  We explore effects of the light vector $U$-boson, which is weakly coupled to
nucleons, on the transition density $\rho_{t}$ and pressure $P_{t}$ at the
inner edge separating the liquid core from the solid crust of neutron stars.
Three methods, i.e., the thermodynamical approach, the curvature matrix
approach and the Vlasov equation approach are used to determine the transition
density $\rho_{t}$ with the Skyrme effective nucleon-nucleon interactions. We
find that the $\rho_{t}$ and $P_{t}$ depend on not only the ratio of coupling
strength to mass squared of the $U$-boson $g^{2}/\mu ^{2}$ but also its mass
$\mu $ due to the finite range interaction from the $U$-boson exchange. In
particular, our results indicate that the $\rho_{t}$ and $P_{t}$ are sensitive
to both $g^{2}/\mu ^{2}$ and $\mu $ if the $U$-boson mass $\mu $ is larger than
about 2 MeV. Furthermore, we show that both $g^{2}/\mu ^{2}$ and $\mu $ can
have significant influence on the mass-radius relation and the crustal fraction
of total moment of inertia of neutron stars. In addition, we study the exchange
term contribution of the $U$-boson based on the density matrix expansion
method, and demonstrate that the exchange term effects on the nuclear matter
equation of state as well as the $\rho_{t}$ and $P_{t}$ are generally
negligible.
