Controlled Loop Expansion for Strained Twisted Bilayer Graphene

We develop a controlled diagrammatic framework for periodic Anderson models,and apply it to heterostrained magic-angle twisted bilayer graphene (MATBG) at charge neutrality using the topological heavy-fermion formulation. Building on arXiv:2604.14278, we organize self-energy insertions and perform a Dyson resummation to any order in the small parameter $s^2$ -- the fraction of the moir\'e Brillouin zone with nontrivial quantum geometry. For strained MATBG, the expansion remains controlled down to arbitrarily low temperatures as long as the strain induced energy scale is not too small. In the flat-chiral limit, an emergent approximate $\rm{U}(1)$ symmetry forbids the leading scattering channel and leaves the Mott bands sharp at order $s^2$. This is in stark contrast to the unstrained case, where the linewidth is of order $N_f s^2 U$ with $U$ the on-site $f$-$f$ Hubbard interaction and $N_f$ the number of $f$ states per site. Away from the chiral limit, the linewidth is non-zero at order $s^2$ but more than an order of magnitude smaller than in the unstrained case. The strain-induced energy scale also imprints itself directly on the spectrum: as an electron-phonon-like kink in the dispersion, and as an additional flat ``trion'' band -- a single-particle excitation bound to a local $f$ particle-hole pair. We use the framework to predict the Quantum Twisting Microscope spectrum at one-loop order for both strained and unstrained MATBG, and compare with recent experiments.