Inter-Layer Correlation of Loop Current Charge Density Wave on the Bilayer Kagom\'e Lattice

Loop current order has been suggested as a promising candidate for the spontaneous time-reversal symmetry breaking $2a_0 \times 2a_0$ charge density wave (CDW) revealed in vanadium-based kagom\'e metals \avs\ ($A$ = K, Rb, Cs) near van Hove filling $n_\text{vH} = 5/12$. Weak-coupling analyses and mean field calculations have demonstrated that nearest-neighbor Coulomb repulsion $V_1$ and next-nearest-neighbor Coulomb repulsion $V_2$ drives, respectively, real and imaginary bond-ordered CDW, with the latter corresponding to time-reversal symmetry breaking loop current CDW. It is important to understand the inter-layer correlation of these bond-ordered CDWs and its consequences in the bulk kagom\'e materials. To provide physical insights, we investigate in this paper the $c$-axis stacking of them, loop current CDW in particular, on the minimal bilayer kagom\'e lattice. The bare susceptibilities for stacking of real and imaginary bond orders are calculated for the free electrons on the bilayer kagom\'e lattice with inter-layer coupling $t_\perp=0.2t$, which splits the van Hove filling to $n_{+\text{vH}}=4.64/12$ and $n_{-\text{vH}}=5.44/12$. While real and imaginary bond-ordered CDWs are still favored, respectively, by $V_1$ and $V_2$, their inter-layer coupling is sensitive to band filling $n$. They tend to stack symmetrically near $n_{\pm\text{vH}}$ with identical bond orders in the two layers and give rise to a $2a_0 \times 2a_0 \times 1c_0$ CDW. On the other hand, they prefer to stack antisymmetrically around $n_\text{vH}$ with opposite bond orders in the two layers and lead to a $2a_0 \times 2a_0 \times 2c_0$ CDW. The concrete bilayer $t$-$t_\perp$-$V_1$-V$_2$ model is then studied. We obtain the mean-field ground states and determine the inter-layer coupling as a function of band filling at various interactions. The nontrivial topological properties of loop current CDWs are studied ...