Dynamical conductivity in interacting quantum chains with disorder

We investigate the optical conductivity of quantum chains with nearest-neighbor repulsive interactions with random chemical potential. When the repulsive interaction is smaller than hopping, the system is in the gapless Tomonaga-Luttinger liquid phase, and when the Luttinger parameter satisfies K<3/2, the system is localized by disorder. We calculate the optical conductivity of this system numerically using the Chebyshev matrix product state method. In the high-frequency region, the optical conductivity decays according to a power law with an exponent that depends on the strength of the interaction. In the weak interaction region, the exponent of the decay agrees with the prediction from the effective field theory obtained by bosonization, while in the strong interaction region, the exponent estimated from the numerical calculation deviates from the analytically expected value. We also determine the characteristic pinning frequency that appears as a peak in the optical conductivity spectrum, which is found to be directly related to the inverse of the localization length. We confirm that the localization length follows a power law with an exponent depending on the interaction strength. In the low-frequency region, the optical conductivity spectrum behaves as ω²(ln ω)², which is the same as in the case of free fermions.