First-Order Self-Energy Calculations

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Hello! Welcome to your next lesson on diagrammatic perturbation theory.

In our last session, we established the Feynman rules in the imaginary-time formalism. We now have a powerful recipe to translate the pictorial language of diagrams into concrete mathematical expressions. We saw how Wick's theorem allows us to expand the interacting Green's function, $G$ , in terms of the non-interacting one, $G_{0}$ , and that each term in this expansion corresponds to a diagram.

Today, we will put this machinery to work. Our goal is to calculate the first-order self-energy for electron-impurity and electron-electron interactions. This is where the abstract formalism begins to yield concrete physical insights, such as energy shifts and finite quasiparticle lifetimes, which are directly relevant to the experimental observables we discussed earlier.

1. The Self-Energy: Summing the Irreducible Parts

As we previewed, the perturbation series for the Green's function contains an infinite number of diagrams. A crucial organizing principle is to classify diagrams as either reducible or irreducible.

An irreducible diagram is one that cannot be cut into two separate pieces by severing a single internal fermion line.
A reducible diagram can be split in this way.

Illustration of reducible vs. irreducible diagrams. Diagram (a) is reducible because it can be cut along the dashed line. Diagram (b) is irreducible.

The self-energy, denoted $Σ (k, i ω_{n})$ , is defined as the sum of all one-particle irreducible (1PI) diagrams, with the two external propagator lines removed. This is a powerful concept because the full Green's function can then be expressed compactly through the Dyson equation, $G = G_{0} + G_{0} Σ G$ , which we will explore in detail in the next lesson.

For now, our task is to calculate the simplest, most fundamental contributions to $Σ$ .

To get a clear understanding of these concepts—irreducible diagrams, the self-energy, and the Dyson equation—in the context of a concrete problem, we'll turn to a set of lecture notes on Green's functions.

Introduction to Green functions and many-body ...

These notes from J. Linder's course provide an exceptionally clear introduction to the self-energy concept using impurity scattering as an example. We will use these notes extensively in the second half of the lesson, but for now, we'll focus on the formal definitions.

Please read Section 4.6, 'Irreducible diagrams, the self-energy, and the Dyson equation' (pages 27-30). Focus on the definitions of irreducible diagrams and the self-energy \Sigma. Observe how the entire perturbation series for G(k) can be restructured in terms of \Sigma, leading to the Dyson equation.

2. Case 1: Electron-Electron Interaction Self-Energy

Let's start with the interacting electron gas. The first-order (in the interaction potential $V$ ) contributions to the self-energy are the familiar Hartree and Fock diagrams.

Hartree-Fock Self-Energy Diagrams and Expression — The two first-order diagrams for the self-energy $\Sigma$. The first is the Hartree term, and the second is the Fock (or exchange) term.

The Hartree Term

The Hartree diagram represents an electron interacting with the average electrostatic potential of all other electrons. Applying the Feynman rules, its contribution involves the interaction potential at zero momentum transfer, $V (q = 0)$ . In the jellium model, a uniform positive background charge is assumed to neutralize the average electron charge. This effectively means that $V (q = 0) = 0$ , so the Hartree term vanishes for the electron gas model.

The Fock Term

The Fock, or exchange, term is more interesting. It arises from the Pauli exclusion principle and is a purely quantum mechanical effect. Let's calculate its contribution to the self-energy, $Σ_{Fock} (k, i ω_{n})$ . The following video provides a detailed, step-by-step derivation.

19- Dyson’s equations and self-energy - Course on Quantum Many-Body Physics

This lecture by Prof. Luis Gregorio Dias meticulously calculates the first-order self-energy for the electron gas. It's an excellent demonstration of applying the Feynman rules in practice.

Please watch the segment from 32:26 to 49:10 for the Hartree term, and from 1:02:02 to 1:07:01 for the Fock term. While watching, pay close attention to: How the Feynman rules from our previous lesson are applied to write down the initial expression. The technique used to evaluate the Matsubara frequency sum in the Hartree term (the contour integration is a standard method). The final expression for the Fock term and its momentum dependence.

As the video shows, after applying the Feynman rules and performing the Matsubara frequency summation, the Fock self-energy is:

Σ_{Fock} (k) = - \int \frac{d ^{3} q}{( 2 π ) ^{3}} V (k - q) n_{F} (ξ_{q})

where $V (q) = \frac{4 π e ^{2}}{q ^{2}}$ is the Fourier transform of the Coulomb potential and $n_{F} (ξ_{q})$ is the Fermi-Dirac distribution, which at $T = 0$ is a step function ( $1$ for $∣ q ∣ < k_{F}$ and $0$ otherwise).

Key properties of the Fock self-energy:

It is real and frequency-independent.
A real self-energy corresponds to a shift in the quasiparticle energy: $ϵ_{k} \to ϵ_{k} + Σ_{Fock} (k)$ .
It depends on momentum $k$ , meaning the energy correction is different for electrons with different momenta.
As discussed in the supplementary notes LINK (Section 2.3), the evaluation of this integral leads to a logarithmic divergence in the derivative of the energy at the Fermi surface ( $d ϵ / d k \to \infty$ at $k = k_{F}$ ). This unphysical result (implying zero effective mass) highlights the limitations of this first-order approximation and points to the importance of screening, which we will study later.

3. Case 2: Electron-Impurity Self-Energy

Now let's consider a different physical situation: non-interacting electrons scattering off a collection of static, randomly distributed impurities. This is the primary mechanism for electrical resistance in metals at low temperatures.

The Hamiltonian is $H = H_{0} + V_{imp}$ , where $V_{imp} (r) = \sum_{j} U (r - R_{j})$ and the $R_{j}$ are the random positions of the impurities. Since any macroscopic measurement averages over the specific random configuration of impurities, we must perform a disorder average over the impurity positions.

This averaging procedure restores translational invariance and has a remarkable consequence: the effect of impurity scattering can be described by diagrams that look like electron-electron interactions, but where the interaction line represents the disorder correlator. For "white noise" disorder, this correlator is $⟨ V_{imp} (q) V_{imp} (- q)⟩ = n_{imp} ∣ u (q) ∣^{2}$ , where $n_{imp}$ is the impurity concentration and $u (q)$ is the potential of a single impurity.

The lowest-order contribution to the self-energy comes from the diagram below:

First-Order Proper Self-Energy Feynman Diagrams — The first-order self-energy diagram for impurity scattering. The dashed line with a cross represents the disorder correlator, $n_{\text{imp}}|u(\mathbf{q})|^2$. The diagram is structurally similar to the Fock diagram.

Let's calculate this contribution. We will return to the excellent notes by J. Linder.

Introduction to Green functions and many-body ...

We will now study the calculation of the first-order self-energy due to impurity scattering. These notes provide a complete and self-contained derivation.

Please read Section 4.7, 'Low-density weak-scattering approximation for the self-energy' (pages 30-32) and Section 4.8, 'The impurity-averaged retarded Green function and its spectral function' (pages 32-33). Pay close attention to: The expression for \Sigma^{(2)} in Eq. (145) (it's the first non-trivial term). The approximation of replacing the density of states D(\xi) with its value at the Fermi level, D(0). The final result for the self-energy, Eq. (148), and its interpretation in terms of a scattering time au. How this imaginary self-energy leads to a Lorentzian spectral function in Eq. (158).

The key result from this calculation, within the weak scattering approximation, is that the self-energy is approximately:

Σ_{imp} (i ω_{n}) \approx - i \frac{sgn ( ω _{n} )}{2 τ}

where $\frac{1}{τ} = 2 π n_{imp} ∣ u ∣^{2} D (0)$ is the elastic scattering rate, proportional to the impurity concentration and the density of states at the Fermi level.

Key properties of the impurity self-energy:

It is purely imaginary and (in this approximation) frequency-dependent only through the sign function.
An imaginary self-energy corresponds to a finite quasiparticle lifetime. When we find the retarded Green's function via analytic continuation ( $i ω_{n} \to ω + i η$ ), we get $Σ^{R} \approx - i / (2 τ)$ .
This leads to a broadened spectral function: $A (k, ω) = - \frac{1}{π} Im G^{R} (k, ω) = \frac{1}{π} \frac{1/ ( 2 τ )}{( ω - ξ _{k} ) ^{2} + ( 1/ ( 2 τ ) ) ^{2}}$

This is a Lorentzian peak of width $1/ τ$ , centered at the original electron energy $ξ_{k}$ . This is exactly the kind of lifetime broadening effect you would observe in spectroscopy experiments like ARPES when a sample has impurities.

Test your understanding!

Contrast the physical effects of the first-order Fock self-energy and the first-order impurity self-energy on a quasiparticle at the Fermi level ( $ξ_{k} = 0$ ). What happens to its energy and its lifetime in each case?

Show answer

Fock Self-Energy: $Σ_{Fock}$ is real. It causes a shift in the quasiparticle's energy but does not affect its lifetime. The quasiparticle remains a perfectly well-defined excitation, just at a different energy.
Impurity Self-Energy: $Σ_{imp}$ is imaginary. It does not shift the quasiparticle's energy (in this approximation), but it gives it a finite lifetime $τ$ . The quasiparticle is no longer a stable excitation; it decays over time due to scattering with impurities. This is reflected in the energy uncertainty (width) of the Lorentzian spectral function.

Conclusion

In this lesson, we have moved from formal rules to concrete physics. By applying the Feynman rules, we calculated the first-order self-energy for two fundamental models in condensed matter physics.

Key Takeaways:

The self-energy $Σ$ is the sum of all one-particle irreducible (1PI) diagrams and encapsulates the effects of interactions.
The first-order electron-electron interaction gives the Hartree-Fock self-energy. The Fock term is real and momentum-dependent, causing a renormalization of the electron's energy.
The first-order electron-impurity interaction (in the weak-scattering approximation) gives a purely imaginary self-energy. This corresponds to a finite quasiparticle lifetime $τ$ and leads to a Lorentzian broadening of the spectral function.
Calculating these terms involves a standard procedure: draw the diagram, apply the rules to write the expression, and evaluate the resulting Matsubara sums and momentum integrals.

Preview of the Next Lesson

We have now calculated the building block, $Σ$ . The next step is to use it. In our next lesson, we will focus on the Dyson equation, $G (k) = G_{0} (k) + G_{0} (k) Σ (k) G (k)$ . We will solve this equation for $G$ and see explicitly how the self-energy "dresses" the bare electron, transforming it into a quasiparticle with a renormalized energy and a finite lifetime. This equation is central to all of many-body theory as it resums an infinite number of diagrams into a single, powerful, self-consistent statement.