Revisiting the Haldane model: features, implications and extensions.

The Haldane model is regarded as one of the most influential minimal models in topological band theory, demonstrating that a two-dimensional lattice system can exhibit a quantized Hall response without Landau levels and with zero net magnetic flux through the unit cell. In this tutorial review, we revisit the Haldane model from a pedagogical and physically transparent perspective, beginning with the connection between Hall conductivity, Berry curvature, and Chern number. We then construct the honeycomb-lattice Hamiltonian step by step, identifying the roles of nearest-neighbor hopping, staggered sublattice potential, and complex next-nearest-neighbor hopping. Particular emphasis is placed on the competition between the inversion-breaking Semenoff mass and the time-reversal-breaking Haldane mass, which pushes a trivial insulator towards a Chern insulator. We discuss the role of valley-dependent mass inversion in producing a nonzero Chern number, the encoding of the underlying band geometry by the Berry curvature, and the appearance of chiral edge states due to bulk-boundary correspondence. Finally, we place the Haldane model in a broader modern context by connecting it to the Kane-Mele model, modified Haldane systems, antichiral edge physics, and contemporary synthetic platforms such as cold atoms, photonic lattices, acoustic systems, and electrical circuits. By emphasizing the connection between lattice hopping, valley mass inversion, Berry curvature, and boundary transport, this review provides a compact tutorial perspective on the Haldane model and its modern extensions.