Extended Haldane model- a modern gateway to topological insulators

The seminal Haldane model brings up a paradigm beyond the quantum Hall effect to look for a plethora of topological phases in the honeycomb and other lattices. Here we dwell into this model considering a full parameter space in the presence of spin-orbit interaction as well as Zeeman field such that the flavour of Kane-Mele model is invoked. Adopting this extended Haldane model as an example, we elucidate, in a transparent manner, a number of topological features in a pedagogical manner. First, we describe various first order topological insulator phases and their characterizations while explaining various anomalous quantum Hall effects and quantum spin Hall effects in the extended Haldane model. Second, we demonstrate the concepts of higher order topological insulator phases along with the topological invariants in the anisotropic limit of the extended Haldane model. At the end, we discuss various open issues involving emergent or extended symmetries that might lead to a broader understanding of various topological phases and the associated criteria behind their emergence.