Quantum embedding theories

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.

• The paper presents a unified exposition of density functional, Green’s function, and density matrix embedding methods for simulating interactions between a subsystem and its broader quantum environment.
• It details how each formalism decouples subsystem computations through unique potentials, hybridization terms, or Schmidt decomposition to reduce computational overhead.
• The work outlines future challenges in modeling excited states and dynamic systems, emphasizing the need for further methodological innovation in quantum embedding.

Quantum Embedding Theories: A Unified Overview

The paper "Quantum Embedding Theories" by Qiming Sun and Garnet Kin-Lic Chan provides an intricate exposition of quantum embedding frameworks. These theories are essential for the study of complex quantum systems where a region of interest interacts with its broader environment. By integrating distinct quantum simulation techniques into a holistic computational protocol, quantum embedding theories facilitate an efficient focus on specific subsystems while considering their interactions with the surroundings.

The authors address three primary formalisms of quantum embedding: Density Functional Embedding (DFE), Green's Function Embedding (GFE), and Density Matrix Embedding Theory (DMET). These frameworks utilize different quantum variables—single-particle density, single-particle Green's function, and single-particle density matrix, respectively—to characterize the target subsystem and its environment.

Density Functional Embedding

In DFE, the authors initiate the discussion with an examination of the embedding potential via the Euler equation. This potential accounts for interactions with the environment primarily through the non-additive kinetic potential. The formalism effectively decouples the subsystem computations from the environment, potentially reducing computational overhead. Novel methodologies, such as wavefunction in DFT embedding, enhance this approach, particularly when the subsystem is treated via high-level wavefunction methods, addressing some limits of DFT in strongly correlated systems.

Green's Function Embedding

Covering GFE, the paper explores its application using Dyson's equation to derive embedding relations analogous to those in DFE. The hybridization term analogous to the DFT embedding potential adds flexibility in capturing more dynamic interactions between subsystem and environment. The utility of GFE has found significant traction in condensed matter physics, but the challenge remains in its substantial computational demand, especially when moving beyond mean-field approximations to more sophisticated self-energy computations.

Density Matrix Embedding Theory

DMET eschews the complexity of Green's function formulations by focusing on the single-particle density matrix, which provides an intermediate level of information richness. The Schmidt decomposition employed in DMET introduces bath orbitals that assist in maintaining computational efficiency while ensuring that subsystem density matrices accurately reflect their entanglements with the environment. Notably, DMET has been effective in strongly correlated lattice models and is increasingly relevant in applications across chemistry and condensed matter physics.

Implications and Future Directions

This unified presentation underscores the versatility of quantum embedding theories across multiple physical systems by harmonizing formalisms based on their structural and informational semantics. The convergence of these methods indicates potential new avenues in the simulation of complex molecular and condensed phase systems, where embedding methods can incorporate additional theoretical constructs such as time-dependence and classical embeddings.

The paper suggests that while each embedding formalism contributes valuable insights, the theoretical challenges remain in deriving efficient and universally applicable embedding methods, particularly for excited states and dynamic simulations. As such, continued methodological innovation and computational advancements are crucial to realizing the full potential of quantum embedding theories in both established and emerging domains of scientific inquiry.