Gross-Neveu-Heisenberg criticality from competing nematic and antiferromagnetic orders in bilayer graphene

We study the phase diagram of an effective model of competing nematic and antiferromagnetic orders of interacting electrons on the Bernal-stacked honeycomb bilayer, as relevant for bilayer graphene. In the noninteracting limit, the model features a semimetallic ground state with quadratic band touching points at the Fermi level. Taking the effects of short-range interactions into account, we demonstrate the existence of an extended region in the mean-field phase diagram characterized by coexisting nematic and antiferromagnetic orders. By means of a renormalization group approach, we reveal that the quantum phase transition from nematic to coexistent nematic-antiferromagnetic orders is continuous and characterized by emergent Lorentz symmetry. It falls into the $(2+1)$-dimensional relativistic Gross-Neveu-Heisenberg quantum universality class, which has recently been much investigated in the context of interacting Dirac systems in two spatial dimensions. The coexistence-to-antiferromagnetic transition, by contrast, turns out to be weakly first order as a consequence of the absence of the continuous spatial rotational symmetry on the honeycomb bilayer. Implications for experiments in bilayer graphene are discussed.

Figure 1

Mean-field phase diagram of interacting electrons on the Bernal-stacked honeycomb bilayer as a function of short-range couplings $g$ and $g^{\prime }$ defined in Sec. 2. Blue and red color codings indicate the magnitudes of the nematic and antiferromagnetic orders, respectively. The electronic spectra near the corners of the hexagonal Brillouin zone are depicted for the different states in the insets. The gray rectangle shows a zoom into the weakly interacting regime, with the dotted gray line indicating the phase boundary between nematic and antiferromagnetic orders in the fermionic RG calculation (Sec. 3). The dashed black line indicates the cut used in Fig. 2. The antiferromagnetic-to-coexistence transition (thick white curve) is weakly first order (Sec. 4). The nematic-to-coexistence transition (thin white curve) is continuous and falls into the Gross-Neveu-Heisenberg universality class (Sec. 5).

Figure 2

(a) Nematic order parameter $\langle n\rangle$ (blue) and antiferromagnetic order parameter $\langle \varphi \rangle$ (red) along the cut through parameter space indicated by the dashed black line in Fig. 1, in the mean-field approximation. In the model with continuous spatial rotational symmetry, both transitions into and out of the coexistence phase are continuous. (b) Nematic order parameter in the vicinity of the antiferromagnet-to-coexistence transition, showing the effects of the $f_3$ term defined in Eq. (10), which breaks the continuous spatial rotational symmetry down to ${120}^{\circ }$ rotations on the honeycomb bilayer. Here, the antiferromagnetic order parameter $\langle \varphi \rangle =0.678{\mathrm{\Lambda }}^2$ has been held constant for simplicity. The inset shows a zoom into the region very close to the transition (gray rectangle), illustrating the fact that finite $f_3\ne 0$ renders the transition weakly first order.

Figure 3

Simplest order-parameter fluctuation correction to the effective potential. The dashed (solid) lines refer to boson (fermion) propagators.

Figure 4

(a) Bosonic and (b) fermionic self-energy Feynman diagrams.

Figure 5

RG flow of Fermi velocities $v_x$ and $v_y$ in units of the boson velocities $c_x$ and $c_y$ for $N_{\text{f}}=4$ and $N_{\text{b}}=3$ at one-loop order. Dark red line represents the rotationally invariant subspace $v:=v_x=v_y$. All points flow ultimately to the relativistic fixed point $v_{★}=1$ (dark red point), even though flow lines initially “fan out” from the $v_x=v_y$ line for small enough initial values.