Eigenfunction structure and scaling of two interacting particles in the one-dimensional Anderson model

TL;DR

This study investigates the eigenfunction structure and scaling behavior of two interacting particles in a one-dimensional Anderson model, confirming the $L_2 o L_1^2$ enhancement and applying one-parameter scaling theory for large systems.

Contribution

The paper provides new large-scale numerical analysis of two-particle localization, confirming the $L_2 o L_1^2$ enhancement and extending understanding of interaction effects across various parameters.

Findings

Confirmation of $L_2 o L_1^2$ localization length enhancement.

Validation of one-parameter scaling theory for large systems.

New insights into interaction dependence outside the band center.

Abstract

The localization properties of eigenfunctions for two interacting particles in the one-dimensional Anderson model are studied for system sizes up to $N=5000$ sites corresponding to a Hilbert space of dimension $\approx 10^7$ using the Green function Arnoldi method. The eigenfunction structure is illustrated in position, momentum and energy representation, the latter corresponding to an expansion in non-interacting product eigenfunctions. Different types of localization lengths are computed for parameter ranges in system size, disorder and interaction strengths inaccessible until now. We confirm that one-parameter scaling theory can be successfully applied provided that the condition of $N$ being significantly larger than the one-particle localization length $L_1$ is verified. The enhancement effect of the two-particle localization length $L_2$ behaving as $L_2\sim L_1^2$ is clearly confirmed for a certain quite large interval of optimal interactions strengths. Further new results for the interaction dependence in a very large interval, an energy value outside the band center, and different interaction ranges are obtained.