Understanding Dirac, Weyl and nodal-line semimetals: a step-by-step guide through model Hamiltonians

Graphene's remarkable two-dimensional nature, protected by time-reversal and space-inversion symmetries, has revolutionized our understanding of electronic band structures. Graphene, as the prototypical semimetal, has inspired a surge of interest in searching for exotic electronic states in three-dimensional materials through simple yet powerful low-energy effective Hamiltonians. In this pedagogical work, we extend these ideas to explore the electronic band structures with the exact solution of density of states, and effective masses of three fascinating classes of topological semimetals-Dirac semimetals (DSM), Weyl semimetals (WSM), and nodal line semimetals (NLS). By adopting model Hamiltonians, we present a clear and accessible framework that invites beginners to engage with the rich physics of these systems, while also connecting theoretical insights to experimental realizations. We have also provided interesting exercises (in the SI) with proper hints to extend the technique further and go beyond what will be learnt here. We hope that this study will not only deepen appreciation for the emergence of novel band structures in modern condensed matter research, but also encourage educators to bring research-inspired perspectives into the classroom. Such integration can enrich undergraduate education by exposing students early to cutting-edge concepts and methodologies, nurturing curiosity and innovation at the interface of learning and discovery.