Obstruction of broken symmetries in topological flat bands

Motivated by the abundance of symmetry breaking states in magic-angle twisted bilayer graphene and other two-dimensional materials, we study superconducting (SC) and charge orders in two-dimensional topological flat bands in the strong correlation regime. By relating the half-filled two-dimensional (2D) topological flat bands to the surface states of 3D topological insulators in symmetry class AIII, we reveal the topological obstruction to the formation of gapped SC and intervalley charge orders without intrinsic topological orders, in the presence of the antiunitary particle-hole symmetry at half filling. This is a generalization of the Li-Haldane arguments for nodal superconductivity to strongly interacting electrons. In contrast to the $\mathbb{Z}$-valued obstruction derived from the noninteracting band topology, the topological obstruction of interacting electrons in half-filled flat bands has a ${\mathbb{Z}}_{8}$ classification, depending on the charge (valley) Chern number of the superconducting (intervalley charge) orders. This is demonstrated by an interacting Hamiltonian for half-filled flat bands with a net Chern number $C=4$, where superconductivity and ${\mathbb{Z}}_{2}$ topological order coexist in a gapped ground state with particle-hole symmetry.