Polynomial sign problem and topological Mott insulator in twisted bilayer graphene

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