We study the chiral Ising, the chiral XY, and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in 4 − ϵ dimensions and compute critical exponents for the GrossNeveu-Yukawa fixed points to order Oðϵ 4 Þ. Further, we provide Padé estimates for the correlation length exponent, the boson and fermion anomalous dimension, as well as the leading correction to scaling exponent in 2 þ 1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with N ¼ 1=4 and N ¼ 1=2 fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

We investigate the properties of the nearest-neighbor singlet pairing and the emergence of dwave superconductivity in the doped honeycomb lattice considering the limit of large interactions and the t − J1 − J2 model. First, by applying a renormalized mean-field procedure as well as slave-boson theories which account for the proximity to the Mott insulating state, we confirm the emergence of d-wave superconductivity in agreement with earlier works. We show that a small but finite J2 spin coupling between next-nearest neighbors stabilizes d-wave symmetry compared to the extended s-wave scenario. At small hole doping, to minimize energy and to gap the whole Fermi surface or all the Dirac points, the superconducting ground state is characterized by a d + id singlet pairing assigned to one valley and a d − id singlet pairing to the other, which then preserves timereversal symmetry. The slightly doped situation is distinct from the heavily doped case (around 3/8 and 5/8 filling) supporting a pure chiral d + id symmetry and breaking time-reversal symmetry. Then, we apply the functional Renormalization Group and we study in more detail the competition between antiferromagnetism and superconductivity in the vicinity of half-filling. We discuss possible applications to strongly-correlated compounds with Copper hexagonal planes such as In3Cu2VO9. Our findings are also relevant to the understanding of exotic superfluidity with cold atoms.

Dirac and Weyl fermions appear as quasi-particle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this work we study the bosonized version of the standard Gross-Neveu model -the Gross-Neveu-Yukawa theory -at three-loop order, and compute critical exponents in 4−ǫ dimensions for general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent super-scaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Padé approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.Introduction. Dirac and Weyl fermions are an abundant form of quasi-particle excitations in condensed matter physics[1, 2] appearing in very different materials ranging from graphene, via d-wave superconductors, to the surface states of topological insulators, or even three-dimensional (3D) materials such as Na 3 Bi and Cd 3 As 2 [3,4]. While the physical origin of the quasirelativistic energy dispersion can be quite different in various materials, it leads to universal low-energy properties shared by all these materials, such as, e.g., the density of states (DOS) and the concomitant thermodynamic properties or various response functions. Dirac and Weyl systems can also undergo transitions from their natural semi-metallic phase to a variety of broken symmetry phases as some parameter is varied. This includes continuous quantum transitions to interaction-induced, ordered, many-body ground states[5-10] and a disorderdriven transition to a diffusive state with finite DOS [11][12][13][14][15][16]. Depending on the broken symmetry of the ordered phase and the fermionic content, the corresponding transition represents a universality class with the concomitant critical behavior. Various of these transitions have been suggested to belong to the universality class defined by the chiral transition appearing in the 3D GrossNeveu (GN) model [7] well-known in the context of highenergy physics and conformal field theories [17,18]. This applies, in particular, to the interaction-induced transition toward a charge density wave of electrons on the 2D honeycomb lattice that breaks the (Ising) sublattice symmetry [7], or the disorder-driven transition toward a diffusive metal in a 3D Weyl semi-metal [15].