A diagrammatic description of the equations of motion, current, and noise within the second-order von Neumann approach

TL;DR

This paper analyzes the second-order von Neumann approach using diagrammatic methods, demonstrating its accuracy for current calculations in non-interacting systems and exploring its limitations for higher-order cumulants and noise predictions.

Contribution

It provides a diagrammatic perspective on the second-order von Neumann approach, clarifying its validity and limitations for current and noise calculations in quantum transport.

Findings

The method reproduces the equation of motion for non-interacting systems.

It accurately predicts current but not higher cumulants.

Discrepancies in noise calculations are analyzed and discussed.

Abstract

We investigate the second-order von Neumann approach from a diagrammatic point-of-view and demonstrate its equivalence with the resonant tunneling approximation. Investigation of higher-order diagrams shows that the method correctly reproduces the equation of motion for the single-particle reduced density matrix of an arbitrary non-interacting many-body system. This explains why the method reproduces the current exactly for such systems. We go on to show, however, that diagrams not included in the method are needed to calculate exactly higher cumulants of the charge transport. This thorough comparison sheds light on the validity of all these self consistent second-order approaches. We analyze the discrepancy between the noise calculated by our method and the exact Levitov formula for a simple non-interacting quantum dot model. Furthermore we study the noise of the canyon of current suppression in a two-level dot, a phenomenon that requires the inclusion of electron-electron interaction as well as higher-order tunneling processes.