Quantum embedding theories

TL;DR

This paper reviews and unifies three rigorous quantum embedding formalisms—density functional, Green's function, and density matrix embedding—highlighting their common principles and recent applications.

Contribution

It provides a unified presentation of the three main quantum embedding theories, clarifying their similarities and differences.

Findings

Unified equations for all three embedding formalisms

Identification of common conceptual foundations

Overview of recent applications and future directions

Abstract

In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calculation on the active region of interest, and a low level calculation on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: it density functional embedding, Green's function embedding, and density matrix embedding, respectively use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used: functional equations for density functional embedding; diagrammatics for Green's function embedding; and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications, and possible future developments.