Many body Fu-Kane indices

In this thesis, we construct a many-body Z₂ index for two-dimensional interacting fermionic lattice systems with a conserved U(1) charge and fermionic time-reversal symmetry. The construction is formulated within the operator-algebraic framework of quantum lattice systems and applies to symmetric short-range entangled states. Motivated by the flux-insertion interpretation of the Fu–Kane–Mele invariant, we define defect states by adiabatically inserting a π-flux through the system and analyze their transformation properties under time reversal. The index takes its non-trivial value precisely when the resulting flux defect transforms as part of a Kramers pair. We prove that the index is independent of all auxiliary choices involved in its definition, including the choice of parent Hamiltonian and flux-insertion geometry. Furthermore, we show that it is multiplicative under stacking and invariant under symmetry-preserving locally generated automorphisms. Consequently, the index depends only on the underlying symmetry-protected topological phase and extends naturally to the class of symmetric stably short-range entangled states. To connect the many-body construction with the established free-fermion theory, we develop a CAR-algebraic formulation of the Fu–Kane–Mele invariant based on flux insertion and spectral flow. We prove that, for symmetric quasi-free states in symmetry class AII, the many-body index coincides with the Fu–Kane–Mele invariant. As a consequence, the interacting index recovers the non-interacting classification and establishes the persistence of the quantum spin Hall phase in the presence of interactions. These results provide a rigorous operator-algebraic characterization of interacting two-dimensional fermionic symmetry-protected topological phases with time-reversal symmetry and furnish a many-body extension of the Fu–Kane–Mele invariant.