Friedel phases and phases of transmission amplitudes in quantum scattering systems

TL;DR

This paper explores the relationship between Friedel phases and transmission amplitude phases in quantum scattering systems, analyzing their behavior and verifying the Friedel sum rule through simple models and complex plane analysis.

Contribution

It clarifies the connection between Friedel phases and transmission phases, highlighting the effects of transmission zeros and providing verification methods for the Friedel sum rule.

Findings

Transmission zeros cause abrupt phase shifts of ±π.

Friedel phase remains continuous despite abrupt changes in transmission phase.

The Friedel sum rule is verified through direct calculations in simple models.

Abstract

We illustrate the relation between the scattering phase appearing in the Friedel sum rule and the phase of the transmission amplitude for quantum scatterers connected to two one-dimensional leads. Transmission zero points cause abrupt phase changes $\pm\pi$ of the phase of the transmission amplitude. In contrast the Friedel phase is a continuous function of energy. We investigate these scattering phases for simple scattering problems and illustrate the behavior of these models by following the path of the transmission amplitude in the complex plane as a function of energy. We verify the Friedel sum rule for these models by direct calculation of the scattering phases and by direct calculation of the density of states.