Understanding Symmetry Breaking in Twisted Bilayer Graphene from Cluster Constraints

TL;DR

This paper investigates the phase diagram of twisted bilayer graphene using an extended Hubbard model, revealing various symmetry-breaking phases and introducing the concept of a cluster rule constraint that influences the system's quantum states.

Contribution

It introduces the cluster rule constraint in the extended Hubbard model for twisted bilayer graphene and explores its impact on phase competition and quantum states.

Findings

Identification of valence-bond-solid, Ne9el-valley antiferromagnetic, and charge-density wave phases.

Development of a perturbative approach projecting into the cluster-rule manifold.

Prediction of local electron density patterns observable via scanning tunneling microscopy.

Abstract

Twisted bilayer graphene is an exciting platform for exploring correlated quantum phases, extremely tunable with respect to both the single-particle bands and the interaction profile of electrons. Here, we investigate the phase diagram of twisted bilayer graphene as described by an extended Hubbard model on the honeycomb lattice with two fermionic orbitals (valleys) per site. Besides the special extended {\it cluster interaction} $Q$, we incorporate the effect of gating through an onsite Hubbard-interaction $U$. Within Quantum Monte Carlo (QMC), we find valence-bond-solid, N\'eel-valley antiferromagnetic or charge-density wave phases. Further, we elucidate the competition of these phases by noticing that the cluster interaction induces an exotic constraint on the Hilbert space, which we dub {\it the cluster rule}, in analogy to the famous pyrochlore spin-ice rule. Formulating the perturbative Hamiltonian by projecting into the cluster-rule manifold, we perform exact diagonalization and construct the fixed-point states of the observed phases. Finally, we compute the local electron density patterns as signatures distinguishing these phases, which could be observed with scanning tunneling microscopy. Our work capitalizes on the notion of cluster constraints in the extended Hubbard model of twisted bilayer graphene, and suggests a scheme towards realization of several symmetry-breaking insulating phases in a twisted-bilayer graphene sheet.