Graphene’s Twisted Layers Reveal New Electronic Properties and Topology

Van der Waals (vdV) materials represent a rapidly evolving area within condensed matter physics, attracting significant attention due to their inherent tunability and potential for novel electronic and quantum phenomena. These materials, characterised by weak interlayer interactions, allow for the creation of heterostructures with tailored properties, offering a platform to explore strongly correlated and topological phases. Recent research focuses on manipulating these materials through twisting, creating moiré patterns that dramatically alter their electronic behaviour and reduce the energy scales required to observe quantum effects.

The emergence of twisted bilayer graphene (TBG) as a model system has been pivotal in advancing this field, enabling researchers to investigate the formation of moiré superlattices with significantly enlarged unit cells when two graphene sheets rotate relative to each other. This geometric modification leads to the formation of flat electronic bands near the Fermi level, enhancing the influence of electron-electron interactions and enabling the observation of unconventional superconductivity and correlated insulating states. The system’s tunability, achieved through external gating and doping, allows for the exploration of phase diagrams resembling those found in high-temperature cuprate superconductors.

Understanding the interplay between these flat bands and higher-energy bands remains a key challenge, as researchers actively investigate how these hybridization processes affect the Chern number. This topological invariant characterises the band structure, and the resulting orbital magnetic energy. Identifying and controlling these interactions is crucial for designing and realising novel quantum devices. The ability to tune topological phase transitions through changes in twist angle, electric field, or pressure further expands the potential of these systems.

Research into twisted bilayer graphene and related moiré systems increasingly relies on sophisticated methodological approaches to unravel the complex interplay between electronic interactions and band topology. Detailed band structure calculations, often employing density functional theory, map the electronic properties of these materials as a function of twist angle. These calculations necessitate careful consideration of the substrate, such as hexagonal boron nitride, as it significantly alters the electronic landscape. Researchers routinely employ techniques like maximally localised Wannier functions, a method developed by Marzari and Vanderbilt, to represent and analyse these complex band structures efficiently.

A key innovation lies in the application of topological concepts to characterise these electronic states, with the calculation of Chern numbers, pioneered by Hatsugai and Suzuki, providing a robust way to quantify the topological properties of the bands, revealing the presence of topological insulators and, crucially, the emergence of fractional Chern insulators. These insulators exhibit fractionalised excitations, a hallmark of strong electron correlation, and are a major focus of current research. The work extends beyond simple Chern numbers, with investigations into higher-order topological insulators, explored by Mizoguchi, Kuno, and Hatsugai, which exhibit topological states not confined to the material’s surface but residing within its bulk.

Recent advancements highlight the importance of considering the quantum geometry and metric of these materials, with the quantum metric, which describes the curvature of the electronic bands, influencing the behaviour of electrons and significantly impacting superconductivity. Researchers demonstrate the dependence of flat band superconductivity on the minimal quantum metric, suggesting that manipulating the quantum geometry could be a pathway to enhance or control superconducting properties. This necessitates the development of theoretical frameworks that go beyond traditional band theory and incorporate the effects of quantum geometry on electronic transport and optical properties. Furthermore, the field increasingly leverages the power of molecular-orbital representations, as developed by Mizoguchi and Hatsugai, to gain a more intuitive understanding of the electronic states and their symmetries.

Twisted bilayer graphene currently attracts significant attention due to its capacity to host a diverse range of electronic phenomena, stemming from the complex interplay between strong electron correlations and non-trivial band topology. Researchers actively investigate the hybridization between flat bands near zero energy and more remote bands, seeking to understand its influence on charge distribution and topological properties. Recent work demonstrates that these interactions significantly impact the system’s behaviour, revealing multiple topological transitions as a function of twist angle. Investigations focus on quantifying topological invariants, such as the Chern number, which characterises the topological nature of electronic bands.

Calculations reveal that the central bands within TBG can exhibit Chern numbers of 2, indicating the presence of previously unreported topological states within twisted bilayer graphene. The quantum metric, a measure of the geometric properties of the band structure, also plays a crucial role, influencing the material’s electronic characteristics and contributing to the emergence of correlated states. Computational techniques, including the use of maximally localised generalised Wannier functions and efficient Chern number calculations, prove essential for accurately modelling and predicting the behaviour of these complex systems.

The field increasingly emphasises the connection between quantum geometry and topology, with researchers exploring how the quantum metric influences the material’s properties and contributes to the emergence of correlated states. Studies demonstrate that the geometric properties of the band structure directly impact the electronic behaviour, offering new avenues for controlling and manipulating the material’s characteristics. This research builds upon foundational work in topological insulators and the Hofstadter butterfly, extending these concepts to the unique context of twisted graphene systems.

A significant trend involves exploring the potential of these materials for quantum information processing and storage, with investigations into thermal quantum information capacity demonstrating a growing interest in leveraging topological properties for quantum technologies. Furthermore, the study of anomalous coherence lengths in superconductors with quantum metrics suggests potential applications in advanced superconducting devices. Future research should prioritise a more comprehensive understanding of the interplay between remote bands and correlated phenomena, and detailed investigations into the impact of strong electron-electron interactions on topological phases. Expanding theoretical models to incorporate more realistic material parameters and exploring the effects of external stimuli, such as strain or magnetic fields, will further refine our understanding. Continued experimental verification, alongside the development of novel characterisation techniques, remains essential for translating theoretical insights into practical applications.