We review the application of lattice QCD techniques, most notably the Hybrid
Monte-Carlo (HMC) simulations, to first-principle study of tight-binding models
of crystalline solids with strong inter-electron interactions. After providing
a basic introduction into the HMC algorithm as applied to condensed matter
systems, we review HMC simulations of graphene, which in the recent years have
helped to understand the semi-metal behavior of clean suspended graphene at the
quantitative level. We also briefly summarize other novel physical results
obtained in these simulations. Then we comment on the applicability of Hybrid
Monte-Carlo to topological insulators and Dirac and Weyl semi-metals and
highlight some of the relevant open physical problems. Finally, we also touch
upon the lattice strong-coupling expansion technique as applied to condensed
matter systems.Comment: 20 pages, 5 figures, Contribution to IJMPA special issue "Lattice
gauge theory beyond QCD". List of references update
