Towards numerically exact computation of conductivity in the thermodynamic limit of interacting lattice models

TL;DR

This paper develops and applies two advanced methods to compute the optical conductivity in the Hubbard model at weak coupling, achieving numerically exact results in the thermodynamic limit without analytical continuation, and reveals persistent vertex corrections.

Contribution

It introduces two state-of-the-art, fully controlled methods for calculating dynamical response functions in the thermodynamic limit, demonstrating the significance of vertex corrections at weak coupling.

Findings

Vertex corrections persist at infinitesimal coupling.

Boltzmann theory's approximations lead to incorrect scaling.

Methods enable numerically exact conductivity calculations without analytical continuation.

Abstract

Computing dynamical response functions in interacting lattice models is a long standing challenge in condensed matter physics. In view of recent results, the dc resistivity $\rho_\mathrm{dc}$ in the weak coupling regime of the Hubbard model is of great interest, yet it is not fully understood. The challenge lies in having to work with large lattices while avoiding analytical continuation. The weak-coupling $\rho_\mathrm{dc}$ results were so far computed at the level of the Boltzmann theory and at the level of the Kubo bubble approximation, which neglects vertex corrections. Neither theory was so far rigorously proven to give exact results even at infinitesimal coupling, and the respective dc resistivity results differ greatly. In this work we develop, cross-check and apply two state-of-the-art methods for obtaining dynamical response functions. We compute the optical conductivity at weak coupling in the Hubbard model in a fully controlled way, in the thermodynamic limit and without analytical continuation. We show that vertex corrections persist to infinitesimal coupling, with a constant ratio to the Kubo bubble. We connect our methods with the Boltzmann theory, and show that the latter applies additional approximations, which lead to quantitatively incorrect scaling of $\rho_\mathrm{dc}$ with respect to the coupling constant.