Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order

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 ${\eta }_{\varphi }$, 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 $1/\nu =1.03\left(15\right), {\eta }_{\varphi }=0.42\left(7\right)$, and ${\eta }_{\psi }=0.180\left(10\right)$ from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods.

Figure 1

Quantum phase diagram of the $(2+1)$-dimensional Gross-Neveu-SO(3) model as a function of tuning parameter $m^2$. The theory exhibits a quantum critical point between a Dirac semimetal and a long-range-ordered phase in which two Dirac cones acquire a mass gap, while one remains gapless, as depicted in the insets. In this work, we provide improved estimates for the universal exponents $1/\nu , {\eta }_{\varphi }$, and ${\eta }_{\psi }$, characterizing this universality class.

Figure 2

Skeleton Schwinger-Dyson two-point functions used to determine ${\eta }_{\psi }$ at $\mathcal{O}(1/N^2)$. Dashed inner lines correspond to critical fermion propagators [Eq. (25)] and wiggly inner lines correspond to critical boson propagators [Eq. (26)].

Figure 3

Leading-order skeleton Schwinger-Dyson three-point function used to determine $\chi$ at $\mathcal{O}(1/N)$.

Figure 4

Diagrams contributing to large-$N$ conformal bootstrap formalism to deduce ${\eta }_{\psi }$ at $\mathcal{O}(1/N^3)$. Black dots refer to Polyakov conformal triangles, see Ref. [9] for details.

Figure 5

Critical exponents of the Gross-Neveu-SO(3) universality class as a function of space-time dimension $D$ for $N=3$ flavors of two-component Dirac fermions from three-loop $4-\epsilon$ expansion, second-order $1/N$ expansion (third-order for ${\eta }_{\psi }$), and FRG in $\mathrm{LPA}{16}^{\prime }$ using linear (lin) and sharp (sh) regulators. $[m/n]$ correspond to different Padé approximants.

Figure 6

Same as Fig. 5 but as a function of two-component Dirac fermion flavors $N$ in fixed space-time dimension $D=3$.

Figure 7

Comparison of the results for the boson and fermion anomalous dimensions obtained from the FRG using both the Taylor expansion in $\mathrm{LPA}{16}^{\prime }$ and the pseudospectral decomposition with $n_T=n_R=9$ collocation points. The curves match almost perfectly in the whole range $2<D<4$. Top panels: Linear (lin) cutoff. Bottom panels: Sharp (sh) cutoff. In the sharp cutoff scheme, there is no stable fixed point between $2.104<D<2.366$.