# Twisted Bilayer Graphene Aligned with Hexagonal Boron Nitride: Anomalous Hall Effect and a Lattice Model

## TL;DR

This paper provides a theoretical analysis of the anomalous Hall effect observed in aligned twisted bilayer graphene with h-BN, proposing models that explain the phenomenon and suggesting experimental tests to distinguish between them.

## Contribution

It introduces a lattice Hubbard model for the system and explores two possible states explaining the Hall effect, advancing understanding of correlated phases in TBG-hBN systems.

## Key findings

- Valence and conduction bands have opposite Chern numbers C=±1.
- Aligned h-BN breaks symmetry and gaps Dirac crossings.
- Proposes experimental methods to distinguish between ferromagnetic insulator and metallic states.

## Abstract

A recent experiment reported a large anomalous Hall effect in Magic Angle Twisted Bilayer Graphene (TBG) aligned with a hexagonal boron nitride(h-BN) substrate at $\frac{3}{4}$ filling of the conduction band. In this paper we study this system theoretically, and propose explanations of this observation. We emphasize that the physics for this new system is qualitatively different from the pure TBG system. The aligned h-BN breaks in-plane two-fold rotation symmetry and gaps out the Dirac crossings of ordinary TBG. The resulting valence and conduction bands of each valley carry equal and opposite Chern numbers $C=\pm 1$. A useful framework is provided by a lattice extended Hubbard model for this system which we derive. An obvious possible explanation of the anomalous Hall effect is that at $3/4$-filling the system is a spin-valley polarized ferromagnetic insulator where the electrons completely fill a Chern band. We also examine an alternate more radical proposal of a compressible valley polarized but spin unpolarized composite ferm liquid metallic state. We argue that either state is compatible with current experiments, and propose ways to distinguish between them in the future. We also briefly discuss the physics at $1/2$ filling.