The study of two-dimensional (2D) materials with flat electronic bands has become a significant area of research, focusing on the interplay between band topology and Coulomb interactions. Graphene-based van der Waals (vdW) heterostructures provide a platform to investigate this interplay due to their Dirac points near the Fermi level. Flat bands can be engineered through heterostructure design and tuned using magnetic fields or gate voltages, enabling control over band topology. Coulomb interactions often lead to spontaneous symmetry breaking, resulting in exotic phases like fractional Chern insulators or unconventional superconductivity. The anomalous Hall effect (AHE), a signature of spontaneously broken time-reversal symmetry, is particularly interesting. Previous studies have reported AHE reflecting orbital ferromagnetism in multilayer vdW heterostructures and Bernal-stacked bilayer graphene. Twisted double bilayer graphene (tDBG), comprising two Bernal-stacked bilayer graphene sheets with a twist angle, offers unique tunability. By controlling electrostatic doping and vertical displacement fields using top and back gates, the band topology, correlations, and broken symmetry phases can be manipulated. This research focuses on investigating the correlated metallic states in tDBG with twist angles around 1.3°, where metallic states with broken spin and valley symmetries have been observed, but their topologies and order parameters remain unclear. The goal is to understand the nature of the spontaneously broken time-reversal symmetry observed in this system.

Extensive prior research has explored the unique properties of twisted bilayer graphene (TBG) and other moiré superlattices. Studies have demonstrated correlated insulator behavior at half-filling in magic-angle graphene superlattices, and unconventional superconductivity in these systems. The interplay between band topology and Coulomb interactions in these flat-band systems has been theoretically explored, connecting them to exotic phases. The engineering of flat bands in different graphene-based heterostructures, including trilayer graphene boron-nitride moiré superlattices, has been reported. Moreover, previous work has investigated the anomalous Hall effect (AHE) as a manifestation of spontaneously broken time-reversal symmetry in various systems, including twisted bilayer graphene aligned with hexagonal boron nitride (hBN) and Bernal-stacked bilayer graphene. These studies provide a context for understanding the behavior of twisted double bilayer graphene (tDBG), particularly regarding the emergence of correlated insulating states and the possibility of AHE due to spontaneous symmetry breaking. The literature establishes a foundation for exploring the tunability of tDBG and the relationship between its band structure and emergent electronic phases.

High-quality twisted double bilayer graphene (tDBG) devices were fabricated using a modified tear and stack technique. Bilayer graphene flakes were exfoliated, pre-cut, and then stacked with hexagonal boron nitride (hBN) layers, with a controlled twist angle around 1.3°. Top and bottom gates were integrated for independent tuning of charge carrier density and displacement field. The devices were patterned into Hall bar geometries for four-terminal resistance (Rxx and Rxy) measurements. Transport measurements were performed in a dilution refrigerator using standard lock-in techniques with a current bias of 1nA. A 3-axis magnet allowed for independent control of in-plane (B||) and out-of-plane (B⊥) magnetic fields. Data were symmetrized and antisymmetrized to minimize mixing between Rxx and Rxy, obtaining longitudinal resistivity (ρxx) and Hall resistance (ρxy). The twist angle was calculated based on the density corresponding to full filling of the moiré miniband. Top and bottom gate voltages were used to independently control charge carrier density and displacement field. The experimental setup facilitated measurements of magnetoresistance under varying conditions of temperature, magnetic field (both in-plane and out-of-plane), and gate voltages.

The researchers observed a strong anomalous Hall effect (AHE) in the correlated metallic state of tDBG, characterized by hysteresis loops spanning hundreds of millitesla in out-of-plane magnetic field (B⊥). This hysteresis persisted for in-plane fields up to several Tesla, strongly suggesting valley (orbital) ferromagnetism. The resistivity was significantly impacted by even milliTesla-scale in-plane magnetic fields, implying spin-valley coupling or direct orbital coupling. A detailed resistivity map revealed correlated insulating states at specific fillings (ν=2 and ν=3), surrounded by a metallic 'halo' region of higher resistance. Within this halo region, the Hall resistance changed sign at integer fillings and at the halo's boundaries, indicating broken four-fold band degeneracy. The most compelling evidence for spontaneously broken time-reversal symmetry came from the magnetoresistance of the correlated metallic states. Hysteresis loops in both ρxx and ρxy were observed, clearly indicating ferromagnetism with coercive fields around 0.3T. The range of filling factors exhibiting ferromagnetic AHE was determined, showcasing multi-domain switching behavior near the single-particle insulator. The AHE hysteresis was largely independent of displacement field but was only present within the halo region. Investigation of in-plane magnetic field effects revealed a sharp negative magnetoresistance peak at B||=0 throughout the AHE region, with rounding on a scale of ~1mT. This peak appeared upon entering the AHE region and grew with increasing filling factor. While the out-of-plane field hysteresis was highly reproducible, the in-plane field response was less so, suggesting a weaker, more complex coupling. The AHE signatures persisted to temperatures around 1.8K, indicating relatively large energy scales associated with valley symmetry breaking. Temperature-dependent measurements showed that the hysteresis loop collapsed with increasing temperature, with a sharp decrease in transition temperature correlated with the onset of AHE.

The observed AHE in tDBG, characterized by its strong anisotropy and persistence in high in-plane magnetic fields, strongly supports valley ferromagnetism as the underlying mechanism. The significant influence of small in-plane magnetic fields suggests a complex interplay between spin and valley degrees of freedom. The results demonstrate a correlated low-temperature ground state with coupled spin and valley order. The unusual sensitivity to low in-plane fields may result from weak coupling of the in-plane field to valley order, or from spin-orbit coupling, although the latter was not intentionally enhanced in these devices. These findings advance our understanding of spontaneously broken symmetries in strongly correlated systems and highlight the importance of considering both spin and valley degrees of freedom in the context of emergent phenomena in twisted van der Waals heterostructures.

This study demonstrates the observation of an anomalous Hall effect signifying orbital magnetic order in AB-AB stacked twisted double bilayer graphene (tDBG). The ferromagnetic state appears only in the strongly interacting regions of the parameter space. Strong magnetic anisotropy indicates valley ferromagnetism, while in-plane field signatures reveal a complex interplay of broken spin/valley symmetries. Future work should focus on a more comprehensive investigation of the in-plane field effects to better understand the underlying mechanisms and explore potential applications of these materials.

The study focused on only two devices with twist angles near 1.3°. While similar behavior was observed in both, further investigation with a wider range of twist angles and device parameters is needed for generalization. The irreproducibility of some in-plane field sweeps suggests that there may be factors influencing the measurements that are not fully understood. The precise nature of the interaction between the in-plane magnetic field and the valley order parameter requires further investigation. Additional theoretical modeling could provide deeper insights into the observed phenomena.