Tomonaga-Luttinger liquid with reservoirs in a multi-terminal geometry

TL;DR

This paper introduces a boundary condition-based formalism to analyze transport in a Luttinger liquid coupled to reservoirs, overcoming limitations of the Landauer-B"uttiker approach, and applies it to multi-terminal geometries and shot noise.

Contribution

It develops a new formalism using boundary conditions for Luttinger liquids coupled to reservoirs, providing exact solutions where traditional methods fail.

Findings

Boundary conditions fully determine transport properties in LL-reservoir systems.

Landauer-B"uttiker formalism can give unphysical negative probabilities in this context.

Application to shot noise reveals new insights into impurity effects in LLs.

Abstract

We propose a formalism which uses boundary conditions imposed on the Luttinger liquid (LL) to describe the transport properties of a LL coupled to reservoirs. The various boundary conditions completely determine linear transport in the joint system reservoirs+LL. As an illustration we consider an exactly solvable microscopic model in a multi-terminal geometry for which such boundary conditions can be explicitly derived; in this model the Landauer-B\"uttiker formalism fails: if it were valid, the relation between the conductance matrix elements and the reflection and transmission coefficients could yield negative probabilities. We then apply our formalism to a discussion of shot noise through an impurity in a LL connected to two reservoirs.