  In a recent contribution [Phys. Rev. B 81, 165104 (2010)] fermionic Projected
Entangled-Pair States (PEPS) were used to approximate the ground state of free
and interacting spinless fermion models, as well as the $t$-$J$ model. This
paper revisits these three models in the presence of an additional next-nearest
hopping amplitude in the Hamiltonian. First we explain how to account for
next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS
algorithms based on simulating time evolution. Then we present benchmark
calculations for the three models of fermions, and compare our results against
analytical, mean-field, and variational Monte Carlo results, respectively.
Consistent with previous computations restricted to nearest-neighbor
Hamiltonians, we systematically obtain more accurate (or better converged)
results for gapped phases than for gapless ones.
