Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

TL;DR

This paper develops an exact hierarchical equations of motion framework for simulating the dynamics and quantum transport in dissipative electronic systems coupled to Fermion baths, capturing non-Markovian effects and arbitrary external fields.

Contribution

It introduces a hierarchically coupled equations of motion approach based on path integral influence functionals for arbitrary electronic systems with Fermion baths, extending previous formalisms.

Findings

Recovers real-time diagrammatic formalism at second tier

Exact for single-particle systems, reproducing Landauer-Büttiker current

Applicable to interacting systems with time-dependent external fields

Abstract

A quantum dissipation theory is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system coupled with grand canonical Fermion bath ensembles. The theoretical construction starts with the second--quantization influence functional in path integral formalism, in which the Fermion creation and annihilation operators are represented by Grassmann variables. Time--derivatives on influence functionals are then performed in a hierarchical manner, on the basis of calculus--on--path--integral algorithm. Both the multiple--frequency--dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting formalism is in principle exact, applicable to interacting systems, with arbitrary time-dependent external fields. It renders an exact tool to evaluate various transient and stationary quantum transport properties of many-electron systems. At the second--tier truncation level the present theory recovers the real--time diagrammatic formalism developed by Sch\"{o}n and coworkers. For a single-particle system, the hierarchical formalism terminates at the second tier exactly, and the Landuer--B\"{u}ttiker's transport current expression is readily recovered.