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$SU(4)$ Valley + Spin Fluctuation Interference Mechanism for Nematic Order in Magic Angle Twisted Bilayer Graphene: Impact of Vertex Corrections

In the magic angle twisted bilayer graphene (MATBG), one of the most remarkable observations is the $C_3$-symmetry-breaking nematic state. We identify that the nematicity in MATBG is the $E$-symmetry ferro bond order, which is the modulation of correlated hopping integrals owing to the $E$-symmetry particle-hole pairing condensation. The nematicity in MATBG originates from prominent quantum interference among $SU(4)$ valley+spin composite fluctuations. This novel "valley + spin fluctuation interference mechanism" is revealed by the density wave equation analysis for realistic multiorbital Hubbard model for MATBG. \color{black}We find that the nematic state is robust once three van Hove singularity points exist in each valley. This interference mechanism also causes novel time-reversal-symmetry-broken valley polarization accompanied by a charge loop current. We discuss interesting similarities and differences between MATBG and Fe-based superconductors.