Dynamical conductivity of disordered quantum chains

TL;DR

This paper investigates the dynamical conductivity of disordered one-dimensional quantum fermionic chains with interactions, using Chebyshev matrix product state methods, revealing power-law decay, localization effects, and agreement with field theory predictions.

Contribution

The study applies CheMPS to compute dynamical conductivity in interacting disordered chains, providing numerical validation of theoretical predictions and exploring the effects of interactions on localization and transport.

Findings

Conductivity decays as a power law at high frequencies with interaction-dependent exponents.

Localization length scales as a power law of disorder strength, matching field theory.

Low frequency behavior is consistent with noninteracting systems, independent of interactions.

Abstract

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.