  The properties of strangeness neutron star are studied within relativistic
mean-field (RMF) model via including a logarithmic interaction as a function of
scalar meson field. This logarithmic interaction, named as the $\sigma$-cut
potential, can largely reduce the attractive contributions of scalar meson
field at high density without any influence on nuclear structure around normal
saturation density. In this work, the TM1 parameter set is chosen as the RMF
interaction, while the strengths of logarithmic interaction are constrained by
the properties of finite nuclei so that we can obtain a reasonable effective
nucleon-nucleon interaction. The hyperons, $\Lambda,~\Sigma$, and $\Xi$ are
also considered in neutron stars within this framework, whose coupling
constants with mesons are determined by the latest hyperon-nucleon and
$\Lambda$-$\Lambda$ potentials extracted from the experimental data of
hypernuclei. The maximum mass of neutron star can be larger than two solar mass
with these hyperons. Furthermore, the nucleon mass at high density will be
saturated due to this additional $\sigma$-cut potential, which is consistent
with the conclusions from the microscopic calculations such as,
Brueckner-Hartree-Fock theory and quark mean-field model.
