We show that for the magic-angle twisted bilayer graphene (TBG) away from the charge neutrality point, although quantum Monte Carlo (QMC) simulations suffer from the sign problem, the computational complexity is at most polynomial at integer fillings of the flat-band limit. For even-integer fillings, the polynomial complexity survives even if an extra intervalley attractive interaction is introduced. This observation allows us to simulate magic-angle TBG and to obtain an accurate phase diagram and dynamical properties. At the chiral limit and filling $\nu =1$, the simulations reveal a thermodynamic transition separating the metallic state and a $C=1$ correlated Chern insulator—topological Mott insulator (TMI)—and the pseudogap spectrum slightly above the transition temperature. The ground state excitation spectra of the TMI exhibit a spin-valley U(4) Goldstone mode and a time-reversal restoring excitonic gap smaller than the single-particle gap. These results are qualitatively consistent with recent experimental findings at zero-field and $\nu =1$ filling in $h$-BN nonaligned TBG devices.

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• Mott insulators
• Topological insulators
• Twisted bilayer graphene
• Computational complexity
• Quantum Monte Carlo