  Pair-density-wave (PDW) superconductivity is a long-sought exotic state with
oscillating superconducting order parameter without the need of applying
external magnetic field. So far it has been rare in establishing a
two-dimensional (2D) microscopic model with such exotic long-range order in its
ground state. Here we propose to study PDW superconductivity in a minimal model
of spinless fermions on the honeycomb lattice with nearest-neighbor (NN) and
next-nearest-neighbor (NNN) interaction $V_1$ and $V_2$, respectively. By
performing a state-of-the-art density-matrix renormalization group (DMRG) study
of this $t$-$V_1$-$V_2$ model at finite doping on six-leg and eight-leg
honeycomb cylinders, we showed that the ground state exhibits PDW ordering
(namely quasi-long-range order with a divergent PDW susceptibility). Remarkably
this PDW state persists on the wider cylinder with 2D-like Fermi surfaces (FS).
To the best of our knowledge, this is probably the first controlled numerical
evidence of PDW in systems with 2D-like FS.
