Phase diagram of twisted bilayer graphene at filling factor $\nu=\pm3$

We study the correlated insulating phases of twisted bilayer graphene (TBG) in the absence of lattice strain at integer filling $\nu=\pm3$. Using the self-consistent Hartree-Fock method on a particle-hole symmetric model and allowing translation symmetry breaking terms, we obtain the phase diagram with respect to the ratio of $AA$ interlayer hopping $(w_0)$ and $AB$ interlayer hopping $(w_1)$. When the interlayer hopping ratio is close to the chiral limit ($w_0/w_1 \lesssim 0.5$), a quantum anomalous Hall state with Chern number $\nu_c = \pm 1$ can be observed consistent with previous studies. Around the realistic value $w_0/w_1 \approx 0.8$, we find a spin and valley polarized, translation symmetry breaking, state with $C_{2z}T$ symmetry, a charge gap and a doubling of the moir\'e unit cell, dubbed the $C_{2z}T$ stripe phase. The real space total charge distribution of this $C_{2z}T$ stripe phase in the flat band limit does not have modulation between different moir\'e unit cells, although the charge density in each layer is modulated, and the translation symmetry is strongly broken. Other symmetries, including $C_{2z}$, $C_{2x}$ and particle-hole symmetry $P$, and the topology of the $C_{2z}T$ stripe phase are also discussed in detail. We observed braiding and annihilation of the Dirac nodes by continuously turning on the order parameter to its fully self-consistent value, and provide a detailed explanation of the mechanism for the charge gap opening despite preserving $C_{2z}T$ and valley $U(1)$ symmetries. In the transition region between the quantum anomalous Hall phase and the $C_{2z}T$ stripe phase, we find an additional competing state with comparable energy corresponding to a phase with a tripling of the moir\'e unit cell.