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

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.

5 + 36 pages

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• "Quantum Geometric Nesting" and Solvable Model Flat-Band Systems
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Interaction-induced crystalline topology of excitons
• Altermagnetic Instabilities from Quantum Geometry
• Exciton Berryology
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
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• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
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• Berry Curvature of Low-Energy Excitons in Rhombohedral Graphene