  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.
