  We analyze the chiral phase transition of the Nambu--Jona-Lasinio model in
the cold and dense region on the lattice developing the Grassmann version of
the anisotropic tensor renormalization group algorithm. The model is formulated
with the Kogut--Susskind fermion action. We use the chiral condensate as an
order parameter to investigate the restoration of the chiral symmetry. The
first-order chiral phase transition is clearly observed in the dense region at
vanishing temperature with $\mu/T\sim O(10^3)$ on a large volume of $V=1024^4$.
We also present the results for the equation of state.
