Motivated by the concept of M\"obius aromatics in organic chemistry, we
extend the recently introduced concept of fragile Mott insulators (FMI) to
ring-shaped molecules with repulsive Hubbard interactions threaded by a
half-quantum of magnetic flux (hc/2e). In this context, a FMI is the
insulating ground state of a finite-size molecule that cannot be adiabatically
connected to a single Slater determinant, i.e., to a band insulator, provided
that time-reversal and lattice translation symmetries are preserved. Based on
exact numerical diagonalization for finite Hubbard interaction strength U and
existing Bethe-ansatz studies of the one-dimensional Hubbard model in the
large-U limit, we establish a duality between Hubbard molecules with 4n and
4n+2 sites, with n integer. A molecule with 4n sites is an FMI in the
absence of flux but becomes a band insulator in the presence of a half-quantum
of flux, while a molecule with 4n+2 sites is a band insulator in the absence
of flux but becomes an FMI in the presence of a half-quantum of flux. Including
next-nearest-neighbor-hoppings gives rise to new FMI states that belong to
multidimensional irreducible representations of the molecular point group,
giving rise to a rich phase diagram
