Many-body quantum theory in condensed matter physics

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Welcome to a sophisticated exploration of the quantum world through the lens of many-body quantum theory in condensed matter physics. In this thorough introduction, we delve into the depths of quantum mechanics and condensed matter physics to provide a comprehensive understanding of the intricate interactions that define this realm.

Detailed Summary of the Book

The book 'Many-body Quantum Theory in Condensed Matter Physics' serves as a crucial resource for those with a keen interest in the quantum mechanics of condensed matter. It starts by establishing foundational knowledge, covering basic quantum mechanical principles and advancing towards more complex concepts involving interacting quantum systems. The text intricately weaves theoretical development with practical application, situated within the broader context of condensed matter. Topics such as Fermi gases, Bose-Einstein condensation, superconductivity, and magnetism are thoroughly explored. We include robust discussions on the Hubbard model and Quantum Hall effects, ensuring a well-rounded grasp of the subject. The book culminates in an exploration of non-equilibrium phenomena, offering insight into development areas such as quantum computing and quantum materials. Designed to engage both seasoned physicists and those newer to the discipline, our book meticulously articulates the mathematical frameworks and experimental methods underpinning modern condensed matter physics.

Key Takeaways

Gain a comprehensive understanding of the fundamental and advanced aspects of many-body quantum theory.
Develop the ability to apply theoretical frameworks to solve complex problems in condensed matter physics.
Understand and analyze significant phenomena such as superconductivity and the Quantum Hall effects.
Explore the intersection of theoretical predictions and experimental research, preparing for future advances in quantum technologies.

Famous Quotes from the Book

"The essence of many-body quantum physics is that understanding the collective behavior of a multitude of particles often reveals fundamentally new physical phenomena, which cannot be deduced simply by extrapolating the properties of individual particles." - Henrik Bruus, Karsten Flensberg

"The strength of many-body theory lies in its ability to provide unified explanations and predictions across a variety of systems, highlighting the universal nature of underlying physics." - Henrik Bruus, Karsten Flensberg

Why This Book Matters

As the field of condensed matter physics continues to grow and evolve, understanding many-body quantum theory becomes increasingly critical. This book offers a robust framework for navigating the complexities of the quantum world, providing insights that extend beyond academia into practical, real-world applications. By bridging the gap between theoretical concepts and practical implementations, the book is an indispensable resource for anyone participating in the advancement of quantum science and technology. Its detailed examination of crucial phenomena in condensed matter physics sets the foundation for innovations in fields such as material science, nanotechnology, and quantum computing. Whether you are a researcher, student, or professional, this book equips you with the necessary tools to contribute to and thrive in one of the most dynamic areas of modern physics.

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Authors:

Karsten Flensberg

Questions about Book

mojvd

How calculate DOS using NEGF?

Book's Answer:

( AI as Henrik Bruus,Karsten Flensberg )

The calculation of the density of states (DOS) using Nonequilibrium Green's Function (NEGF) formalism involves evaluating the retarded Green's function. Here is a general outline of the process:

1. **Hamiltonian Setup**: Define your system's Hamiltonian, which may include the central region and the coupling to leads or other subsystems.

2. **Retarded Green's Function**: Calculate the retarded Green's function \(\mathbf{G}^r(E)\), which is defined as:
\[
\mathbf{G}^r(E) = [E \mathbf{I} - \mathbf{H} - \mathbf{\Sigma}^r(E)]^{-1}
\]
where \(\mathbf{I}\) is the identity matrix, \(\mathbf{H}\) is the Hamiltonian of the central region, and \(\mathbf{\Sigma}^r(E)\) is the self-energy accounting for the coupling to leads.

3. **Self-Energy Calculation**: Determine the self-energy \(\mathbf{\Sigma}^r(E)\), which represents the effect of the leads and is dependent on the specific system setup, including the nature of the coupling.

4. **DOS Calculation**: Once you have the retarded Green's function, the local density of states (LDOS) at a given energy \(E\) can be calculated using the formula:
\[
\rho(E) = -\frac{1}{\pi} \text{Im} \{ \text{Tr} [ \mathbf{G}^r(E) ] \}
\]
Here, \(\text{Im}\) denotes the imaginary part and \(\text{Tr}\) is the trace over the degrees of freedom of the central region.

By following these steps with the system's specific parameters and Hamiltonian, you can compute the DOS using the NEGF formalism. Note that detailed examples, boundary conditions, and specific system setups might be further elaborated in the context of the book and specific quantum systems.

mojvd

How calculate DOS using NEGF?

Book's Answer:

( AI as Henrik Bruus,Karsten Flensberg )

The calculation of the density of states (DOS) using non-equilibrium Green's functions (NEGF) involves analyzing the retarded Green's function. The DOS is related to the imaginary part of the retarded Green's function, \(G^R(\mathbf{k}, \omega)\), for the system. The general expression for DOS is given by:

\[
D(\omega) = -\frac{1}{\pi} \sum_{\mathbf{k}} \text{Im} \, \text{Tr} \, G^R(\mathbf{k}, \omega)
\]

In the NEGF formalism, the retarded Green's function \(G^R(\mathbf{k}, \omega)\) is calculated using the Dyson equation or from the relation:

\[
G^R = \left[(\omega + i\eta)I - H - \Sigma^R\right]^{-1}
\]

where:
- \(H\) is the Hamiltonian of the system,
- \(\Sigma^R\) is the retarded self-energy that accounts for interactions and coupling to leads or other parts of the system,
- \(\omega\) is the energy,
- \(i\eta\) is an infinitesimal positive imaginary component introduced for causality.

Once \(G^R(\mathbf{k}, \omega)\) is determined, the DOS can be calculated by taking the imaginary part and performing the sum over the Brillouin zone for all wave vectors \(\mathbf{k}\). This calculation needs to be performed either numerically or analytically, depending on the complexity of the system.

This method is crucial for studying quantum systems, especially out of equilibrium, and provides significant insights into the electronic properties of materials.

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