Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene (1808.02482v3)

Abstract: Correlated insulators and superconductivity have been observed in "magic-angle" twisted bilayer graphene, when the nearly flat bands close to neutrality are partially filled. While a momentum-space continuum model accurately describes these flat bands, interaction effects are more conveniently incorporated in tight-binding models. We have previously shown that no fully symmetric tight-binding model can be minimal, in the sense of capturing just the flat bands, so extended models are unavoidable. Here, we introduce a family of tight-binding models that capture the flat bands while simultaneously retaining all symmetries. In particular, we construct three concrete models with five, six, or ten bands per valley and per spin. These models are also faithful, in that the additional degrees of freedom represent energy bands further away from neutrality, and they serve as optimal starting points for a controlled study of interaction effects. Furthermore, our construction demonstrates the "fragile topology" of the nearly flat bands; i.e., the obstruction to constructing exponentially localized Wannier functions can be resolved when a particular set of trivial bands is added to the model.

• The paper introduces novel tight-binding frameworks that accurately capture nearly flat bands and fragile topology in magic-angle bilayer graphene.
• It overcomes topological obstructions by constructing multi-band models that respect symmetry and enable detailed interaction analyses.
• The study connects theoretical predictions with experimental observations, offering new insights into superconductivity and correlated insulator behavior.

Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene: An Overview

This paper presents a comprehensive investigation of the tight-binding models for understanding the phenomena observed in magic-angle twisted bilayer graphene (TBG), with a particular focus on the concept of fragile topology. The paper stems from recent experimental observations of correlated insulators and superconductivity in TBG, specifically when graphene sheets are twisted at a magic angle, close to 1.05 degrees. The phenomena are primarily attributed to the formation of nearly flat bands at these angles, which play a significant role in emergent electronic properties.

The theoretical underpinning for TBG typically involves a momentum-space continuum model to describe these flat bands. However, the incorporation of interaction effects is more practical in a real-space tight-binding framework. A fundamental challenge lies in defining a minimal tight-binding model that captures only the flat bands, which proves particularly complex due to the band topology, leading to obstructions in creating exponentially localized Wannier functions for the two bands under scrutiny.

The authors propose a novel family of tight-binding models constructed to represent the flat bands in TBG while maintaining all requisite symmetries. The focus centers on three specific models consisting of five, six, or ten bands per valley and spin. These models demonstrate robustness by including additional energy bands that extend beyond the flat bands themselves. The extended models remain faithful to the electronic states' symmetries and offer optimal starting points for examining interaction effects, thereby providing a tangible approach to studying the exotic phases observed experimentally.

A significant highlight of the paper is the introduction and exploration of the "fragile topology" of nearly flat bands in TBG. Fragile topology implies a subtle form of topological obstruction, where the non-trivial nature of the band topology can effectively be neutralized by adding trivial, or atomic, bands to the model. This differs from stable topological phases, where non-trivial topology cannot be neutralized by any trivial means. This concept is particularly relevant for the construction of tight-binding models for TBG since it allows for models that integrate topological bands without the need for inversion or breaking original symmetries.

The ten-band model, as elaborated in the paper, offers a distinctive resolution to achieving the symmetry requirements while including all complimentary and energy-filling bands incorporating particle-hole symmetry, which is known to be an approximate local symmetry in TBG's higher-energy states. To construct the model, the authors introduce inventive quasi-orbitals that emulate required energy and symmetry features of the actual TBG band structure.

Moreover, the theoretical implications of these models extend to the broader discussion of correlated electron models akin to those traditionally used for cuprates, like the Hubbard model. By drawing parallels between TBG and other high-temperature superconductors, the paper represents an instrumental advancement in understanding how minimal models for complex materials should be formulated when taking into account both kinetic and interaction effects as well as intricate topological characteristics.

Looking forward, the presented models pave the way for more comprehensive theoretical and computational investigations, raising potential questions about interactions and emergent properties in fragile topological phases. These could feed into practical device applications where the manipulation of electronic states through mechanical means (such as twisting) provides a new paradigm in material design and application.

In conclusion, this paper sets a foundational context for studying TBG through tight-binding models that do not only appropriately summarize empirical observations but also challenge the existing understanding of topological effects in material science. As these models are further honed, they may significantly contribute both to theoretical advances in quantum materials and to practical applications in electronic technologies that capitalize on novel topological properties.