Interacting fermions on the honeycomb bilayer: From weak to strong coupling

Many-body instabilities of the half-filled honeycomb bilayer are studied using weak-coupling renormalization group (RG) as well as strong-coupling expansion. For spinless fermions and assuming parabolic degeneracy, there are four independent four-fermion contact couplings. While the dominant instability depends on the microscopic values of the couplings, the broken symmetry state is typically a gapped insulator with either broken inversion symmetry or broken time-reversal symmetry, with a quantized anomalous Hall effect. Under certain conditions, the dominant instability may appear in the particle-particle (pairing) channel. For some nongeneric fine-tuned initial conditions, weak-coupling RG trajectories flow into the noninteracting fixed point, although generally we find runaway flows which we associate with ordering tendencies. Additionally, a tight-binding model with nearest-neighbor hopping and nearest-neighbor repulsion is studied in weak and strong couplings and in each regime a gapped phase with inversion symmetry breaking is found. In the strong-coupling limit, the ground-state wave function is constructed for vanishing in-plane hopping but finite interplane hopping, which explicitly displays the broken inversion symmetry and a finite difference between the number of particles on the two layers. Finally, we discuss the spin-1/2 case and use Fierz identities to show that the number of independent four-fermion contact couplings is 9. The corresponding RG equations in the spin-1/2 case are also presented, and used to show that, just as in strong coupling, the most dominant weak-coupling instability of the repulsive Hubbard model (at half filling) is an antiferromagnet.

Figure 1

(i) Schematic representation of the $A\text{−}B$ stacked bilayer. The low-energy wave function near $\mathbf{K}$ is also sketched, with $\omega =e^{i\pi /3}=\frac{1}{2}+i\frac{\sqrt{3}}{2}$. The primitive lattice vectors are ${\mathbf{R}}_1=\sqrt{3}a\widehat{x}$ and ${\mathbf{R}}_2=\frac{\sqrt{3}}{2}a\widehat{x}+\frac{3}{2}a\widehat{y}$. The area of the unit cell is $A_{uc}=\widehat{z}\cdot \left({\mathbf{R}}_1\times {\mathbf{R}}_2\right)=\frac{3\sqrt{3}}{2}{a}^2$. (ii) Schematic representation of the (reciprocal) $\mathbf{k}$ space.

Figure 2

Diagrams appearing at one-loop RG. Each vertex can be represented by either a $4\times 4$ matrix (spinless) or $8\times 8$ matrix (spin-1/2) case.

Figure 3

(Color online) Flow diagram in the coupling constant ratio space assuming that $g_{A_2}<0$ (generic behavior). There are two sinks given by Eqs. (73, 75) in the text and two mixed fixed ratios [Eqs. (74, 76)]. For $g_{A_2}>0$ the flow is reversed, and generically, the divergent coupling constant ratios simply mean that $g_{A_2}$ has shrunk and crossed 0. After this point $g_{A_2}$ becomes negative and the directionality shown here is restored.

Figure 4

Schematic representation of the strong-coupling state for the spinless $t\text{−}V$ model. For $t=0$ but finite $t_{\perp }$, $V$, and $V_{\perp }$, the sublattice $b_1$ is empty, while sublattice $b_2$ is fully occupied. The $a_1\text{−}{a}_2$ dimer is singly occupied, with an electron partially delocalized onto $a_2$ despite the repulsion from the occupied $b_2$ sites. Such gapped state breaks inversion symmetry between the layers 1 and 2.

Figure 5

(Color online) Susceptibility vs $\text{ln}s$ [Eq. (108)] for the Hubbard model with initial $\frac{m}{4\pi A_c}U=0.01$, in the channels $C_1^{\left(s\right)}–B_4^{\left(s\right)}$ (left to right). The fastest divergence appears in the antiferromagnetic channel $C_1\otimes \vec{\sigma }$. Altogether 32 particle-hole channels have been analyzed (Table ); the channels not shown are either symmetry related to the ones shown, or $d\text{ln}\Delta /d\text{ln}s-2$ vanishes (or is negative).