  We use the DMRG to study the correlated electron states favored by the
Coulomb interaction projected onto the narrow bands of twisted bilayer graphene
within a spinless one-valley model. The Hilbert space of the narrow bands is
constructed from a pair of hybrid Wannier states with opposite Chern numbers.
Depending on the parameters in the BM model, the DMRG in this basis determines
the ground state at one particle per unit cell to be either QAH state or a
state with no Hall effect which is nearly a product state. Based on this form,
we then apply the variational method to study their competition, thus
identifying three states: the QAH, a gapless $C_2T$ symmetric nematic, and a
gapped $C_2T$ symmetric stripe. All three states are nearly degenerate at the
realistic parameters of the BM model. The single particle spectrum of the
nematic contains either a quadratic node or two close Dirac nodes near
$\Gamma$. Motivated by the Landau level degeneracy found in this state, we
propose it to be the state observed at the charge neutrality point once spin
and valley degeneracies are restored. The optimal period for the $C_2T$ stripe
state is found to be $2$ unit cells. In addition, using the fact that the
topological charge of the nodes in the $C_2T$ nematic phase is no longer
described simply by their winding numbers once the translation symmetry is
broken, but rather by certain elements of a non-Abelian group that was recently
pointed 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, they
can be annihilated after braiding with other nodes connecting them to adjacent
(folded) bands. Therefore, if the translation symmetry is broken, the gap at
one particle per unit cell can open even if the system preserves the $C_2T$ and
valley $U(1)$ symmetries, and the gap to remote bands remains open.
