Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants

TL;DR

This paper extends real space topological classification to interacting 2D fermionic systems, introducing many-body invariants that identify fragile and stable topological phases beyond single-particle descriptions.

Contribution

It develops many-body local Real Space Invariants (RSIs) for interacting 2D states, linking boundary symmetry operators to global topological indices and topological quantum field theory.

Findings

Many-body RSIs can distinguish fragile topological states from trivial states.

Global many-body RSIs reduce to Chern numbers in the non-interacting limit.

Identifies strongly correlated topological phases with no single-particle analogs.

Abstract

The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained (in part) from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs 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, which we use to identify which single-particle fragile states are many-body topological or 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 strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.