Is The Replacement I Ω N → Ω + I Η I\omega_n\rightarrow \omega+i\eta I Ω N ​ → Ω + I Η In Matsubara Green Function Valid?

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Introduction

In many-body theory, the Matsubara Green function plays a crucial role in understanding the behavior of interacting systems. The Matsubara Green function is a mathematical tool used to describe the dynamics of particles in a system, and it is an essential component in the calculation of various physical properties, such as the density of states and the spectral function. However, the Matsubara Green function is defined in imaginary time, and to obtain the retarded Green function, a replacement of $i\omega_n$ with $\omega+i\eta$ is typically made. In this article, we will discuss the validity of this replacement and its implications for the calculation of physical properties.

Matsubara Green Function

The Matsubara Green function is defined as:

$$\mathcal{G}(i\omega_n) = \frac{1}{i\omega_n - H}$$

where $H$ is the Hamiltonian of the system, and $i\omega_n$ is the Matsubara frequency. The Matsubara frequency is a discrete variable that takes on values $i(2n+1)\pi T$, where $n$ is an integer and $T$ is the temperature.

Retarded Green Function

The retarded Green function is defined as:

$$G^R(\omega) = \frac{1}{\omega - H + i\eta}$$

where $\omega$ is the real frequency, and $\eta$ is a small positive quantity that represents the broadening of the spectral function.

Replacement of $i\omega_n$ with $\omega+i\eta$

To obtain the retarded Green function from the Matsubara Green function, a replacement of $i\omega_n$ with $\omega+i\eta$ is typically made. This replacement is based on the idea that the Matsubara Green function can be analytically continued to the real frequency axis, and the resulting expression is the retarded Green function.

Validity of the Replacement

The validity of the replacement $i\omega_n\rightarrow \omega+i\eta$ has been a topic of debate in the many-body community. Some authors argue that this replacement is valid only in the limit of zero temperature, while others argue that it is valid for all temperatures.

Analytic Continuation

The analytic continuation of the Matsubara Green function to the real frequency axis is a complex process that involves the use of contour integration and the residue theorem. The resulting expression is the retarded Green function, which is a function of the real frequency $\omega$.

Residue Theorem

The residue theorem is a mathematical tool used to evaluate the integral of a function over a closed contour. In the context of the Matsubara Green function, the residue theorem is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.

Contour Integration

Contour integration is a mathematical technique used to evaluate the integral of a function over a closed contour. In the context of the Matsubara Green function, contour integration is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.

Implications for Physical Properties

The replacement $i\omega_n\rightarrow \omega+i\eta$ has important implications for the calculation of physical properties, such as the density of states and the spectral function. The density of states is a measure of the number of states available to a particle in a system, and it is an essential component in the calculation of various physical properties.

Conclusion

In conclusion, the replacement $i\omega_n\rightarrow \omega+i\eta$ in the Matsubara Green function is a complex process that involves the use of analytic continuation, the residue theorem, and contour integration. While this replacement is widely used in many-body theory, its validity has been a topic of debate in the community. Further research is needed to fully understand the implications of this replacement for the calculation of physical properties.

References

• [1] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, 1975.
• [2] G. D. Mahan, Many-Particle Physics, Plenum Press, 1990.
• [3] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover Publications, 2003.

Appendix

A detailed derivation of the replacement $i\omega_n\rightarrow \omega+i\eta$ can be found in the appendix. The derivation involves the use of analytic continuation, the residue theorem, and contour integration.

Derivation of the Replacement

The derivation of the replacement $i\omega_n\rightarrow \omega+i\eta$ involves the following steps:

1. Analytic Continuation: The Matsubara Green function is analytically continued to the real frequency axis using the residue theorem.
2. Residue Theorem: The residue theorem is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.
3. Contour Integration: Contour integration is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.

The resulting expression is the retarded Green function, which is a function of the real frequency $\omega$.

Mathematical Derivation

The mathematical derivation of the replacement $i\omega_n\rightarrow \omega+i\eta$ involves the following steps:

1. Matsubara Green Function: The Matsubara Green function is defined as:

$$\mathcal{G}(i\omega_n) = \frac{1}{i\omega_n - H}$$

2. Analytic Continuation: The Matsubara Green function is analytically continued to the real frequency axis using the residue theorem.

$$\mathcal{G}(\omega) = \frac{1}{\omega - H + i\eta}$$

3. Residue Theorem: The residue theorem is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.

$$\int_{-\infty}^{\infty} \mathcal{G}(\omega) d\omega = 2\pi i \sum_{n} \mathcal{G}(i\omega_n)$$

4. Contour Integration: Contour integration is used to evaluate the integral of the Green function over a contour that encircles the real frequency axis.

$$\int_{-\infty}^{\infty} \mathcal{G}(\omega) d\omega = 2\pi i \sum_{n} \frac{1}{i\omega_n - H}$$

The resulting expression is the retarded Green function, which is a function of the real frequency $\omega$.

Conclusion

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Q: What is the Matsubara Green function?

A: The Matsubara Green function is a mathematical tool used to describe the dynamics of particles in a system. It is an essential component in the calculation of various physical properties, such as the density of states and the spectral function.

Q: What is the purpose of the replacement $i\omega_n\rightarrow \omega+i\eta$?

A: The purpose of the replacement $i\omega_n\rightarrow \omega+i\eta$ is to obtain the retarded Green function from the Matsubara Green function. This replacement is based on the idea that the Matsubara Green function can be analytically continued to the real frequency axis, and the resulting expression is the retarded Green function.

Q: Is the replacement $i\omega_n\rightarrow \omega+i\eta$ valid for all temperatures?

A: The validity of the replacement $i\omega_n\rightarrow \omega+i\eta$ has been a topic of debate in the many-body community. Some authors argue that this replacement is valid only in the limit of zero temperature, while others argue that it is valid for all temperatures.

Q: What is the mathematical derivation of the replacement $i\omega_n\rightarrow \omega+i\eta$?

A: The mathematical derivation of the replacement $i\omega_n\rightarrow \omega+i\eta$ involves the use of analytic continuation, the residue theorem, and contour integration. The resulting expression is the retarded Green function, which is a function of the real frequency $\omega$.

Q: What are the implications of the replacement $i\omega_n\rightarrow \omega+i\eta$ for the calculation of physical properties?

A: The replacement $i\omega_n\rightarrow \omega+i\eta$ has important implications for the calculation of physical properties, such as the density of states and the spectral function. The density of states is a measure of the number of states available to a particle in a system, and it is an essential component in the calculation of various physical properties.

Q: What are some common applications of the Matsubara Green function?

A: The Matsubara Green function has many applications in condensed matter physics, including the calculation of the density of states, the spectral function, and the transport properties of systems.

Q: What are some common criticisms of the replacement $i\omega_n\rightarrow \omega+i\eta$?

A: Some common criticisms of the replacement $i\omega_n\rightarrow \omega+i\eta$ include the fact that it is not a well-defined mathematical operation, and that it can lead to incorrect results in certain situations.

Q: What are some alternative methods for obtaining the retarded Green function?

A: Some alternative methods for obtaining the retarded Green function include the use of the Keldysh formalism, the use of the Schwinger-Dyson equations, and the use of the functional renormalization group.

Q: What are open questions in the field of many-body theory?

A: Some open questions in the field of many-body theory include the development of a more rigorous mathematical framework for the Matsubara Green function, the development of more accurate methods for calculating physical properties, and the development of a deeper understanding of the relationship between the Matsubara Green function and the retarded Green function.

Conclusion

In conclusion, the replacement $i\omega_n\rightarrow \omega+i\eta$ in the Matsubara Green function is a complex process that involves the use of analytic continuation, the residue theorem, and contour integration. While this replacement is widely used in many-body theory, its validity has been a topic of debate in the community. Further research is needed to fully understand the implications of this replacement for the calculation of physical properties.