Haldane model and Chern insulators
Complex next-neighbor hopping can generate a quantum Hall-like phase without net magnetic flux per unit cell.
$$H(k)=d_0(k)I+\mathbf d(k)\cdot\boldsymbol\sigma$$
What this means
Complex next-neighbor hopping can generate a quantum Hall-like phase without net magnetic flux per unit cell.
Deep dive: mathematical and physical mechanism
On a honeycomb lattice, staggered sublattice mass and complex next-nearest-neighbor hoppings compete to set Dirac masses at two valleys. Their signs determine whether the occupied band has zero or nonzero Chern number. A topological transition requires a gap closing at a valley.
How it appears in quantum phenomena
Cold-atom optical lattices and engineered photonic lattices emulate Haldane-type models.
Assumptions, caveats and limitations
A vanishing net unit-cell flux does not mean microscopic time-reversal symmetry is preserved.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.