  The conventional tensor-network states employ real-space product states as
reference wave functions. Here, we propose a many-variable variational Monte
Carlo (mVMC) method combined with tensor networks by taking advantages of both
to study fermionic models. The variational wave function is composed of a pair
product wave function operated by real space correlation factors and tensor
networks. Moreover, we can apply quantum number projections, such as spin,
momentum and lattice symmetry projections, to recover the symmetry of the wave
function to further improve the accuracy. We benchmark our method for one- and
two-dimensional Hubbard models, which show significant improvement over the
results obtained individually either by mVMC or by tensor network. We have
applied the present method to hole doped Hubbard model on the square lattice,
which indicates the stripe charge/spin order coexisting with a weak $d$-wave
superconducting order in the ground state for the doping concentration less
than 0.3, where the stripe oscillation period gets longer with increasing hole
concentration. The charge homogeneous and highly superconducting state also
exists as a metastable excited state for the doping concentration less than
0.25.
