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Pair-density-wave (PDW) superconductivity is a long-sought exotic state withoscillating superconducting order parameter without the need of applyingexternal magnetic field. So far it has been rare in establishing atwo-dimensional (2D) microscopic model with such exotic long-range order in itsground state. Here we propose to study PDW superconductivity in a minimal modelof spinless fermions on the honeycomb lattice with nearest-neighbor (NN) andnext-nearest-neighbor (NNN) interaction $V_1$ and $V_2$, respectively. Byperforming a state-of-the-art density-matrix renormalization group (DMRG) studyof this $t$-$V_1$-$V_2$ model at finite doping on six-leg and eight-leghoneycomb cylinders, we showed that the ground state exhibits PDW ordering(namely quasi-long-range order with a divergent PDW susceptibility). Remarkablythis 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 numericalevidence of PDW in systems with 2D-like FS.

Pair density wave superconductivity: a microscopic model in two dimensions