Quantum Embedding Theories.

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Complex systems often require simulating only a small region of interest within a larger environment.
• Quantum embedding methods combine high-level calculations on active regions with low-level calculations on their environment.
• Existing rigorous formalisms for quantum embedding include density functional, Green's function, and density matrix embedding, each using different quantum variables.

Purpose of the Study:

• To provide a unified presentation of three major quantum embedding formalisms.
• To highlight the common theoretical underpinnings and intellectual strands connecting these methods.
• To offer a theoretical perspective with an overview of recent applications and future developments.

Main Methods:

• Introduces density functional embedding via the Euler equation, discussing potentials, mean-field, and wave function approaches.
• Presents Green's function embedding using the Dyson equation, drawing parallels with density functional methods and introducing concepts like hybridization and bath representation.
• Explains density matrix embedding, utilizing Schmidt decomposition to represent entanglement and connecting to dynamical mean-field theory.

Main Results:

• Demonstrates the shared theoretical foundations across density functional, Green's function, and density matrix embedding formalisms.
• Provides parallel equations and conceptual links between the different embedding approaches.
• Outlines recent advancements and potential future directions for each embedding method.

Conclusions:

• Quantum embedding methods, despite differing mathematical languages, share fundamental theoretical connections.
• A unified understanding facilitates the application and development of these powerful simulation techniques.
• This work serves as a foundational resource for researchers in quantum embedding and complex system simulations.