RPA Dielectric Function and Plasmon Dispersion

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Hello! Welcome back to our study of many-body quantum mechanics.

In our previous lesson, we accomplished a major goal: we derived the Random Phase Approximation (RPA) for the dielectric function, $ϵ_{RPA} (q, ω) = 1 - V (q) Π_{0} (q, ω)$ . We saw that this approximation can be understood both as a self-consistent field theory and as a summation of an infinite series of "bubble" diagrams. This function describes how the electron gas screens an external potential.

Today, we will apply this powerful result to uncover one of its most important predictions: the existence of plasmons, which are collective, quantized oscillations of the entire electron gas. Your background in condensed matter physics has introduced you to collective excitations like phonons; today, you will see how a fundamentally different type of collective mode emerges from the long-range Coulomb interaction. Our goal is to use the RPA dielectric function to calculate the plasmon dispersion relation, $ω_{p} (q)$ , and understand its physical properties.

1. The Condition for Collective Excitations

How do we find collective modes from a response function like $ϵ (q, ω)$ ? A collective mode is a natural, self-sustaining oscillation of the system. This means it can exist even without an external driving potential.

Recall the relationship between the external potential $ϕ_{ext}$ and the total, screened potential $ϕ_{tot}$ :

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

A self-sustaining oscillation corresponds to a situation where we have a finite response ( $ϕ_{tot} \neq = 0$ ) in the absence of an external driving field ( $ϕ_{ext} = 0$ ). From the equation above, this is only possible if the denominator vanishes:

ϵ (q, ω) = 0

This simple condition is the key to finding collective excitations. For the RPA, our condition becomes:

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

Since the Lindhard function $Π_{0} (q, ω)$ is complex, the solutions $ω (q)$ for this equation will generally be complex. A complex frequency $ω = ω_{p} - i Γ$ represents a damped oscillation with energy $ℏ ω_{p}$ and lifetime $τ = 1/Γ$ .

A stable, long-lived collective excitation—a true quasiparticle—should not be damped. We are therefore looking for solutions where $ω$ is purely real, in a region where the system has no mechanism to absorb energy. The primary mechanism for energy absorption at this level is the creation of electron-hole pairs, which is quantified by the imaginary part of the polarizability, $Im [Π_{0} (q, ω)]$ .

Thus, for an undamped plasmon, we must satisfy two conditions simultaneously:

The real part of the dielectric function must be zero: $1 - V (q) Re [Π_{0} (q, ω)] = 0$ .
The imaginary part must also be zero, which, since $V (q)$ is real, implies $Im [Π_{0} (q, ω)] = 0$ .

2. The Particle-Hole Continuum

The condition $Im [Π_{0} (q, ω)] = 0$ tells us where to look for undamped plasmons. As we discussed when calculating the Lindhard function, $Im [Π_{0} (q, ω)]$ represents the density of states for creating an electron-hole pair with momentum $ℏ q$ and energy $ℏ ω$ . A non-zero value means an excitation with these quantum numbers can decay by creating an electron-hole pair. This decay mechanism is a form of collisionless damping known as Landau damping.

The region in the $(q, ω)$ plane where $Im [Π_{0} (q, ω)] \neq = 0$ is called the particle-hole continuum. Undamped plasmons, if they exist, must lie outside this region.

region-ph-continuum.png

This diagram shows the region in the momentum-frequency plane where the electron gas can absorb energy by creating electron-hole pairs (shaded area), based on the Lindhard function. Undamped collective excitations can only exist outside this region. The red curve, labeled "PLASMONS", shows the dispersion relation we are about to calculate. Note that it starts at a finite energy at q=0 and eventually merges with the continuum at a critical wavevector.

3. Plasmon Frequency at Long Wavelengths ( $q \to 0$ )

Let's solve the plasmon condition in the most important limit: long wavelengths ( $q \to 0$ ). We need the corresponding limits for the Coulomb potential $V (q)$ and the real part of the Lindhard function $Re [Π_{0} (q, ω)]$ .

Coulomb Potential (3D): The Fourier transform of the potential $e^{2} / r$ is $V (q) = \frac{4 π e ^{2}}{q ^{2}}$ . (We'll work in Gaussian units, which are common in many-body literature).
Lindhard Function Limit: In the last lesson, we saw that for small $q$ , the real part of the polarizability has the limit:
$q \to 0 lim Re [Π_{0} (q, ω)] = \frac{n q ^{2}}{m ω ^{2}}$
where $n$ is the electron density.

Now, we substitute these into our condition $1 = V (q) Re [Π_{0} (q, ω)]$ :

1 = (\frac{4 π e ^{2}}{q ^{2}}) (\frac{n q ^{2}}{m ω ^{2}})

A remarkable thing happens: the $q^{2}$ from the long-range nature of the Coulomb potential exactly cancels the $q^{2}$ from the polarizability. This cancellation is a deep and essential feature of charge screening.

Solving for $ω$ , we find the plasmon frequency at zero momentum transfer:

ω_{p}^{2} (0) = \frac{4 π n e ^{2}}{m}

This is the famous plasma frequency. The fact that $ω_{p} (0)$ is finite is a profound result. Unlike acoustic phonons in a solid, whose frequency goes to zero as $q \to 0$ , the collective charge oscillations require a finite amount of energy (a "gap") even at infinite wavelength. This energy gap is a direct signature of the long-range Coulomb interaction. Any short-range interaction would have $V (q) \to const$ as $q \to 0$ , and the plasmon frequency would vanish.

The dielectric function of jellium in the RPA

The following notes provide a clear, step-by-step derivation of this long-wavelength limit and discuss its physical interpretation.

Please read the section titled 'Dynamical limiting case' on page 5. This section shows exactly the calculation we just performed, leading to the plasma frequency \omega_p(0). It also nicely explains the interpretation as a collective excitation and connects it to experimental observations in metals like Aluminum.

4. Plasmon Dispersion and Damping

What happens when we move to small but finite $q$ ? We need to include the next term in the expansion of $Re [Π_{0} (q, ω)]$ . A more detailed expansion gives:

Re [Π_{0} (q, ω)] \approx \frac{n q ^{2}}{m ω ^{2}} (1 + \frac{3}{5} \frac{v _{F}^{2} q ^{2}}{ω ^{2}})

where $v_{F} = ℏ k_{F} / m$ is the Fermi velocity.

Plugging this into our condition $1 = V (q) Re [Π_{0} (q, ω)]$ gives:

ω^{2} \approx ω_{p}^{2} (0) (1 + \frac{3}{5} \frac{v _{F}^{2} q ^{2}}{ω ^{2}})

Since the second term is a small correction, we can approximate $ω^{2}$ inside the parenthesis with $ω_{p}^{2} (0)$ to get the plasmon dispersion relation:

ω_{p}^{2} (q) \approx ω_{p}^{2} (0) + \frac{3}{5} v_{F}^{2} q^{2}

The plasmon dispersion is positive and quadratic for small $q$ . The plasmon energy increases as its wavelength decreases.

As we increase $q$ , the plasmon energy $ℏ ω_{p} (q)$ continues to rise, while the upper boundary of the particle-hole continuum also rises. Eventually, at a critical wavevector $q_{c}$ , the plasmon dispersion curve enters the continuum (as shown in the figure above).

For $q < q_{c}$ , the plasmon is a stable, well-defined quasiparticle.
For $q > q_{c}$ , the plasmon can decay into an electron-hole pair. It is no longer a stable quasiparticle but a broadened resonance. This is Landau damping in action. The plasmon "dissolves" into the sea of single-particle excitations.

This behavior is not just a theoretical curiosity; it is directly observed in experiments.

RPA in infinite systems

These notes beautifully illustrate the plasmon's properties and its relation to the particle-hole continuum, including the crucial point where it enters the continuum and becomes damped.

Please read the section titled 'Plasmons in the electron gas' (page 13). Pay close attention to the eigenvalue equation for the plasmon energy, 1 - 2V(Q) ext{Re}\Pi^{(0)}(Q, E_p) = 0, and the figure on the next page ('Location of plasmon'), which is a concise graphical representation of everything we've discussed.

Test your understanding!

Acoustic phonons are collective modes with a dispersion $ω (q) = c_{s} q$ for small $q$ , where $c_{s}$ is the speed of sound. Why is the plasmon dispersion fundamentally different, starting at a finite $ω_{p} (0)$ ?

Show answer

The fundamental difference lies in the range of the underlying interaction. Phonons arise from short-range inter-atomic forces. A long-wavelength distortion ( $q \to 0$ ) costs very little energy, so $ω (q) \to 0$ . Plasmons, however, arise from the long-range $1/ r$ Coulomb interaction. A long-wavelength charge displacement creates large-scale electric fields that cost a significant amount of energy to establish, even as $q \to 0$ . This is mathematically reflected in the cancellation of the $1/ q^{2}$ divergence in the Coulomb potential $V (q)$ with the $q^{2}$ dependence of the polarizability $Π_{0} (q, ω)$ , leading to a finite, non-zero energy $ℏ ω_{p} (0)$ .

Finally, let's look at the excellent agreement between this RPA theory and experimental measurements.

The dielectric function of jellium in the RPA

To see how well this theory works, let's look at some real experimental data.

Please go to page 11 of this resource. It shows the measured plasmon dispersion curves for Aluminum, Sodium, and Cesium from inelastic X-ray scattering experiments, plotted against the theoretical RPA prediction. Notice how well the theory captures the quadratic dispersion and the overall energy scale.

Conclusion

In this lesson, we put the RPA dielectric function to work and extracted its most striking prediction: the existence of plasmons. This exercise bridged the gap between the formal diagrammatic theory and tangible, measurable physics in real materials.

Key Takeaways:

Collective excitations correspond to the zeros of the dielectric function, $ϵ (q, ω) = 0$ .
In the long-wavelength limit ( $q \to 0$ ), this condition yields a finite plasma frequency, $ω_{p} (0) = 4 π n e^{2} / m$ , a direct consequence of the long-range Coulomb force.
The plasmon dispersion relation for small $q$ is quadratic: $ω_{p} (q) \approx ω_{p} (0) + A q^{2}$ , where $A > 0$ .
Plasmons are stable quasiparticles only for wavevectors below a critical value $q_{c}$ . For $q > q_{c}$ , the plasmon enters the particle-hole continuum and becomes a damped resonance via Landau damping.
RPA provides a remarkably accurate description of plasmon dispersion in simple metals.

Preview of the Next Lesson:

We have now analyzed a specific quasiparticle—the plasmon—in great detail. This provides a perfect foundation for the next module, which generalizes the concept of quasiparticles. We will begin by introducing Landau Fermi Liquid Theory, a powerful phenomenological framework that describes how the low-energy excitations of an interacting electron system can still be understood in terms of "dressed" electrons, or quasiparticles, that behave much like free electrons but with modified properties like an effective mass.