Dyson's Equation and the Dressed Propagator

V(k-k') \theta(k_F-|k'|) ]. This gives \Sigma_F(k) = - \int_{|k'|<k_F} \frac{d^3k'}{(2\pi)^3} V(k-k')`. This also matches Mahan's result after frequency integration.
* So, the summary is correct.
* Hartree: ${\Sigma }_H=V(q=0)n$. This is a constant energy shift, independent of the particle's momentum. It represents the average potential from all other electrons.
* Fock: ${\Sigma }_F(k)=-{\int }_{|k^{\prime }|<k_F}\frac{d^3{k}^{\prime }}{(2\pi {)}^3}V(k-k^{\prime })$.thought
Hello! Welcome to our next lesson on diagrammatic many-body theory.

In our previous session, we introduced Dyson's equation, $G(k,\omega )=(\omega -{ϵ}_k-\Sigma (k,\omega ){)}^{-1}$, a cornerstone result that elegantly reformulates the problem of an infinite perturbation series into finding a single, powerful function: the self-energy, $\Sigma (k,\omega )$.

Today, we will transition from formal derivations to practical application. Our goal is to use the Dyson equation to calculate the dressed propagator (the full Green's function, $G$) for a physically crucial system: the interacting electron gas, described by the Jellium model. This process involves calculating an approximation for the self-energy. We will compute the first-order diagrams for $\Sigma$, which corresponds to the famous Hartree-Fock approximation, and explore the physical consequences for the electron propagator.

This lesson directly connects the abstract diagrammatic formalism to the concrete physics of electrons in materials, a central theme in your study of condensed matter.

1. Approximating the Self-Energy: The Hartree-Fock Level

The exact self-energy $\Sigma$ is the sum of all one-particle irreducible (1PI) diagrams—an infinite and unsolvable task. The power of the formalism, however, lies in our ability to form systematic approximations by selecting specific classes of diagrams.

The most fundamental approximation is to include only the diagrams that are first-order in the interaction potential $V$. There are exactly two such 1PI diagrams.

Therefore, in the Hartree-Fock approximation, the self-energy is:

Our task is to translate these two diagrams into mathematical expressions and analyze their physical meaning.

2. Calculating the First-Order Self-Energy

We will now evaluate these diagrams for the Jellium model. The following video provides a very clear, step-by-step walkthrough of this calculation. It's a substantial derivation that showcases how the Feynman rules we've learned are applied in practice, including important techniques like frequency summation.

Summary of the Calculation

Let's consolidate the key results from the video. The Feynman rules allow us to write down the expressions for the two self-energy diagrams. After performing the frequency summations (as shown in the video using contour integration), we arrive at the following results for the self-energy at zero temperature:

1. Hartree Self-Energy (${\Sigma }_H$)

This term corresponds to the tadpole-like diagram where an electron interacts with the average potential created by all other electrons.

where $V(q=0)$ is the zero-momentum component of the interaction, $n$ is the electron density, and the factor of 2 accounts for spin.

• Physical Interpretation: The Hartree self-energy is a constant energy shift, independent of the electron's momentum $k$ or energy $\omega$. It represents the mean-field electrostatic potential experienced by an electron due to the average charge density of all other electrons. For the Jellium model, where the interacting electrons are immersed in a uniform positive background charge, this term is exactly canceled by the electron-background interaction. Therefore, for Jellium, ${\Sigma }_H=0$.

2. Fock Self-Energy (${\Sigma }_F(k)$)

This term arises from the exchange diagram and is a purely quantum mechanical effect due to the Pauli exclusion principle.

where the integration is over all occupied states within the Fermi sphere.

• Physical Interpretation: Unlike the Hartree term, the Fock self-energy depends on the electron's momentum $k$. This is because the exchange interaction is non-local; an electron at momentum $k$ feels a potential that depends on its own state, as it only avoids other electrons with the same spin. This $k$-dependence is crucial and directly modifies the particle's dispersion relation.

3. The Dressed Propagator in the Hartree-Fock Approximation

Now we can insert our calculated self-energy into Dyson's equation to find the dressed propagator, $G_{HF}(k,\omega )$.

where ${ϵ}_k^0=\frac{{\hslash }^2{k}^2}{2m}$ is the bare electron dispersion.

Since ${\Sigma }_H=0$ for Jellium, ${\Sigma }_{HF}(k)={\Sigma }_F(k)$. The propagator becomes:

This result is highly significant. The poles of the Green's function define the energy of the system's excitations (the quasiparticles). In this approximation, we find a new, "renormalized" energy dispersion ${ϵ}_k$:

This is precisely the dispersion relation for a quasiparticle in the Hartree-Fock approximation. We have re-derived a classic result of many-body theory, but this time through the powerful and systematic machinery of Green's functions and Feynman diagrams.

4. Physical Consequences and Limitations

The momentum-dependent Fock term has profound physical consequences. To see this, one can explicitly calculate the integral for a Coulomb interaction $V(q)=4\pi e^2/q^2$. The result for ${\Sigma }_F(k)$ leads to a logarithmic divergence in the slope of the dispersion ${ϵ}_k$ at the Fermi momentum, $k_F$.

This implies that the group velocity $v_g=\frac{1}{\hslash }\frac{d{ϵ}_k}{dk}$ goes to infinity at the Fermi surface, and consequently, the density of states $N(E)\propto (dE/dk{)}^{-1}$ goes to zero. This is a famously unphysical prediction of the Hartree-Fock approximation for the electron gas.

The key takeaway is that the Hartree-Fock approximation fails for the electron gas because it improperly handles the long-range nature of the Coulomb interaction. In reality, the presence of other mobile electrons screens the interaction, effectively making it a short-range force at long distances. A proper description requires summing an infinite class of diagrams that capture this screening effect, which leads to the Random Phase Approximation (RPA).

Conclusion

In this lesson, we have made a significant leap from formalism to application. We used Dyson's equation as a practical tool, not just a formal identity.

Key Takeaways:

• Dyson's equation shifts the problem of finding the full propagator $G$ to finding an approximation for the self-energy $\Sigma$.
• The first-order approximation for $\Sigma$ is the Hartree-Fock approximation, consisting of the direct (Hartree) and exchange (Fock) diagrams.
• Applying this to the Jellium model, we found the dressed propagator and the renormalized quasiparticle energy ${ϵ}_k={ϵ}_k^0+{\Sigma }_F(k)$.
• The Hartree-Fock self-energy is real and frequency-independent, implying stable, infinite-lifetime quasiparticles.
• This approximation leads to unphysical results for the electron gas due to its failure to account for screening of the long-range Coulomb interaction.

Preview of the Next Lesson:

The failure of the Hartree-Fock approximation sets the stage for our next topic. To correctly describe the electron gas, we need to go beyond first-order perturbation theory. We will see how to do this by calculating the polarization bubble (the Lindhard function), which is the fundamental building block for describing screening effects and leads to the Random Phase Approximation (RPA). This will be our first encounter with summing an infinite series of diagrams to capture collective behavior.