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We use the DMRG to study the correlated electron states favored by theCoulomb interaction projected onto the narrow bands of twisted bilayer graphenewithin a spinless one-valley model. The Hilbert space of the narrow bands isconstructed from a pair of hybrid Wannier states with opposite Chern numbers.Depending on the parameters in the BM model, the DMRG in this basis determinesthe ground state at one particle per unit cell to be either QAH state or astate with no Hall effect which is nearly a product state. Based on this form,we then apply the variational method to study their competition, thusidentifying three states: the QAH, a gapless $C_2T$ symmetric nematic, and agapped $C_2T$ symmetric stripe. All three states are nearly degenerate at therealistic parameters of the BM model. The single particle spectrum of thenematic contains either a quadratic node or two close Dirac nodes near$\Gamma$. Motivated by the Landau level degeneracy found in this state, wepropose it to be the state observed at the charge neutrality point once spinand valley degeneracies are restored. The optimal period for the $C_2T$ stripestate is found to be $2$ unit cells. In addition, using the fact that thetopological charge of the nodes in the $C_2T$ nematic phase is no longerdescribed simply by their winding numbers once the translation symmetry isbroken, but rather by certain elements of a non-Abelian group that was recentlypointed out, we identify the mechanism of the gap opening within the $C_2T$stripe state. Although the nodes at the Fermi energy are locally stable, theycan be annihilated after braiding with other nodes connecting them to adjacent(folded) bands. Therefore, if the translation symmetry is broken, the gap atone particle per unit cell can open even if the system preserves the $C_2T$ andvalley $U(1)$ symmetries, and the gap to remote bands remains open.

Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene