What is the simplest possible topological Bloch function?

Kohmoto (1985) pointed out in Topological Invariant and the Quantization of the Hall Conductance how TKNN's calcuation of Hall conducance is related to topology, in which topologically nontriviality is said to be equivalent to impossiblility choosing a global phase of Bloch function $u_k (r)$ in Brillouin zone. As shown in the Figure, we can choose two distinct gauges in sector I and II, and the curvature is the loop integral of phase mismatch on boundary $\partial H$.

What is the simplest possible Bloch function that is

• topologically nontrivial, and
• an eigenstate of Bloch Hamiltonian?

Bloch Hamiltonian: $H(k_x,k_y) = \frac{1}{2m}(-i\partial + {\bf k}+e{\bf A}(x,y))^2 + U(x,y)$ where $U$ is lattice periodic.

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