Inter-Layer Correlation of Loop Current Charge Density Wave on the Bilayer Kagomé Lattice

Abstract: 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 kagomé 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 kagomé 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 kagomé lattice. The bare susceptibilities for stacking of real and imaginary bond orders are calculated for the free electrons on the bilayer kagomé 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 ...

• The paper demonstrates that nn and nnn Coulomb interactions induce distinct real and loop current CDWs with specific stacking patterns in bilayer kagomé lattices.
• The analysis employs a mean-field Hamiltonian to uncover ground-state configurations like SD-SD, ISD-ISD, FM-LC, and AF-LC, heavily influenced by band filling.
• The study also reveals topological features of loop current CDWs, assigning nontrivial Chern numbers and suggesting potential anomalous Hall effects.

Inter-Layer Correlation of Loop Current Charge Density Wave on the Bilayer Kagomé Lattice

Introduction

The paper investigates the inter-layer correlation of loop current charge density waves (CDWs) in vanadium-based kagomé materials, with a focus on the bilayer kagomé lattice. These materials, specifically $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs), manifest intriguing electronic properties related to spontaneous time-reversal symmetry breaking charge density waves, which are potentially driven by loop current order. Understanding the stacking and interactions between these phenomena in the bulk material is crucial for developing comprehensive theoretical models that can also be reconciled with experimental observations.

Theoretical Model and Susceptibility Calculations

The study employs a bilayer tight-binding model with inter-layer coupling $t_{\perp}$ to analyze the CDW instabilities in kagomé lattices. The primary focus is on two interactions: nearest-neighbor (nn) Coulomb repulsion $V_1$ and next-nearest-neighbor (nnn) Coulomb repulsion $V_2$. These interactions respectively drive real and imaginary bond-ordered CDWs, with the latter manifesting as loop current orders. The paper reports on the calculation of the bare susceptibilities for symmetric and antisymmetric stacking of these CDWs, revealing sensitivity to band filling, which suggests different stacking preferences.

Mean-Field Study of Inter-Layer Correlations

The mean-field Hamiltonian derived from the model considers $t$-$t_{\perp}$-$V_1$-$V_2$ interactions. This approach allows the identification of possible ground-state configurations of CDWs under symmetric or antisymmetric stacking, such as SD-SD (Star-of-David) and ISD-ISD (inverse Star-of-David), as well as loop current CDWs like FM-LC (ferromagnetic loop current) and AF-LC (antiferromagnetic loop current). Calculations show that near particular band fillings, the system prefers either a $2a_0 \times 2a_0 \times 1c_0$ or a $2a_0 \times 2a_0 \times 2c_0$ stacking pattern, dictated by the nature of the charge density wave and its interaction-mediated instabilities.

Results and Discussion

The analysis illustrates that for sufficient nn Coulomb repulsion $V_1$, real bond-ordered CDWs are favored, whereas significant nnn Coulomb repulsion $V_2$ stabilizes loop current CDWs. The transition from real to complex bond orders reflects the sensitivity to band filling, effectively highlighting the role of multiband interactions in determining electronic ground states. Additionally, the presence of loop currents leads to nontrivial topological properties and possible implications for experimental observables like the anomalous Hall effect (AHE).

Topological Properties

Further investigation into the band topology of these CDWs reveals that states like FM-LC and FE-LC (ferrimagnetic loop current) are topologically nontrivial, characterized by nonzero Chern numbers assigned to isolated electronic bands. This intrinsic topological nature is a direct consequence of breaking time-reversal symmetry via loop currents, showcasing roots in the kagomé lattice's unique electronic configurations.

Conclusion

This paper provides a detailed treatment of the interplay between electronic interactions and CDW order in bilayer kagomé lattices, offering insights into both symmetric and asymmetric stacking phenomena. The theoretical model and resulting predictions underscore the complex nature of vanadium-based kagomé metals and open pathways for future experimental exploration and theoretical refinement of CDW and loop current interactions in these and similar materials. The study contributes to understanding the potential for novel quantum phases and their application to both fundamental research and technological advancements in materials science.