  Two-dimensional spin-orbital magnets with strong exchange frustration have
recently been predicted to facilitate the realization of a quantum critical
point in the Gross-Neveu-SO(3) universality class. In contrast to previously
known Gross-Neveu-type universality classes, this quantum critical point
separates a Dirac semimetal and a long-range-ordered phase, in which the
fermion spectrum is only partially gapped out. Here, we characterize the
quantum critical behavior of the Gross-Neveu-SO(3) universality class by
employing three complementary field-theoretical techniques beyond their leading
orders. We compute the correlation-length exponent $\nu$, the order-parameter
anomalous dimension $\eta_\phi$, and the fermion anomalous dimension
$\eta_\psi$ using a three-loop $\epsilon$ expansion around the upper critical
space-time dimension of four, a second-order large-$N$ expansion (with the
fermion anomalous dimension obtained even at the third order), as well as a
functional renormalization group approach in the improved local potential
approximation. For the physically relevant case of $N=3$ flavors of
two-component Dirac fermions in 2+1 space-time dimensions, we obtain the
estimates $1/\nu = 1.03(15)$, $\eta_\phi = 0.42(7)$, and $\eta_\psi =
0.180(10)$ from averaging over the results of the different techniques, with
the displayed uncertainty representing the degree of consistency among the
three methods.
