  We discuss possible existence of inhomogeneous phase in low temperature and
high density region in the 1+1 dimensional chiral Gross-Neveu ( $\chi$GN$_2$ )
model on the lattice. First we investigate the phase structure of the
$\chi$GN$_2$ model at vanishing chemical potential, changing temperature. From
behavior of $\Delta^2 = \Sigma^2 + \Pi^2$ as a function of Monte Carlo time, a
candidate of an order parameter for chiral symmetry in the $\chi$GN$_2$ model
is $\Delta$. At vanishing chemical potential, we observe restoration of chiral
symmetry at high $T$. In low temperature and high chemical potential region, we
find existence of inhomogeneous phase in spatial correlation functions of
$\sigma$ and $\pi$. The signal of inhomogeneous phase in correlation function
of $\Delta$ is more clearly than that of $\sigma$ or $\pi$.
