Multiband bosonic generalization of the Hopf invariant

Determine whether the delicate multiband construction that restores a \mathbb{Z} invariant for all N-band Hopf insulators with N\geq2 by resolving each band into its own gap and using the classifying space SU(N)/U(1)^{N-1} carries over to the bosonic Krein-space setting.

Background

The paper establishes that the symplectic Hopf invariant is nontrivial for an isolated bosonic Bogoliubov band only when the unit cell contains exactly two bosonic modes. For three or more modes, the relevant Krein-positive projective space has trivial third homotopy group, so the single-band Hopf invariant disappears while weak symplectic Chern invariants remain.

The authors identify a known fermionic multiband route as a possible way to restore an integer Hopf classification by separating all bands with individual gaps. Whether an analogous construction exists for bosonic Bogoliubov-de Gennes systems with an indefinite Krein metric is left unresolved.

References

Several further directions remain open. First, it will be interesting to explore whether the delicate multiband route of~\Ccite{Lapierre2021}, that restores the $\mathbb{Z}$ invariant for all $N$-band Hopf insulators ($N\geq 2$) by resolving every band into its own gap (classifying space $SU(N)/U(1){N-1}$), carries over to the bosonic Krein-space setting.

— Symplectic Hopf Insulator: Delicate Topology in Bosonic Bogoliubov-de Gennes Systems  (2609.10541 - Tesfaye et al., 9 Sep 2026) in Section Summary and outlook (Section sec:summary-outlook)

Further, another open task is to come up with pairing-generated BBdG Hopf phases with no Hermitian parent, that is BBdG models of the type~eq:bog-hamiltonian-main where the Hopf topology is entirely carried by the off-diagonal non-particle-number conserving pairing blocks.

— Symplectic Hopf Insulator: Delicate Topology in Bosonic Bogoliubov-de Gennes Systems  (2609.10541 - Tesfaye et al., 9 Sep 2026) in Section Summary and outlook (Section sec:summary-outlook)