Interaction-Induced Dirac Fermions from Quadratic Band Touching in Bilayer Graphene

We revisit the effect of local interactions on the quadratic band touching (QBT) of the Bernal honeycomb bilayer model using renormalization group (RG) arguments and quantum Monte Carlo (QMC) simulations. We present a RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead, they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased QMC simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite $U/t$ despite the $U=0$ hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with $(2+1)\mathrm{D}$ Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.

Figure 1

Two schematics for the renormalization group flows for the Bernal stacked bilayer in the space of $\alpha$ (linear term in the dispersion) and $g$ (generic quartic interactions) [cf. Eq. (1) for definitions]. The fixed points are $Q$, quadratic band touching; $D$, Dirac linear dispersion; and GN, Gross-Neveu. (a) The RG flow implied by previous studies. In this scenario, weak interactions always flow to strong coupling when one begins with a QBT, i.e., along $\alpha =0$. (b) RG flow advocated for in this work, when $\alpha$ arises in a RG flow due to interactions. Note that crucially in (b), $\alpha$ is generated even when one begins on the $\alpha =0$ line. The point marked as $D$ indicates one of a set of fixed points with $\alpha \ne 0$ which all have a linear Dirac dispersion at low energy but different details of the electronic structure.

Figure 2

Structure factor $S_{\mathrm{AFM}}(\mathbf{k})$ and order parameter $m^2$ data for $H_{\mathrm{BSB}}$ with $t=t_{\perp }=1$ (fixed throughout the Letter). In (a) and (b), we show the structure factor in $\mathbf{k}$ space for an $L=18$ lattice and its peak at $\mathbf{\Gamma }$. Note the dramatic appearance of a Bragg peak structure as one changes from $U=2$ to $U=3$, indicative of a phase transition. (c) Finite size scaling of the order parameter $m^2\equiv S_{\mathrm{AFM}}(\mathbf{\Gamma })/L^2$, showing a finite $m^2$ for $U=3.6$ and 3.2 and then a dramatic suppression for smaller $U$.

Figure 3

Correlation ratio $R_{m^2}$ [Eq. (3)] close to the phase transition. The inset has a broad range of $U$ showing how $R_{m^2}$ reaches its asymptotes of 0 in the nonmagnetic phase and 1 in the magnetic phase. At a continuous transition, $R_{m^2}$ is expected to be volume independent and cross at a universal value. The main panel show an enlargement close to the critical point. The crossing point is at $U_c\approx 2.5$ with very small drifts on the largest system sizes.

Figure 4

Collapse of the magnetic data $m^2$ and $R_{m^2}$ close to the critical point. Parameters used for the collapse are $U_c=2.6$, $\nu =0.9$, and $a=0.3$.

Figure 5

The collapse of the single-particle gap $L{\mathrm{\Delta }}_{\mathrm{sp}}$ close to the phase transition with the same critical parameters as in Fig. 4 and $z=1$. The inset shows the single-particle gap $L{\mathrm{\Delta }}_{\mathrm{sp}}$ as a function of $U$ which exhibits a crossing at a coupling of $U_c\approx 2.5$, consistent with the expected scaling form with $z=1$.