On the Su-Schrieffer-Heeger model of electron transport: low-temperature optical conductivity by the Mellin transform

TL;DR

This paper analytically investigates the low-temperature optical conductivity of a 1D electron system modeled by the Su-Schrieffer-Heeger Hamiltonian, focusing on the behavior as the energy gap closes, using Mellin transform techniques.

Contribution

It introduces an analytical method employing Mellin transforms to evaluate the optical conductivity of the SSH model near the gap-closing point.

Findings

Analytical expressions for optical conductivity near the bandgap closure.

Good agreement between analytical results and numerical simulations.

Identification of singularities in the Mellin transform related to conductivity behavior.

Abstract

We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su-Schrieffer-Heeger \emph{tight-binding} Hamiltonian for non-interacting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo-type formulation for the optical conductivity with a nonzero damping due to microscopic collisions; (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone; and (iii) evaluation of this integral in terms of elementary functions via the three-dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.