Dielectric Screening and Polarization Bubbles

Hello! Welcome to the fourth lesson in our module on many-body formalism.

In our last lesson, we introduced the single-particle Green's function, a powerful tool that describes the propagation of quasiparticles—electrons "dressed" by their interactions with the surrounding medium. We saw that all the complex interaction effects are bundled into the self-energy, $\Sigma (k,\omega )$. Our ultimate goal is to find ways to approximate this self-energy.

However, before we can calculate how an electron interacts with its environment, we must first understand the environment itself. The bare Coulomb interaction, $V_C(r)=e^2/r$, which is long-ranged and powerful, is not what electrons in a solid actually experience. The presence of all other mobile electrons fundamentally alters this interaction. This lesson is dedicated to understanding this phenomenon.

Our learning objective is to introduce the concept of dielectric screening and the reducible polarization bubble within linear response theory. We will see how the collective response of the electron gas "screens" the bare Coulomb force and develop the formal framework to describe this effect.

1. The Physical Picture of Screening

Imagine introducing a single negative test charge into a uniform sea of electrons (an electron gas). The mobile electrons will be repelled by the test charge, creating a region of lower electron density around it. This depletion of negative charge leaves behind a net positive charge from the background ions.

This "screening cloud" of net positive charge cancels the field of the test charge at large distances. The result is that the effective potential felt by another charge far away is much weaker and falls off much more rapidly than the bare $1/r$ Coulomb potential. This is the essence of dielectric screening.

To formalize this physical intuition, we need a tool to calculate how a system responds to an external perturbation. This tool is linear response theory.

2. Linear Response Theory and the Kubo Formula

Linear response theory provides a general framework for calculating the change in the expectation value of an observable, $\hat{A}$, when a small, time-dependent perturbation, ${\hat{H}}^{\prime }(t)$, is applied to a system initially in equilibrium.

The central result, known as the Kubo formula, states that the change in $⟨\hat{A}(t)⟩$ is linearly proportional to the perturbation. The proportionality "constant" is a retarded correlation function involving the commutator of the observable $\hat{A}$ and the perturbation ${\hat{H}}^{\prime }$.

The following video provides a rigorous derivation of the Kubo formula. Given your background in quantum mechanics and QFT, you should find the derivation using the interaction picture straightforward.

The key result from the video is that the change in the expectation value of operator $\hat{A}$ is:

$$\delta ⟨\hat{A}(t)⟩=-i{\int }_{-\infty }^{t}d{t}^{\prime }⟨[{\hat{A}}_I(t),{\hat{H}}_I^{\prime }(t^{\prime })]⟩$$

where the subscript $I$ indicates operators in the interaction picture. This powerful formula connects the macroscopic response of a system to its microscopic correlation functions.

3. The Dielectric Function

Now, let's apply linear response theory to our screening problem.
The perturbation is an external electrostatic potential ${\varphi }_{ext}(r,t)$. This potential couples to the charge density operator $\hat{\rho }(r)$. The perturbation Hamiltonian is:

$${\hat{H}}^{\prime }(t)=\int d^{3}r\hat{\rho }(r){\varphi }_{ext}(r,t)$$

We want to find the response, which is the induced charge density, ${\rho }_{ind}(r,t)=\delta ⟨\hat{\rho }(r,t)⟩$.

Combining these with the Kubo formula, we find that the induced charge density is related to the external potential via a density-density correlation function. This function, which measures the correlation of charge density fluctuations at different points in space and time, is often called the polarizability or polarization function, denoted $\chi$.

${\rho }_{ind}=\chi {\varphi }_{ext}$ (schematically)

The induced charge ${\rho }_{ind}$ creates its own induced potential, ${\varphi }_{ind}$, governed by the Poisson equation: ${\nabla }^2{\varphi }_{ind}=-4\pi {\rho }_{ind}$. In Fourier space, this becomes ${\varphi }_{ind}(q)=V(q){\rho }_{ind}(q)$, where $V(q)=4\pi e^2/q^2$ is the Fourier transform of the bare Coulomb interaction.

The total potential inside the material is the sum of the external and induced potentials:

$${\varphi }_{tot}={\varphi }_{ext}+{\varphi }_{ind}$$

The dielectric function, $ϵ(q,\omega )$, is defined as the factor that relates the total potential to the external one. It quantifies how much the external field is weakened (screened) by the medium.

$${\varphi }_{tot}(q,\omega )=\frac{{\varphi }_{ext}(q,\omega )}{ϵ(q,\omega )}$$

The following resource provides a concise, self-consistent derivation of the dielectric function within this framework.

As derived in the notes, the key idea is to assume that the induced density responds not to the external potential, but to the total potential felt by the electrons. This is a mean-field type of approximation. The response is described by the non-interacting density-density response function, ${\chi }_0$:

$${\rho }_{ind}(q,\omega )={\chi }_0(q,\omega ){\varphi }_{tot}(q,\omega )$$

(Note: we are absorbing the sign from electron charge into the definition of ${\chi }_0$ for simplicity, so ${\chi }_0$ is formally the charge-charge response function).

Substituting this into our equations:

1. ${\varphi }_{tot}={\varphi }_{ext}+{\varphi }_{ind}$
2. ${\varphi }_{ind}=V{\rho }_{ind}=V{\chi }_0{\varphi }_{tot}$

Combining them gives ${\varphi }_{tot}={\varphi }_{ext}+V(q){\chi }_0(q,\omega ){\varphi }_{tot}$. Rearranging for ${\varphi }_{tot}$ yields:

$${\varphi }_{tot}(q,\omega )=\frac{{\varphi }_{ext}(q,\omega )}{1-V(q){\chi }_0(q,\omega )}$$

By comparing with the definition of the dielectric function, we arrive at one of the central results of this field:

$$ϵ(q,\omega )=1-V(q){\chi }_0(q,\omega )$$

This expression is the heart of the Random Phase Approximation (RPA). It tells us that the entire dielectric screening property of an electron gas is determined by the bare Coulomb interaction $V(q)$ and the non-interacting response function ${\chi }_0(q,\omega )$.

4. The Polarization Bubble

What is this object ${\chi }_0$? It represents the response of a non-interacting Fermi gas to a potential. Physically, the potential excites an electron from an occupied state $|k⟩$ to an unoccupied state $|k+q⟩$, creating an electron-hole pair. ${\chi }_0$ is the sum over all possible such pairs.

In the language of Feynman diagrams, ${\chi }_0$ is represented by a simple loop, known as the polarization bubble.

As shown in the resource, the polarization bubble ${\chi }_0$ consists of two Green's function lines, one for the electron and one for the hole, propagating and then recombining.

This brings us to the concept of reducibility.

• Irreducible Polarization Bubble ($\Pi \equiv {\chi }_0$): This is the "bare" bubble. It is irreducible because you cannot cut it into two separate polarization diagrams by severing a single interaction line $V$. In RPA, the irreducible polarization is simply ${\chi }_0$.
• Reducible Polarization Bubble ($\chi$): This is the full density-density response function of the interacting system. It is called reducible because, as shown in the diagram above for ${\chi }_{RPA}$, it can be cut into pieces by severing an interaction line.

The RPA equation for the dielectric function can be understood as summing an infinite geometric series of these bubble diagrams. The effective interaction between two charges is not just the bare Coulomb line, but the bare line plus a line mediated by one bubble, plus a line mediated by two bubbles, and so on. This infinite sum is what gives us the screened interaction.

Test your understanding!

The RPA expression for the dielectric function is $\epsilon(q, \omega) = 1 - V(q) \chi_0(q, \omega)$. In the static, long-wavelength limit ($\omega \to 0, q \to 0$), $\chi_0(0,0)$ is proportional to the density of states at the Fermi level, $D(E_F)$. For a simple metal, $D(E_F) > 0$, while for an insulator, $D(E_F) = 0$. What does this imply about the screening of a static charge in a metal versus an insulator?

Show answer

In a metal, $D(E_F) > 0$, so $\chi_0(0,0)$ is finite and positive. Since $V(q) = 4\pi e^2/q^2$ diverges as $q \to 0$, the dielectric function $\epsilon(q,0)$ diverges. This means that $\phi_{tot}(q\to 0) = \phi_{ext}(q\to 0)/\epsilon(q\to 0) \to 0$. The potential from a static charge is perfectly screened at very long distances.

In an insulator, $D(E_F)=0$, so ${\chi }_0(0,0)=0$. Therefore, $ϵ(q\to 0,0)\to 1$. This implies there is no screening from mobile charge carriers at long distances, and the long-range $1/r$ Coulomb interaction persists (though it can be weakened by a constant dielectric constant from bound-electron polarization).

Conclusion

Today, we've built the foundation for understanding interactions in many-electron systems. We've moved from a simple picture of bare Coulomb repulsion to a more realistic model of screened interactions.

Key Takeaways:

• Dielectric screening is the process where mobile charges in a medium rearrange to weaken an external electric field, modifying the effective interaction.
• Linear response theory provides the mathematical framework to calculate this response, culminating in the Kubo formula, which relates the response to a correlation function.
• The dielectric function $ϵ(q,\omega )$ quantifies screening by relating the total potential to the external potential: ${\varphi }_{tot}={\varphi }_{ext}/ϵ$.
• Within the Random Phase Approximation (RPA), the dielectric function is given by $ϵ=1-V{\chi }_0$, where $V$ is the bare Coulomb interaction and ${\chi }_0$ is the non-interacting density-density response function.
• ${\chi }_0$ is known as the irreducible polarization bubble, representing the creation of electron-hole pairs. The full, or reducible, polarization function $\chi$ is approximated in RPA by summing an infinite series of these bubbles.

Preview of the Next Lesson

We have now defined the dielectric function $ϵ$ in terms of the polarization bubble ${\chi }_0$. Our next step will be to use this to formally define the effective interaction within the RPA by summing the bubble diagrams. This screened interaction, $W(q,\omega )=V(q)/ϵ(q,\omega )$, is the object we need to start building realistic approximations for the electron self-energy, which will eventually connect back to the Green's functions and quasiparticles we discussed previously.