Unconventional Gross-Neveu quantum criticality: Interaction-induced SO(3)-biadjoint insulator and emergent SU(3) symmetry

I point out that Gross-Neveu theory with SO(3) isospin in three space-time dimensions—proposed recently, for instance, as an effective description of the Néel transition in certain spin-orbital liquids—also hosts quantum criticality of a more exotic kind. The ordered phase breaks SO(3) spontaneously, but the SO(3)-Néel order parameter vanishes. The fermionic bilinear order parameter is instead a biadjoint with respect to SO(3); unlike its Néel cousin, it constitutes an interaction-induced insulator. Furthermore, I show that the Néel and biadjoint order parameters can be combined to transform as an adjoint under SU(3) symmetry; the symmetry is emergent at the critical point separating the symmetric semimetal and the biadjoint insulator, but only if the flavor number is small enough, suggesting order-parameter fluctuations and the interplay between different channels play a crucial role in stabilizing the enlarged symmetry. In candidate SO(3) spin-orbital liqiuds, thermodynamic critical exponents carry fingerprints of “spinons.” The existence of an independent universality class in addition to the Néel transition opens the possibility of posing further constraints on spinon properties from thermodynamic measurements near criticality alone.

Figure 1

Top row: phase diagrams as a function of four-Fermi couplings [see Eq. (4) for definitions] for interaction-induced transitions from symmetric semimetal (SM) to Lorentz invariant SO(3) isospin-broken phases in three-dimensional Gross-Neveu theory; insets show the fermions' single-particle spectrum, along with the number of Nambu-Goldstone bosons (NGBs) where applicable. Bottom row: corresponding RG phase portraits, with fixed points [projected to the $(g_{01},g_{02})$ plane] denoted by solid circles and (projections of) RG-invariant subspaces denoted by heavy dashed lines; flow lines are schematic and not to scale. Left: $N_{\text{f}}=9$. In this regime, the mean-field picture is qualitatively valid. In addition to the usual Néel $[≔SO{\left(3\right)}_1]$ transition, there is a transition to an SO(3)-biadjoint insulator $[≔SO{\left(3\right)}_2]$ governed by the finite-$N_{\text{f}}$ descendant of the large-$N_{\text{f}}$ fixed point; the SU(3) symmetric subspace is IR unstable. Right: $N_{\text{f}}=6$. While the stability of ${\mathrm{SO}\left(3\right)}_1$ remains unchanged, SU(3) symmetry emerges for $N_{\text{f}}<N_{\text{f},\text{cr}}\approx 6.5$ in the IR at a putative GN-${\mathrm{SO}\left(3\right)}_2$ transition. The fermions' single-particle spectrum is qualitatively the same, but robust qualitative differences in the ${\mathrm{SO}\left(3\right)}_2$ vis-à-vis SU(3) scenarios include a distinct number of NGBs. There is a bicritical fixed point (white) not adiabatically connected to any mean-field limit; for $9<N_{\text{f}}<6$, it collides first with GN-${\mathrm{SO}\left(3\right)}_2$ and then GN-SU(3), realizing thus a transfer of stability between the two fixed points, before settling down between GN-SU(3) and GN-${\mathrm{SO}\left(3\right)}_1$ to form the phase boundary between the ${\mathrm{SO}\left(3\right)}_1$ and SU(3) phases.

Figure 2

Plot of the effective potential of the order parameter $\varphi ={\varphi }_A{Q}_A$ for $g_{02}<g_{02,\text{cr}}$ (left) and $g_{02}>g_{02,\text{cr}}$ (right) in the frame $\left({\varphi }_A\right)=(0,0,0,{\varphi }_4,{\varphi }_5)$ in units of $V_{\text{eff}}\left({\varphi }^{\text{ref}}\right)-V_{\text{eff}}\left(\langle \varphi \rangle \right)$, where ${\varphi }^{\text{ref}}=(0,0,0,1/4,1/4)$ is an arbitrarily chosen reference point. The minimum is located at $(0,\pm v)$ up to rotations in the $({\varphi }_4,{\varphi }_5)$ plane by $n\pi /6$ ($n\in \mathbb{Z}$); note that the corresponding operation on $(Q_4,Q_5)$ sends the matrix $Q_5$ to one that is unitarily equivalent. The critical value of the coupling $g_{02,\text{cr}}$ can only be determined from an explicit computation; see discussion below Eq. (8). All dimensionful quantities are measured in units of the UV cutoff $k_{\text{UV}}$, which corresponds (roughly) to the lattice constant.

Figure 3

Values of four-Fermi couplings at fixed points ${\mathcal{Q}}_{{\text{SO(3)}}_{\ell }}$ ($\ell =1,2$) and ${\mathcal{Q}}_{\text{SU(3)}}$, showing the nearly single-channel nature of these fixed points persists down to small $N_{\text{f}}$.