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].

The effect of particle-hole fluctuations for the BCS-BEC crossover is investigated by use of functional renormalization. We compute the critical temperature for the whole range in the scattering length a. On the BCS side for small negative a we recover the Gorkov approximation, while on the BEC side of small positive a the particle-hole fluctuations play no important role, and we find a system of interacting bosons. In the unitarity limit of infinite scattering length our quantitative estimate yields Tc/TF = 0.264. We also investigate the crossover from broad to narrow Feshbach resonances -for the later we obtain Tc/TF = 0.204 for a −1 = 0. A key ingredient for our treatment is the computation of the momentum dependent four-fermion vertex and its bosonization in terms of an effective bound-state exchange.

Gapless Dirac fermions appear as quasiparticle excitations in various condensed-matter systems. They feature quantum critical points with critical behavior in the 2+1 dimensional Gross-Neveu universality class. The precise determination of their critical exponents defines a prime benchmark for complementary theoretical approaches, such as lattice simulations, the renormalization group and the conformal bootstrap. Despite promising recent developments in each of these methods, however, no satisfactory consensus on the fermionic critical exponents has been achieved, so far. Here, we perform a comprehensive analysis of the Ising Gross-Neveu universality classes based on the recently achieved four-loop perturbative calculations. We combine the perturbative series in 4 − spacetime dimensions with the one for the purely fermionic Gross-Neveu model in 2+ dimensions by employing polynomial interpolation as well as two-sided Padé approximants. Further, we provide predictions for the critical exponents exploring various resummation techniques following the strategies developed for the three-dimensional scalar O(n) universality classes. We give an exhaustive appraisal of the current situation of Gross-Neveu universality by comparison to other methods. For large enough number of spinor components N ≥ 8 as well as for the case of emergent supersymmetry N = 1, we find our renormalization group estimates to be in excellent agreement with the conformal bootstrap, building a strong case for the validity of these values. For intermediate N as well as in comparison with recent Monte Carlo results, deviations are found and critically discussed. arXiv:1806.04977v1 [cond-mat.str-el]

In view of the measured Higgs mass of 125 GeV, the perturbative renormalization group evolution of the Standard Model suggests that our Higgs vacuum might not be stable. We connect the usual perturbative approach and the functional renormalization group which allows for a straightforward inclusion of higher-dimensional operators in the presence of an ultraviolet cutoff. In the latter framework we study vacuum stability in the presence of higher-dimensional operators. We find that their presence can have a sizable influence on the maximum ultraviolet scale of the Standard Model and the existence of instabilities. Finally, we discuss how such operators can be generated in specific models and study the relation between the instability scale of the potential and the scale of new physics required to avoid instabilities.

We investigate the instabilities of interacting electrons on the honeycomb bilayer by means of the functional renormalization group for a range of interactions up to the third-nearest neighbor. Besides a novel instability toward a gapless charge-density wave we find that using interaction parameters as determined by ab-initio calculations for graphene and graphite puts the system close to the boundary between antiferromagnetic and quantum spin Hall instabilities. Importantly, the energy scales for these instabilities are large such that imperfections and deviations from the basic model are expected to play a major role in real bilayer graphene, where interaction effects seem to be seen only at smaller scales. We therefore analyze how reducing the critical scale and small doping of the layers affect the instabilities.Comment: 5 pages, 4 figure

We study the triviality and hierarchy problem of a Z2-invariant Yukawa system with massless fermions and a real scalar field, serving as a toy model for the standard-model Higgs sector. Using the functional RG, we look for UV stable fixed points which could render the system asymptotically safe. Whether a balancing of fermionic and bosonic contributions in the RG flow induces such a fixed point depends on the algebraic structure and the degrees of freedom of the system. Within the region of parameter space which can be controlled by a nonperturbative next-to-leading order derivative expansion of the effective action, we find no non-Gaußian fixed point in the case of one or more fermion flavors. The fermion-boson balancing can still be demonstrated within a model system with a small fractional flavor number in the symmetry-broken regime. The UV behavior of this small-N f system is controlled by a conformal Higgs expectation value. The system has only two physical parameters, implying that the Higgs mass can be predicted. It also naturally explains the heavy mass of the top quark, since there are no RG trajectories connecting the UV fixed point with light top masses.