take advantage of pumped charge being invariant as long as the energy gap is preserved. This means that we have the freedom to change the energy dispersion \(E(k_x,k_y)\) arbitrarily, as long as we do not close the gap.

It is convenient to make the energy dispersion completely flat along the \(k_x\) direction for \(k_y=0\), analogous to the case of Landau levels. In this way, since at fixed \(k_y\) all the wave functions have the same energy, we can choose our initial quantum state to be localized in a single unit cell in the \(x\) direction,

\[\left|\psi(n,t=0)\right\rangle=\int_0^{2\pi} dk_x\, e^{i k_x n}\,\left|\psi(k_x, k_y=0)\right\rangle\,.\]

Starting from this state, after one adiabatic cycle we obtain

\[\left|\psi(n,t=T)\right\rangle=\int_0^{2\pi} dk_x\, e^{i k_x n}\,\exp\,\left[i\gamma(k_x)-i\theta(k_x)\right]\,\left|\psi(k_x, k_y=2\pi)\right\rangle,\]

where \(\theta(k_x)=\int_0^T E[k_x, k_y(t)]\,d t\) is the dynamical phase. Now we notice something strange. While \(\theta(k_x)\) is a truly periodic function of \(k_x\) because \(E(k_x)=E(k_x+2\pi)\), the only restriction on the Berry phase \(\gamma(k_x)\) is to be periodic modulo \(2\pi\). That is, we can have \(\gamma(k_x+2\pi)=\gamma(k_x)+2\pi W\) with \(W\) an integer number.

Let’s try to deform the dispersion along \(k_y\) in order to make the combination \(\gamma(k_x)-\theta(k_x)\) as large as possible (just like before, this is allowed as long as we do not close the gap). The best we can do is choose \(\theta(k_x)\) so that

\[\gamma(k_x)-\theta(k_x)=W k_x.\]

Plugging this in to the form of the wave-function we see that

\[\left|\psi(n,t=T)\right\rangle=\int dk_x e^{i k_x (n+W)}\,\left|\psi(k_x, k_y=0)\right\rangle,\]

which means that every wave function is shifted over by \(W\) unit cells. Thus the system with the wave functions \(\left|\psi(\mathbf{k})\right\rangle\) pumps \(W\) units of charge if the Berry phase satisfies

\[\gamma(k_x+2\pi)-\gamma(k_x)=2\pi W.\]

The quantity \(W\) is called the Chern number and is the topological invariant characterizing the bandstructure of two dimensional quantum Hall systems. Because it is an integer, it cannot be changed by any continuous deformation of the Hamiltonian, provided the gap does not close. The Chern number is in fact the bulk topological invariant for all insulators with broken time-reversal symmetry. If \(W=0\), we have a topologically trivial insulator with no chiral edge states. If \(W=n\) there are \(n\) chiral edge states at the boundary of the insulator.