# Momentum dependence of quantum critical Dirac systems

## TL;DR

This paper investigates quantum critical behavior in 2+1D fermionic systems, focusing on the Gross-Neveu and Thirring models, using an advanced FRG approach with momentum-dependent couplings to clarify their universality and fixed points.

## Contribution

It introduces a novel FRG analysis with momentum-dependent couplings for fermionic models, providing new insights into the Thirring model's critical behavior and fixed points.

## Key findings

- Gross-Neveu universality class results are stable across methods.
- Momentum dependencies are crucial in Thirring models at small flavor numbers.
- Non-Gaussian fixed points exist for large flavor numbers, enabling a continuum limit.

## Abstract

We analyze fermionic criticality in relativistic 2+1 dimensional fermion systems using the functional renormalization group (FRG), concentrating on the Gross-Neveu (chiral Ising) and the Thirring model. While a variety of methods, including the FRG, appear to reach quantitative consensus for the critical regime of the Gross-Neveu model, the situation seems more diverse for the Thirring model with different methods yielding vastly different results. We present a first exploratory FRG study of such fermion systems including momentum-dependent couplings using pseudo-spectral methods. Our results corroborate the stability of results in Gross-Neveu-type universality classes, but indicate that momentum dependencies become more important in Thirring-type models for small flavor numbers. For larger flavor numbers, we confirm the existence of a non-Gaussian fixed point and thus a physical continuum limit. In the large-$N$ limit, we obtain an analytic solution for the momentum dependence of the fixed-point vertex.