Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field

We derive the Hamiltonian for trilayer moir\'e systems with Coulomb interaction projected onto bands near charge neutrality point. Motivated by latest experimental results, we focus on twisted symmetric graphene (TSTG) a mirror-symmetry respect to middle layer. provide full symmetry analysis of non-interacting perpendicular displacement field coupling band structure made otherwise bilayer (TBG) and high velocity Dirac fermions, identify hidden non-local problem. In presence this field, construct an approximate single-particle model, akin tripod model TBG, capturing essence TSTG. also more quantitative perturbation schemes low-energy physics TSTG obtaining corresponding eigenstates. This allows us obtain in active wavefunctions many-body system. efficient parameterization interacting Hamiltonian. Finally, show that discrete symmetries at level promote $\mathrm{U} \left( 2 \right) \times \mathrm{U} \right)$ spin-valley enlarged groups problem under different limits. The part exhibits large 4 chiral limit. Moreover, identifying new which dub spatial conjugation, is around neutrality.