What are the difference about the concept of polarization and screening in fundamental electromagnetics and many-body physics?

Question

I found that some concepts, such as polarization and screening, met firstly in fundamental electromagnetics, are used in the context of many-body Green's functions in condensed matter physics. I am curious about the subtleties between both. Is there a deep connection between both? For completeness, let me first explain the two physical picture in detail.

1. Viewpoint from the fundamental electromagnetics

As shown above, if we applied an external electric field $\vec{E}_0$ to a homogeneous (constant dielectric function) and infinite dielectric materials (ignoring boundary conditions), then charges will move in response to $\vec{E}_0$. Along with charge moves, the dipole moment will be formed. Correspondingly, the dipole moment per unit volume is defined as dielectric polarization $\vec{P}$. The average charge density is still zero, but the induced (depolarizing) electric field $\vec{E}_1$ will weaken the applied $\vec{E}_0$. The local electric field will be introduced: $$\vec{E}_{local}=\vec{E}_0+\vec{E}_1 ,$$ which plays the role of screening. If we consider only linear effects, one can found further the following relation: $$\vec{P}=\epsilon_0\chi_e\vec{E}.$$ The dielectric displacement is defined as: $$\vec{D}=\epsilon_0\vec{D}+\vec{P}$$ then we find $$\vec{D}=\epsilon_0\epsilon\vec{E}$$ with $\epsilon=1+\chi_e$.

2. Viewpoint from many-body Green's function:

The physical picture from the viewpoint of many-body Green's function is explained in the above figure. One can see the concepts of polarization and screening again. In particular, the polarization will be translated to polarization response function and screening will be represented by the dielectric function.

I cannot believe there is a coincidence here. So the many-body Green's function borrows some concepts from fundamental electromagnetics? Or two viewpoints for the same physics just from macroscopic and microscopic view?

Answer 1

The terms polarization and screening are two different concepts used in different context. Let me try to explain both concepts.

As you explained well, the local electric field is introduced to have a more exact value of the electric field that acts at the site of an atom. Indeed, the local field is often significantly different from the applied macroscopic field $E_0$, mainly due to the depolarization field $E_1$. This local electric field is for example used to define the polarizability $\alpha$ of an atom by $$p=\alpha E_{local}$$ This depolarization field $e_1$ that in some sense screens the macroscopic field is due to dipoles in the material re-orientating themselves in such a way that they create an electric field opposing $E_0$.

This picture above is often used to describe the polarizability of dielectric materials. However, the phenomena of screening is something that takes place in metals where there is a high concentration of carriers (electrons). The screening is also rather easy understood. So if no electric field is present the electrons in the metal are more less in equilibrium with positive charged ion lattice. However, applying an electric field will results the re-organization of the electrons in such a way that they screen the electric field. In this way the Coulomb force will no longer have a $1/r$-dependence but rather an exponential dependence. You can check Thomas-Fermi screening model on this.

So notice that the difference here is that in the first case, we were treating dielectrics where it is the dipole moment that screens/lowers the electric field. This dipole moment often arises from the electron cloud around the atom. Whereas, in this second case, we have the free electron which organize in such a way that they change the r-dependence of the Coulomb interaction such that we have a screened Coulomb potential!

At last, to explain your last figure in the context of the slides you sent. This part discuss on how an interacting many-body system can be treated as a collection of non-interacting single-particle excitations, called quasi-particles. These quasi-particles can be considered as a single-particle together with some different properties due to their interaction with its surrounding. For example, in the figure they illustrate a quasiparticle as a hole combined with the electron cloud around it. So in this case we can think of the quasiparticle hole as as simple hole but with a larger mass and lower charge.

Answer 2

In condense matter many body greens function, we deal with the individual particle wave function. It is a bit like having bunch of +ve and -ve charges rearranging themselves, after an external radiation creates a perturbation.

Whereas, for a dielectric medium we look at the overall field that arises due to rearrangement of molecules, to screen the external fields. Indeed both the process talk about screening or lets say dynamics for minimizing the free energy, I wonder if looking a many body condense matter system in thermodynamic and classical limit with constant external electric field applied to the system, give same answer as Maxwell's law.