  We develop a formalism for constructing particle-number-conserving Gaussian
fermionic projected entangled pair states [U(1)-GfPEPS] and show that these
states can describe ground states of band insulators and gapless fermions with
band touching points. When using them as variational Ans\"{a}tze for two Dirac
fermion systems ($\pi$-flux model on the square lattice and $[0,\pi]$-flux
model on the kagome lattice), we find that the U(1)-GfPEPS, even with a
relatively small bond dimension, can accurately approximate the Dirac Fermi sea
ground states. By applying Gutzwiller projectors on top of these U(1)-GfPEPS,
we obtain PEPS representation of U(1)-Dirac spin liquid states for spin-1/2
systems. With state-of-the-art tensor network numerics, the critical exponent
in the spin-spin correlation function of the Gutzwiller-projected $\pi$-flux
state is estimated to be $\eta \approx 1.7$.
