  We study a system of strongly interacting one-dimensional (1D) bosons on a
ring pierced by a synthetic magnetic flux tube. By the Fermi-Bose mapping, this
system is related to the system of spin-polarized non-interacting electrons
confined on a ring and pierced by a solenoid (magnetic flux tube). On the ring
there is an external localized delta-function potential barrier $V(\phi)=g
\delta(\phi-\phi_0)$. We study the Berry phase associated to the adiabatic
motion of delta-function barrier around the ring as a function of the strength
of the potential $g$ and the number of particles $N$. The behavior of the Berry
phase can be explained via quantum mechanical reflection and tunneling through
the moving barrier which pushes the particles around the ring. The barrier
produces a cusp in the density to which one can associate a missing charge
$\Delta q$ (missing density) for the case of electrons (bosons, respectively).
We show that the Berry phase (i.e., the Aharonov-Bohm phase) cannot be
identified with the quantity $\Delta q/\hbar \oint \mathbf{A}\cdot
d\mathbf{l}$. This means that the missing charge cannot be identified as a
(quasi)hole. We point out to the connection of this result and recent studies
of synthetic anyons in noninteracting systems. In addition, for bosons we study
the weakly-interacting regime, which is related to the strongly interacting
electrons via Fermi-Bose duality in 1D systems.
