Absence of SO(4) quantum criticality in Dirac semimetals at two-loop order

TL;DR

This paper investigates whether a specific quantum field theory model exhibits critical behavior in 2+1 dimensions and finds that, even at two-loop order, the model does not support a stable fixed point, suggesting no true quantum criticality.

Contribution

The study extends renormalization-group analysis to two-loop order in four dimensions and shows the absence of a stable fixed point in the physical case, clarifying the nature of quantum criticality in Dirac semimetals.

Findings

No stable RG fixed point at two-loop order for Nf=2

Critical flavor numbers approach 2 slowly with epsilon expansion

Continuum theory likely does not exhibit true quantum criticality

Abstract

Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding ($2+1$)-dimensional Gross--Neveu--Yukawa field theory with $N_f=2$ four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors $N_f$, which disappears for a broad range of flavor numbers around the physical case, $N_f=2$, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4-\epsilon$ spacetime dimensions. We determine the $\epsilon$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case $N_f=2$. However, this only happens very slowly, such that an extrapolation to $\epsilon=1$ still suggests the absence of criticality in $2+1$ dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.