  We study the transport properties of a one dimensional quantum system with
disorder. We numerically compute the frequency dependence of the conductivity
of a fermionic chain with nearest neighbor interaction and a random chemical
potential by using the Chebyshev matrix product state (CheMPS) method. As a
benchmark, we investigate the noninteracting case first. Comparison with exact
diagonalization and analytical solutions demonstrates that the results of
CheMPS are reliable over a wide range of frequencies. We then calculate the
dynamical conductivity spectra of the interacting system for various values of
the interaction and disorder strengths. In the high frequency regime, the
conductivity decays as a power law, with an interaction dependent exponent.
This behavior is qualitatively consistent with the bosonized field theory
predictions, although the numerical evaluation of the exponent shows deviations
from the analytically expected values. We also compute the characteristic
pinning frequency at which a peak in the conductivity appears. We confirm that
it is directly related to the inverse of the localization length, even in the
interacting case. We demonstrate that the localization length follows a power
law of the disorder strength with an exponent dependent on the interaction, and
find good quantitative agreement with the field theory predictions. In the low
frequency regime, we find a behavior consistent with the one of the
noninteracting system $\omega^{2}(\ln\omega)^{2}$ independently of the
interaction. We discuss the consequences of our finding for experiments in cold
atomic gases.
