Non-equilibrium Green's function formalism and the problem of bound states

TL;DR

This paper examines the limitations of the non-equilibrium Green's function formalism in accurately describing bound states, highlighting issues with the uniqueness of the density matrix without additional equilibration mechanisms.

Contribution

It identifies the problem of non-uniqueness in the density matrix for bound states within the formalism and proposes auxiliary reservoirs as a potential solution.

Findings

Correct density for continuum states obtained

Non-uniqueness of bound state density matrix identified

Auxiliary reservoirs restore uniqueness in equilibrium

Abstract

The non-equilibrium Green's function formalism for infinitely extended reservoirs coupled to a finite system can be derived by solving the equations of motion for a tight-binding Hamiltonian. While this approach gives the correct density for the continuum states, we find that it does not lead, in the absence of any additional mechanisms for equilibration, to a unique expression for the density matrix of any bound states which may be present. Introducing some auxiliary reservoirs which are very weakly coupled to the system leads to a density matrix which is unique in the equilibrium situation, but which depends on the details of the auxiliary reservoirs in the non-equilibrium case.