Antiferromagnetism in the Hubbard Model on the Bernal-Stacked Honeycomb Bilayer

Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin moments in the ordered ground state, with enhanced moments at the threefold coordinated sites. Furthermore, we find the antiferromagnetic ground state to be robust with respect to enhanced interlayer couplings and extended Coulomb interactions.

Figure 1

(a) Bernal stacking of the honeycomb bilayer with intra- (inter)layer hopping $t$ ($t_{\perp }$) between the sublattices $A$, $B$ and $A^{\prime }$, $B^{\prime }$ ($A^{\prime }$, $B$). Within the sublattices an equal number of sites have a coordination number $z=3$ or 4. (b) Patching scheme of the Brillouin zone in the FRG. Dots denote the momenta at which the vertex function is evaluated.

Figure 2

Single particle gap ${\Delta }_{\mathrm{sp}}$ from QMC calculations for different system sizes and the finite size extrapolation to the TDL using a polynomial fit function, along with the FRG critical scale ${\Lambda }_c$ as a function of the local Coulomb repulsion $U/t$. The inset on the left shows the same data vs $t/U$ in a semilog scale, exhibiting an exponential onset of the gap in the large $t/U$ range. The inset on the right shows the FRG data along with MFT results for system sizes $L=129$, 258, and 516.

Figure 3

The sublattice magnetization vs $U/t$ for sites with coordination number $z=3$ (open symbols) and $z=4$ (filled symbols) from (a) MFT in the TDL, (b) QMC simulations for different system sizes, and (c) in the Heisenberg limit vs the interlayer AF exchange coupling $J_{\perp }/J$. The data in (c) result from SSE QMC simulations, after extrapolation to the TDL; also shown in (c) is the overall staggered magnetization $m$. Part (d) shows the local dynamical spin susceptibility ${\chi }_{\sigma }(\omega )$ for sites with $z=3$ and $4$ for $L=12$, $U/t=4$.

Figure 4

Finite size behavior of (a) the spin gap ${\Delta }_{\sigma }$ and (b) the AF structure factor $S_{\mathrm{AF},3}$ on lattice sites with coordination number $z=3$. The inset in (b) illustrates the discretization effects on the free dispersion relation, a dominant source of finite size effects.