821 Haldane-Model and numerical determination of the Chern number
Before the experimental isolation of graphene in 2004 by Nobel Prize winners Andre Geim and Konstantin Novoselov [1], the honeycomb lattice only existed as a theoretical model for a single layer graphite crystal. Nevertheless, it proved to be highly useful since in a tight-binding approach the honeycomb lattice features a band touching at the two unequivalent Brillouin-zone corners K and K′, leading to so-called Dirac cones and low-energy excitations that behave as massless Dirac Fermions. This single layer graphite lattice also stroked the interest of the condensed-matter community in 1988, when F. D. M. Haldane published a paper [2] in which he proposes a tight-binding model that exhibits the quantized Hall effect, or Quantum Hall effect(QHE), without the consideration of Landau levels. By introducing a complex next-nearest neighbor (NNN) hopping amplitude which breaks the time reversal symmetry of the system, Haldane showed that a gapped band structure can carry a nontrivial topological invariant, the Chern number. Thus, the model is the first realization of a topological insulator without Landau levels. The goal of this theoretical advanced lab project is to understand the model which Haldane proposed, whilst deepening the understanding of a tight-binding approach in more complex systems, as well as an introduction to the field of topological solid-state theory. In 2016 F. D. M. Haldane received the Nobel Prize for all his notable contributions to the field of topological condensed matter theory.
