Wilson-Fisher fixed points in presence of Dirac fermions

Wilson-Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the $\phi ^4$ field theories that describe low-energy physics of the canonical models such as Ising, XY, and Heisenberg. Here I review its application to a class of the Gross-Neveu-Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the $\phi^4$ field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term, and which still exhibit critical fixed points in the suitably formulated Wilson-Fisher $\epsilon$-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-N\'eel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art $\epsilon$-expansion, quantum Monte Carlo, large-N, and functional renormalization group calculations can be made.