  The doped Mott insulator in one dimension has been studied based on the phase
Hamiltonian with the Umklapp scattering process, in which the charge degree of
freedom is described by the quantum sine-Gordon model. The well-known
equivalence between the quantum sine-Gordon model and the massive Thirring
model for the spinless fermion makes it clear that the Mott-Hubbard gap
originates from the Umklapp scattering process as was indicated by Emery and
Giamarchi. Compressibility, density-density correlation function, frequency
dependence of optical conductivity and Drude weight have been calculated in the
presence of the impurity scattering treated in the self-consistent Born
approximation. It is seen that there exists a crossover behavior in the
spectral weight of charge excitations: the acoustic mode is dominant in small
wave number region while the optical excitations across the Mott-Hubbard gap
lie in large wave number region and that this crossover wave number is reduced
as the Mott transition is approached.
