Many-Body Topological Quantum Chemistry Indices in 2D with Real Space Invariants

The topological phases of non-interacting fermions are understood from the perspective of topological quantum chemistry, and more recently from a spatially local formulation using Real Space Invariants (RSIs). This work generalizes the real space classification to interacting 2D states at integer fillings. We construct many-body local RSIs (the Noether charges of discrete symmetries) as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, identifying which single-particle fragile states are many-body topological and which are trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit but also identify many strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body RSIs appear as quantized coefficients of Wen-Zee terms in the topological response theory describing the phase.