  We propose a new framework for investigating two-flavor lattice QCD with
finite temperature and density. We consider the Karsten-Wilczek fermion
formulation, in which a species-dependent imaginary chemical potential term can
reduce the number of species to two without losing chiral symmetry. This
lattice discretization is useful for study on finite-($T$,$\mu$) QCD since its
discrete symmetries are appropriate for the case. To show its applicability, we
study strong-coupling lattice QCD with temperature and chemical potential. We
derive the effective potential of the scalar meson field and obtain a critical
line of the chiral phase transition, which is qualitatively consistent with the
phenomenologically expected phase diagram. We also discuss that $O(1/a)$
renormalization of imaginary chemical potential can be controlled by adjusting
a parameter of a dimension-3 counterterm.
