Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry

Abstract

We investigate a class of relativistic fermion theories in 2 < d < 4 space-time dimensions with continuous chiral U(N f ) × U(N f ) symmetry. This includes a number of well-studied models, e.g., of Gross-Neveu and Thirring type, in a unified framework. Within the limit of pointlike interactions, the RG flow of couplings reveals a network of interacting fixed points, each of which defines a universality class. A subset of fixed points are "critical fixed points" with one RG relevant direction being candidates for critical points of second-order phase transitions. Identifying invariant hyperplanes of the RG flow and classifying their attractive/repulsive properties, we find evidence for emergent higher chiral symmetries as a function of N f . For the case of the Thirring model, we discover a new critical flavor number that separates the RG stable large-N f regime from an intermediate-N f regime in which symmetry-breaking perturbations become RG relevant. This new critical flavor number has to be distinguished from the chiral-critical flavor number, below which the Thirring model is expected to allow spontaneous chiral symmetry breaking, and its existence offers a resolution to the discrepancy between previous results obtained in the continuum and the lattice Thirring models. Moreover, we find indications for a new feature of universality: details of the critical behavior can depend on additional "spectator symmetries" that remain intact across the phase transition. Implications for the physics of interacting fermions on the honeycomb lattice, for which our theory space provides a simple model, are given.

Four loop renormalization of the Gross-Neveu model

“…Such a study would not be an isolated investigation. For instance, given the functional renormalization group analyses of more general Gross-Neveu models, [66], having the four loop perturbative renormalization group functions for that generalization would be useful for refining the fixed point structure.…”

Momentum dependence of quantum critical Dirac systems

“…(18b). This fact is used in many studies of the Gross-Neveu model to restrict the theory space to just the Gross-Neveu vertex, even though, in principle, also the Thirring vertex has the same symmetries as the irreducible Gross-Neveu model; in the case of the reducible Gross-Neveu model, there are even three additional vertices compatible with the reduced symmetries of the model 42 . With regard to the flow equations (13) of the momentum-dependent model, we already observe that this property no longer holds true in the general momentum-dependent case, because there are two diagrams of order g 2 GN contributing to the flow of g Th .…”

“…In fact, the FRG as formulated in terms of the Wetterich equation 87 has already widely been used for the two models under consideration. Common approximation schemes focus on local fermionic interactions 31,32,42,82 , or use partial-bosonization techniques together with a derivative expansion to also account for the emerging composite bosonic degrees of freedom 14,17,24,40,50,65,68,74,[88][89][90] . Whereas these methods work very well for the Gross-Neveu model, exhibiting apparent convergence for the quantitative estimates of critical exponents 68 , the Thirring model already on this level is more involved, requiring composite scalar and vector boson fields 74 as well as dynamical bosonization techniques [91][92][93] .…”

Four-Fermion Theories with Exact Chiral Symmetry in Three Dimensions

“…CSB is possible for N f = 1, but the existence of the lattice artefact phase does not allow for a final conclusion. Motivated by [19] we also studied a model with interaction ± …”