# Superconductor versus insulator in twisted bilayer graphene

## TL;DR

This paper introduces a simplified model capturing the competition between superconducting and insulating phases in twisted bilayer graphene, highlighting how electron interactions and external fields influence these states.

## Contribution

The model demonstrates the emergence of superconductivity at incommensurate fillings and insulating states at commensurate fillings, aligning with experimental findings.

## Key findings

- Superconductivity occurs at generic incommensurate fillings.
- Insulating states appear at commensurate fillings due to electron interactions.
- External Zeeman fields can turn insulating states into gapless states.

## Abstract

We present a simple model that we believe captures the key aspects of the competition between superconducting and insulating states in twisted bilayer graphene. Within this model, the superconducting phase is primary, and arises at generic fillings, but is interrupted by the insulator at commensurate fillings. Importantly, the insulator forms because of electron-electron interactions, but the model is agnostic as to the superconducting pairing mechanism, which need not originate with electron-electron interactions. The model is composed of a collection of crossed one-dimensional quantum wires whose intersections form a superlattice. At each superlattice point, we place a locally superconducting puddle which can exchange Cooper pairs with the quantum wires. We analyze this model assuming weak wire-puddle and wire-wire couplings. We show that for a range of repulsive intrawire interactions, the system is superconducting at `generic' incommensurate fillings, with the superconductivity being `interrupted' by an insulating phase at commensurate fillings. We further show that the gapped insulating states at commensurate fillings give way to gapless states upon application of external Zeeman fields. These features are consistent with experimental observations in magic-angle twisted bilayer graphenes despite the distinct microscopic details. We further study the full phase diagram of this model and discover that it contains several distinct correlated insulating states, which we characterize herein.