Block-determinant formalism for an action of a multi-terminal scatterer

TL;DR

This paper develops a generalized block-determinant formalism for the action of multi-terminal quantum scatterers, enhancing the theoretical framework for electron transport, counting statistics, and superconducting correlations in complex junctions.

Contribution

It introduces a comprehensive derivation of a multi-terminal action expressed as a block-determinant of the scattering matrix, extending existing two-terminal formalisms.

Findings

Derived a general block-determinant form of the action for multi-terminal scatterers

Reproduced basic results of counting statistics in multi-terminal junctions

Analyzed chiral anomaly in one-dimensional channels

Abstract

The scattering theory of electron transport allows for a compact and powerful description in terms of $\check{g}^2 = 1$ Green functions, so-called circuit theory of quantum transport. A scatterer in the theory is characterized by an action, most generally a Keldysh one, that can be further used as a building bock of theories describing statistics of electron transport, superconducting correlations, time-dependent and interaction effects. The action is usually used in the form suitable for a two-terminal scatterer. Here we provide a comprehensive derivation of a more general form of the action that is especially suitable and convenient for general multi-terminal scatterers. The action is expressed as a determinant of a block of the scattering matrix obtained by projection on the positive eigenvalues of the Green functions characterizing the reservoirs. We start with traditional Green function formalism introducing $\check{g}^2 = 1$ matrices and give a first example of multi-terminal counting statistics. Further we consider one-dimensional channels and discuss chiral anomaly arising in this context. Generalizing on many channels and superconducting situation, we arrive at the block-determinant relation. We give the necessary elaborative examples reproducing basic results of counting statistics and super-currents in multi-terminal junctions.