Deriving the RPA Dielectric Function

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Hello! Welcome to the next lesson in our exploration of many-body quantum mechanics.

In the previous lesson, we undertook the crucial calculation of the polarization bubble, $Π_{0} (q, ω)$ , which we identified as the Lindhard function. We saw that this function describes the response of a non-interacting electron gas to a perturbation and contains rich physics, from Thomas-Fermi screening to the origin of Friedel oscillations. We concluded by stating that the Lindhard function is the key ingredient in the Random Phase Approximation (RPA) for the dielectric function, $ϵ_{RPA} (q, ω) = 1 - V (q) Π_{0} (q, ω)$ .

Today, our goal is to formally derive this central result of many-body theory. We will solidify our understanding of what the Random Phase Approximation truly is by approaching it from two powerful and complementary perspectives:

A self-consistent field method based on linear response theory.
A diagrammatic summation using the concept of the irreducible polarizability.

By the end of this lesson, you will not only have the formula for the RPA dielectric function but also a deep physical intuition for the approximation it represents and why it is so fundamental to understanding collective phenomena in electron systems.

1. The Dielectric Function and Screening

Let's begin by formalizing the concept of screening. When we introduce an external potential $ϕ_{ext} (q, ω)$ into an electron gas, the electrons redistribute themselves, creating an induced potential $ϕ_{ind} (q, ω)$ . The total potential felt by any given electron is the sum of these two:

ϕ_{tot} (q, ω) = ϕ_{ext} (q, ω) + ϕ_{ind} (q, ω)

The dielectric function, $ϵ (q, ω)$ , is formally defined as the ratio of the external potential to the total potential:

ϵ (q, ω) \equiv \frac{ϕ _{ext} ( q , ω )}{ϕ _{tot} ( q , ω )}

A dielectric function greater than one signifies that the total potential is weaker than the external one—the medium has screened the external perturbation. Our task is to find an expression for $ϵ (q, ω)$ based on the microscopic physics of the electron gas.

2. Derivation via the Self-Consistent Field Method

This first derivation is particularly intuitive. It treats screening as a self-consistent process: the total potential induces a charge density, which in turn generates the induced potential. The RPA provides the crucial simplifying assumption for this feedback loop.

The key steps are as follows:

The Induced Potential: The induced potential $ϕ_{ind}$ is generated by the induced charge density $ρ_{ind}$ via the Coulomb interaction $V (q)$ . In Fourier space, this is simply:
$ϕ_{ind} (q, ω) = V (q) ρ_{ind} (q, ω)$
The RPA Assumption: Here is the core of the approximation. We assume that the induced charge density responds linearly to the total potential $ϕ_{tot}$ . Furthermore, we assume this response is governed by the polarizability of a non-interacting electron gas. This non-interacting polarizability is precisely the Lindhard function $Π_{0} (q, ω)$ that we calculated in the previous lesson.
$ρ_{ind} (q, ω) = Π_{0} (q, ω) ϕ_{tot} (q, ω)$
This is the "mean-field" aspect of RPA: each electron responds to an average, total potential, and the response itself is calculated as if the electrons were non-interacting.
Self-Consistency: We now substitute these relations into our starting equation for $ϕ_{tot}$ .
$ϕ_{tot} = ϕ_{ext} + ϕ_{ind} = ϕ_{ext} + V Π_{0} ϕ_{tot}$
We have a self-consistent equation for the total potential.
Solving for $ϵ$ : Rearranging the equation to solve for $ϕ_{tot}$ gives:
$ϕ_{tot} (1 - V (q) Π_{0} (q, ω)) = ϕ_{ext} (q, ω)$ $ϕ_{tot} (q, ω) = \frac{ϕ _{ext} ( q , ω )}{1 - V ( q ) Π _{0} ( q , ω )}$

By comparing this result directly with the definition $ϕ_{tot} = ϕ_{ext} / ϵ$ , we arrive at the celebrated RPA formula for the dielectric function:

ϵ_{RPA} (q, ω) = 1 - V (q) Π_{0} (q, ω)

This derivation elegantly shows how the RPA emerges from a simple, physically motivated picture of self-consistent screening.

Screening, RPA, GW and GW+EDMFT: an Introduction

The following lecture slides by Antoine Georges, a prominent figure in many-body physics, present this self-consistent derivation very clearly.

Please read the sections titled 'II. SCREENING AND RPA', starting from 'A. Introduction' and continuing through 'B. Linear Response Derivation' and 'C. Dielectric Function and Screened Potential'. This covers the exact logic we just followed. Note on conventions: The author defines the susceptibility \chi_0 such that ( ho_{ ext{ind}} = -\chi_0 \phi_{ ext{tot}}). This means their \chi_0 is the negative of our \Pi_0 (i.e., \chi_0^{ ext{Georges}} = -\Pi_0). This leads to their final formula being \epsilon = 1 + V\chi_0, which is mathematically identical to our \epsilon = 1 - V\Pi_0. This kind of sign convention difference is common in the literature, so it's good to be aware of it.

3. Derivation via Diagrammatic Summation

A second, equally powerful way to understand the RPA is through Feynman diagrams. This perspective makes it clear that RPA is an infinite-order perturbation theory, but one that only includes a specific class of diagrams.

Let's define our terms diagrammatically:

The bare Coulomb interaction, $V (q)$ , is a wavy line.
The fully screened, effective interaction, $W (q, ω)$ , is a double wavy line.
The proper polarizability, $Π (q, ω)$ , is the sum of all "one-particle irreducible" diagrams that connect an incoming interaction line to an outgoing one. "Irreducible" means the diagram cannot be cut in two by snipping a single wavy interaction line. The simplest such diagram is our friend, the single bubble $Π_{0}$ .

The screened interaction $W$ can be expressed as an infinite geometric series involving the bare interaction $V$ and the proper polarizability $Π$ :

W = V + V Π V + V Π V Π V + \dots

This is a Dyson-like equation for the interaction, which can be formally summed to:

W (q, ω) = \frac{V ( q )}{1 - V ( q ) Π ( q , ω )}

This equation is exact! It's just a re-packaging of the full perturbation series. All the complexity of the many-body problem is now hidden inside $Π (q, ω)$ .

The Random Phase Approximation is now defined simply and precisely: we approximate the full proper polarizability $Π$ with its simplest possible form—the single, non-interacting bubble $Π_{0}$ .

Π (q, ω) \approx Π_{0} (q, ω)

Substituting this approximation into the Dyson equation gives the RPA screened interaction:

W_{RPA} (q, ω) = \frac{V ( q )}{1 - V ( q ) Π _{0} ( q , ω )}

Finally, we use the definition of the dielectric function in terms of interactions: the screened interaction is just the bare interaction divided by the dielectric function, $W = V / ϵ$ .
Comparing our two expressions for $W$ immediately gives us the RPA dielectric function, once again:

ϵ_{RPA} (q, ω) = 1 - V (q) Π_{0} (q, ω)

561 F 2005 Lecture 8

These lecture notes provide a concise overview of this diagrammatic derivation.

Please read sections 2 ('Random Phase Approximation for screening...') and 3 ('RPA form for the polarization self-energy...'). Focus on how the screened interaction W is defined using the 'proper polarizability' \Pi^\star (their notation for our \Pi) and how the RPA is defined as the approximation \Pi^\star \approx \Pi_0 (the single bubble diagram).

Test your understanding!

From the diagrammatic perspective, what physical processes are being neglected in the Random Phase Approximation?

Show answer

The RPA approximates the full proper polarizability $Π$ with just the bare bubble $Π_{0}$ . This means we are neglecting all other irreducible diagrams. These diagrams represent more complex interactions. For example, we neglect diagrams where the electron and the hole forming the bubble interact with each other (these are known as vertex corrections or ladder diagrams). We also neglect interactions of the propagating electron or hole with the surrounding electron gas. The RPA essentially treats the propagation of each electron-hole pair as being independent of all others.

4. What Does "Random Phase" Mean?

The name of the approximation comes from early work by Bohm and Pines. They analyzed the collective oscillations of the electron gas (plasmons). In their analysis, the approximation that leads to the RPA result is equivalent to assuming that the phases of the density fluctuations at different wavevectors $q$ are uncorrelated, or "random."

Self-Consistent View: The total potential felt by an electron is a sum of the fields from all other electrons. The RPA assumes this electron responds only to this average field, ignoring the specific correlations between the electron's position and the positions of others.
Diagrammatic View: We sum an infinite series of bubble diagrams. This means we account for an external charge polarizing the medium, that polarization creating a new potential, which in turn polarizes the medium further, and so on, ad infinitum. However, we assume each polarization "bubble" is independent and doesn't interact with the others, apart from contributing to the mean field.

The RPA is most accurate in the high-density limit (small $r_{s}$ ), where the kinetic energy dominates over the potential energy, making the non-interacting gas a good starting point. Despite its simplicity, it successfully captures the most important qualitative feature of the interacting electron gas: the screening of the long-range Coulomb interaction.

Conclusion

In this lesson, we formally derived the expression for the dielectric function within the Random Phase Approximation. We saw how this fundamental result can be reached through two different but equivalent lines of reasoning, deepening our understanding of the physics involved.

Key Takeaways:

The RPA dielectric function is given by $ϵ_{RPA} (q, ω) = 1 - V (q) Π_{0} (q, ω)$ , where $Π_{0}$ is the Lindhard function.
The self-consistent field derivation shows that RPA treats screening as a linear response of non-interacting electrons to the total (screened) potential.
The diagrammatic derivation reveals that RPA is equivalent to approximating the full polarizability with the simplest bubble diagram, $Π_{0}$ , and summing the resulting geometric series of interactions.
Physically, RPA neglects correlations between electron-hole pairs, an approximation that is well-justified in the high-density limit.

Preview of the Next Lesson:

Now that we have derived the RPA dielectric function, we can use it to explore its most dramatic consequence: the existence of collective charge oscillations known as plasmons. In the next lesson, we will analyze the zeros of $ϵ_{RPA} (q, ω)$ to derive the plasmon dispersion relation, $ω_{p} (q)$ , and understand its physical significance.