Lindhard Function for Electron Screening

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back.

In our last lesson, we calculated the ground-state energy of the electron gas in the Hartree-Fock (HF) approximation. We found that the competition between kinetic energy and the attractive exchange energy gives rise to a stable, bound system. However, we also noted that the HF approximation has severe deficiencies, such as predicting a zero density of states at the Fermi level, because it fails to account for how mobile electrons screen the long-range Coulomb interaction.

Today, we will directly address this by going beyond the first-order approximation for the interaction. We will calculate the fundamental object that describes how the electron gas polarizes in response to a charge perturbation. This object is represented by a key Feynman diagram: the polarization bubble, $Π_{0} (q, ω)$ . The result of this calculation is the famous Lindhard function, which is central to understanding screening, plasmons, and other collective phenomena in metals.

1. The Physical Picture of Screening

Imagine placing a positive test charge into the electron gas (the Jellium model). The mobile electrons will be attracted to it, creating a local region of excess negative charge that surrounds the test charge. From a distance, this "cloud" of electrons partially cancels the field of the test charge, effectively screening it.

This means that the bare Coulomb interaction, $V (q) = 4 π e^{2} / q^{2}$ , is replaced by a weaker, short-range effective interaction, $V_{eff} (q, ω)$ . The relationship between them is defined by the frequency and momentum-dependent dielectric function, $ϵ (q, ω)$ :

V_{eff} (q, ω) = \frac{V ( q )}{ϵ ( q , ω )}

Our goal is to calculate this dielectric function. In diagrammatic terms, the effective interaction is not just the bare wavy line, but an infinite series of processes where the interaction creates virtual particle-hole pairs.

The simplest such process is the polarization bubble, $Π_{0}$ . Within the Random Phase Approximation (RPA), we approximate the effective interaction by summing a geometric series of these bubble diagrams:

$V_{eff} = V + V Π_{0} V + V Π_{0} V Π_{0} V + \dots$

This series can be summed exactly, giving:

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

Comparing this with the definition of the dielectric function, we find the RPA expression for $ϵ (q, ω)$ :

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

This powerful result shows that if we can calculate the polarization bubble $Π_{0}$ , we can understand screening within this widely-used approximation.

2. Calculating the Polarization Bubble

The polarization bubble is a diagram representing the creation and subsequent annihilation of an electron-hole pair. It is a closed fermion loop connected to an interaction line. We can now use the Feynman rules we've learned to write down its mathematical expression.

Chap 6 Green function theory

The following resource, 'Chap 6 Green function theory', shows how to write down the expression for the polarization bubble diagram and connects it to the dielectric function.

Please read the section 'A. Feynman rules' and focus on the example calculation immediately following it for the 'bubble diagram in photon self-energy' (equation 114). Note how the expression for \Pi_0(q) is constructed from two Green's functions and how it relates to the effective interaction V_{eff}(q) and the dielectric function \epsilon(q) (equations 115 and 116).

As derived in the resource, applying the Feynman rules (a fermion loop gives a factor of -1, sum over internal momenta and spin) gives the expression for the polarization bubble:

Π_{0} (q) = - 2 i \int \frac{d ^{4} k}{( 2 π ) ^{4}} G_{0} (k) G_{0} (k + q)

Here, $q = (q_{0}, q)$ is the 4-momentum transfer, $G_{0} (k)$ is the non-interacting Green's function, and the factor of 2 comes from summing over spin. The task now is to evaluate this integral. This involves substituting the expression for $G_{0}$ and performing a contour integral over the frequency component $k_{0}$ .

The full calculation is a classic but lengthy exercise in many-body theory. The resource LINK walks through it step-by-step.

Chap 6 Green function theory

This section provides the complete derivation of the Lindhard function starting from the Green's function expression for \Pi_0(q).

Please study section 'VI. ELECTRIC POLARIZATION'. Follow the main steps: (1) substituting the expression for G_0(k) (Eq. 117); (2) performing the frequency integral dk_0 using the residue theorem to get Eq. (119); and (3) using the Plemelj formula (Eq. 122) to separate the real and imaginary parts (Eqs. 123 and 124). Don't worry about memorizing every detail, but focus on understanding the logical flow.

After performing the frequency integration, we arrive at the central result, known as the Lindhard function:

Π_{0} (q, ω) = 2 \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{n _{k} - n _{k + q}}{ω + ( ϵ _{k} - ϵ _{k + q} ) + i η}

where $n_{k}$ is the Fermi-Dirac distribution (a step function at T=0), $ϵ_{k} = ℏ^{2} k^{2} /2 m$ is the free electron energy, and $η$ is a positive infinitesimal.

Note that this same function can be derived from a different starting point using linear response theory (the Kubo formula), where it appears as the non-interacting density-density response function $χ_{ρ}^{0} (q, ω)$ . The resource LINK (sections C and C.1) shows this alternative derivation. The fact that diagrammatic perturbation theory and linear response theory give the same answer is a powerful cross-check of the formalism.

3. Physical Properties of the Lindhard Function

The Lindhard function $Π_{0} (q, ω)$ contains a wealth of information about the electron gas. We can extract its physical meaning by examining its real and imaginary parts. Using the Plemelj formula, $\frac{1}{x + i η} = P (\frac{1}{x}) - iπ δ (x)$ , we find:

Imaginary Part: Electron-Hole Excitations
$Im Π_{0} (q, ω) = - 2 π \int \frac{d ^{3} k}{( 2 π ) ^{3}} (n_{k} - n_{k + q}) δ (ω - (ϵ_{k + q} - ϵ_{k}))$
The imaginary part is non-zero only when the energy $ω$ and momentum $q$ of the perturbation can create an electron-hole pair that conserves energy. This process—an excitation absorbing energy from the perturbation—is a form of collisionless damping known as Landau damping. The region in $(q, ω)$ space where $Im Π_{0} \neq = 0$ is called the electron-hole continuum.
Real Part: Screening
$Re Π_{0} (q, ω) = 2 P \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{n _{k} - n _{k + q}}{ω - ( ϵ _{k + q} - ϵ _{k} )}$
The real part describes the polarization, or reactive response, of the gas. It determines the screening properties via the dielectric function $ϵ = 1 - V Π_{0}$ .

The following image provides a visualization of the complex structure of a susceptibility function, which is directly analogous to the Lindhard function we are studying.

Susceptibility Functions for Electron Systems — Figure 1: 3D surface plots of the real ($\chi'$, panels b, d) and imaginary ($\chi''$, panels a, c) parts of a susceptibility function vs. in-plane wavevector $q$. The real part corresponds to screening, while the imaginary part corresponds to absorption (electron-hole pair creation). The complex features, such as ridges and peaks, reflect the underlying electronic structure and the constraints of energy-momentum conservation in creating excitations, which are precisely the physics captured by the Lindhard function.

4. Important Limits and Consequences

Let's analyze the static case, $ω = 0$ , which describes the screening of a stationary charge.

4.1. The Static Lindhard Function and Thomas-Fermi Screening

For $ω = 0$ , the integral for $Re Π_{0}$ can be solved analytically. As shown in equation (126) of LINK, the result is:

Π_{0} (q, 0) = - D (E_{F}) [\frac{1}{2} + \frac{4 k _{F}^{2} - q ^{2}}{4 q k _{F}} ln \frac{2 k _{F} + q}{2 k _{F} - q}]

where $D (E_{F}) = m k_{F} / (π^{2} ℏ^{2})$ is the density of states at the Fermi energy for both spin directions.

In the long-wavelength limit ( $q \to 0$ ), this function simplifies significantly.

q \to 0 lim Π_{0} (q, 0) = - D (E_{F})

Plugging this into the RPA dielectric function gives:

ϵ (q, 0) \approx 1 - \frac{4 π e ^{2}}{q ^{2}} (- D (E_{F})) = 1 + \frac{4 π e ^{2} D ( E _{F} )}{q ^{2}} = 1 + \frac{k _{T F}^{2}}{q ^{2}}

Here, $k_{T F} = 4 π e^{2} D (E_{F})$ is the Thomas-Fermi screening wavevector. This result perfectly recovers the simpler Thomas-Fermi theory of screening, but now derived from our more general quantum mechanical framework.

4.2. The $q = 2 k_{F}$ Singularity: Friedel Oscillations and Kohn Anomalies

Look again at the full expression for $Π_{0} (q, 0)$ . The logarithmic term has a peculiar behavior. While the function itself is finite at $q = 2 k_{F}$ , its derivative, $d Π_{0} / d q$ , diverges logarithmically.

Physical Origin: This singularity arises from the geometry of the Fermi sphere. For $q < 2 k_{F}$ , there is a large phase space of electron-hole pairs that can be created across the Fermi surface. When $q > 2 k_{F}$ , it's no longer possible to connect two points on the Fermi surface with the vector $q$ . The number of available final states for low-energy excitations changes abruptly at $q = 2 k_{F}$ , leading to the singularity.

This singularity in the response function has profound physical consequences in condensed matter physics:

Friedel Oscillations: It causes the screened potential around an impurity not to decay exponentially (like in Thomas-Fermi theory), but with a slower power law modulated by oscillations of wavelength $π / k_{F}$ . The form is $δ ρ (r) \sim \frac{c o s ( 2 k _{F} r )}{r ^{3}}$ .
Kohn Anomalies: In a real crystal, the electron-ion interaction can be screened by this mechanism. The singularity at $q = 2 k_{F}$ in the electronic response can cause a kink or anomaly in the phonon dispersion curve $ω (q)$ for the lattice vibrations. Your current studies in condensed matter physics will likely cover this phenomenon.

Test your understanding!

Why is the Lindhard theory of screening considered an improvement over the simpler Thomas-Fermi theory? What new physical phenomenon does it predict that Thomas-Fermi misses?

Show answer

The Lindhard theory is an improvement because it is valid for all wavevectors $q$ , not just small $q$ . Thomas-Fermi theory is the long-wavelength limit of Lindhard theory. The most important new phenomenon predicted by the Lindhard function is the singularity at $q = 2 k_{F}$ , which gives rise to Friedel oscillations—the long-range, oscillating screening of a charge impurity. Thomas-Fermi theory only predicts a simple exponential decay of the screened potential.

Conclusion

In this lesson, we made a significant step beyond the Hartree-Fock approximation by calculating the polarization bubble, $Π_{0} (q, ω)$ , also known as the Lindhard function. This is the cornerstone of the Random Phase Approximation.

Key Takeaways:

Screening in an electron gas is described by the dielectric function $ϵ (q, ω)$ , which relates the bare and effective interactions.
The polarization bubble, $Π_{0}$ , is the lowest-order diagrammatic contribution to screening, representing the creation of virtual electron-hole pairs.
The RPA sums the infinite series of bubble diagrams to give $ϵ_{RPA} = 1 - V (q) Π_{0} (q, ω)$ .
The imaginary part of the Lindhard function describes energy dissipation via the creation of real electron-hole pairs (Landau damping).
The static ( $ω = 0$ ) Lindhard function recovers Thomas-Fermi screening in the long-wavelength limit ( $q \to 0$ ) but also predicts a singularity in its derivative at $q = 2 k_{F}$ , which is the origin of Friedel oscillations and Kohn anomalies.

Preview of the Next Lesson:

We have now calculated the central ingredient of the RPA. In the next lesson, "Derive the Random Phase Approximation (RPA) for the dielectric function," we will consolidate these ideas. We will use the Lindhard function to formally analyze the RPA dielectric function, explore its role in determining the plasmon dispersion relation, and calculate the RPA contribution to the ground-state correlation energy, which is the part of the energy that Hartree-Fock theory misses entirely.