Wilson-Fisher fixed points in the 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 ϕ4 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 ϕ4 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 ϵ-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-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 ϵ-expansion, quantum Monte Carlo, large <italic>N</italic>, and functional renormalization-group calculations can be made. [ABSTRACT FROM AUTHOR]