Kekulé Spiral Order at All Nonzero Integer Fillings in Twisted Bilayer Graphene

We study magic angle graphene in the presence of both strain and particle-hole symmetry breaking due to non-local inter-layer tunneling. perform a self-consistent Hartree-Fock that incorporates these effects alongside realistic interaction substrate potentials, explore comprehensive set competing orders including those break translational at arbitrary wavevectors. find all non-zero integer fillings very small strains, comparable measured scanning tunneling experiments, stabilize fundamentally new type time-reversal symmetric spatially non-uniform order. This order, which we dub 'incommensurate Kekul\'e spiral' (IKS) spontaneously breaks emergent valley-charge conservation moir\'e translation symmetries, but preserves modified $\hat{T}'$ -- simultaneously shifts spatial coordinates rotates $U(1)$ characterizes spontaneous inter-valley coherence. discuss phenomenological microscopic properties this argue our findings are consistent with experimental observations reported so far, suggesting unified explanation global phase diagram terms IKS