Dyson Equation and Green's Functions

Hello! Let's continue our exploration of diagrammatic perturbation theory.

In our last lesson, we took a major step by calculating the first-order self-energy $\Sigma$ for both electron-electron and electron-impurity interactions. We saw that $\Sigma$ acts as a repository for the effects of interactions, with its real part shifting energies and its imaginary part introducing finite lifetimes. We defined it as the sum of all one-particle irreducible (1PI) diagrams.

Today, we will address the central purpose of the self-energy. Our learning outcome is to use the Dyson equation to relate the bare and dressed Green's functions via the self-energy. This equation is arguably the most important in many-body perturbation theory. It allows us to go beyond simple first-order corrections and package the effects of an infinite number of interaction processes into a single, elegant, and powerful self-consistent equation.

1. From Infinite Series to a Single Equation

In principle, the full or "dressed" Green's function $G$ is the sum of an infinite number of Feynman diagrams. This is a daunting prospect. The power of the self-energy concept is that it allows us to re-organize this infinite series in a much more manageable way.

The key idea is to classify every diagram contributing to $G$ based on its 1PI components.

• The simplest diagram is just the bare propagator, $G_0$, which has zero 1PI parts.
• The next class of diagrams consists of two $G_0$ lines connected by a single 1PI block (the self-energy $\Sigma$). This can be written schematically as $G_0\Sigma G_0$.
• The next class consists of three $G_0$ lines connected by two $\Sigma$ blocks: $G_0\Sigma G_0\Sigma G_0$.
• And so on...

The full Green's function $G$ is the sum of all these classes of diagrams:

$$G=G_0+G_0\Sigma G_0+G_0\Sigma G_0\Sigma G_0+G_0\Sigma G_0\Sigma G_0\Sigma G_0+\dots$$

This structure—a bare propagator followed by all possible chains of self-energy insertions—is illustrated beautifully in the following video.

As the video demonstrates, the infinite series above is a geometric series. We can write it compactly by factoring out the initial $G_0$ term:

$$G=G_0(1+\Sigma G_0+(\Sigma G_0{)}^2+(\Sigma G_0{)}^3+\dots )$$

Summing the geometric series in the parenthesis gives $(1-\Sigma G_0{)}^{-1}$. So, we have $G=G_0(1-\Sigma G_0{)}^{-1}$.

While correct, this form is a bit unwieldy. A more common and intuitive form is found by noticing that the series $G=G_0+G_0\Sigma (G_0+G_0\Sigma G_0+\dots )$ can be written as:

$$G=G_0+G_0\Sigma G$$

This is the Dyson equation in its integral form. In momentum-frequency space, where these objects become matrices or simple functions and multiplication implies matrix multiplication or simple products (not convolutions), we can solve for $G$.

Starting from $G=G_0+G_0\Sigma G$:

$$G-G_0\Sigma G=G_0$$ $$(1-G_0\Sigma )G=G_0$$ $$G=(1-G_0\Sigma {)}^{-1}{G}_0$$

Multiplying by $G_0^{-1}$ from the left on both sides of $(1-G_0\Sigma )G=G_0$ gives $G_0^{-1}G-\Sigma G=1$, or $(G_0^{-1}-\Sigma )G=1$. This leads to the most useful form of the Dyson equation:

$$G^{-1}=G_0^{-1}-\Sigma$$

This algebraic equation is the primary result. It elegantly states that the inverse of the full Green's function is the inverse of the bare Green's function corrected by the self-energy.

2. An Alternate Derivation: The Equation of Motion

Your preference for formal derivations is well-served by an alternative route to the Dyson equation using the equation of motion (EOM) method. This approach is less pictorial but more algebraically direct. The idea is to write down the time-derivative of the Green's function, which, via Heisenberg's equation of motion, introduces the Hamiltonian. The interaction part of the Hamiltonian generates higher-order Green's functions, and the self-energy is formally defined as the object that connects this higher-order term back to the original Green's function, thereby closing the equation.

Both the diagrammatic and EOM approaches arrive at the same fundamental equation, which underscores its centrality to the theory.

3. Physics of the Dyson Equation: Dressing the Electron

Let's now use the Dyson equation to see how interactions "dress" a bare particle. In momentum-frequency space, the bare retarded Green's function for a particle with dispersion ${ϵ}_{\mathbf{k}}^0$ is:

$$G_0^R(\mathbf{k},\omega )=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0+i\eta }$$

Its inverse is simply $G_0^{R^{-1}}(\mathbf{k},\omega )=\omega -{ϵ}_{\mathbf{k}}^0+i\eta$.

Using the Dyson equation, the full, dressed Green's function is:

$$G^R(\mathbf{k},\omega )=\frac{1}{G_0^{R^{-1}}(\mathbf{k},\omega )-{\Sigma }^R(\mathbf{k},\omega )}=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-{\Sigma }^R(\mathbf{k},\omega )}$$

where we've absorbed the infinitesimal $i\eta$ into the imaginary part of the self-energy.

This equation is profound. The properties of the dressed particle (the quasiparticle) are no longer determined by the poles of $G_0$ but by the poles of $G$. The new poles are the solutions to:

$$\omega -{ϵ}_{\mathbf{k}}^0-{\Sigma }^R(\mathbf{k},\omega )=0$$

Let's revisit the two cases from our last lesson:

Case 1: Impurity Scattering
We found ${\Sigma }_{\text{imp}}^R\approx -i/(2\tau )$. Plugging this in:

$$G^R(\mathbf{k},\omega )=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-(-i/2\tau )}=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0+i/(2\tau )}$$

The pole is at $\omega ={ϵ}_{\mathbf{k}}^0-i/(2\tau )$. The quasiparticle energy is (approximately) unchanged, but it has acquired a finite lifetime $\tau$. The spectral function $A(\mathbf{k},\omega )=-\frac{1}{\pi }\text{Im}{G}^R$ becomes the Lorentzian we derived previously, confirming that the Dyson equation correctly incorporates lifetime broadening.

Case 2: Fock Self-Energy
We found ${\Sigma }_{\text{Fock}}(\mathbf{k})$ was real and frequency-independent. Plugging this in:

$$G^R(\mathbf{k},\omega )=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-{\Sigma }_{\text{Fock}}(\mathbf{k})}$$

The pole is at $\omega ={ϵ}_{\mathbf{k}}^0+{\Sigma }_{\text{Fock}}(\mathbf{k})$. The quasiparticle has a new, renormalized energy ${ϵ}_{\mathbf{k}}={ϵ}_{\mathbf{k}}^0+{\Sigma }_{\text{Fock}}(\mathbf{k})$. Since the self-energy is real, the pole is real, and the lifetime is still infinite.

The General Quasiparticle
In a general interacting system, the self-energy ${\Sigma }^R(\mathbf{k},\omega )$ will be a complex function of both momentum and frequency. We write it as ${\Sigma }^R=\text{Re}{\Sigma }^R+i\text{Im}{\Sigma }^R$. The full Green's function is:

$$G^R(\mathbf{k},\omega )=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-\text{Re}{\Sigma }^R(\mathbf{k},\omega )-i\text{Im}{\Sigma }^R(\mathbf{k},\omega )}$$

This shows the complete picture:

• The quasiparticle energy is renormalized by the real part of the self-energy.
• The quasiparticle lifetime is determined by the imaginary part of the self-energy. Specifically, the lifetime ${\tau }_k$ is given by $\frac{1}{2{\tau }_k}\approx -Z_k\text{Im}{\Sigma }^R(\mathbf{k},{ϵ}_k)$, where $Z_k$ is the quasiparticle weight, a factor we'll discuss later.

This framework is incredibly powerful. Any physical process that affects an electron—scattering off phonons, other electrons, magnons, etc.—can be calculated as a contribution to $\Sigma$. The Dyson equation then tells us how that process modifies the electron's properties.

Test your understanding!

Suppose you are studying a system where the first-order self-energy due to some interaction is purely real, ${\Sigma }^R(\mathbf{k},\omega )={\Delta }_k$, and the second-order self-energy is purely imaginary, ${\Sigma }^{R(2)}(\mathbf{k},\omega )=-i\Gamma$. Assuming other contributions are negligible, write down the dressed Green's function $G^R(\mathbf{k},\omega )$. What are the resulting quasiparticle energy and lifetime (in terms of $\Gamma$)?

Show answer

The total self-energy is ${\Sigma }^R={\Delta }_k-i\Gamma$. Using the Dyson equation:

$$G^R(\mathbf{k},\omega )=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-{\Sigma }^R(\mathbf{k},\omega )}=\frac{1}{\omega -{ϵ}_{\mathbf{k}}^0-({\Delta }_k-i\Gamma )}=\frac{1}{(\omega -({ϵ}_{\mathbf{k}}^0+{\Delta }_k))+i\Gamma }$$

The pole is at $\omega =({ϵ}_{\mathbf{k}}^0+{\Delta }_k)-i\Gamma$.

• The quasiparticle energy is shifted to ${ϵ}_{\mathbf{k}}={ϵ}_{\mathbf{k}}^0+{\Delta }_k$.
• The decay rate is $\Gamma$, so the lifetime is $\tau =1/(2\Gamma )$.

Conclusion

This lesson established the central role of the Dyson equation in many-body theory. It provides the formal and practical link between the microscopic interactions, encapsulated in the self-energy, and the resulting properties of the quasiparticles that dominate the low-energy physics of the system.

Key Takeaways:

• The Dyson equation reorganizes the infinite perturbation series for the Green's function into a compact, self-consistent form.
• The two most common forms are the integral equation $G=G_0+G_0\Sigma G$ and the algebraic equation $G^{-1}=G_0^{-1}-\Sigma$.
• Solving the algebraic form gives the dressed Green's function: $G(\mathbf{k},\omega )=[\omega -{ϵ}_{\mathbf{k}}^0-\Sigma (\mathbf{k},\omega ){]}^{-1}$.
• The self-energy $\Sigma$ shifts the poles of the Green's function away from the real axis. The real part of $\Sigma$ renormalizes the quasiparticle energy, while its imaginary part gives the quasiparticle a finite lifetime.

Preview of the Next Lesson

We have now constructed the full interacting Green's function $G$. The final piece of the puzzle is to connect this theoretical object to experimental reality. In the next lesson, we will derive the Lehmann representation of the Green's function and connect the spectral function to the density of states. This will lead us to one of the most important results in experimental condensed matter physics: the direct relationship between the spectral function $A(\mathbf{k},\omega )$ and the intensity measured in Angle-Resolved Photoemission Spectroscopy (ARPES), solidifying the link between our diagrams and real-world measurements.