Just as in title: What is the difference between Fermi level and Fermi edge? My friend makes some research about XPS and he encountered this term. He knows what the Fermi level is, but never heard about Fermi edge.
[Physics] the difference between Fermi level and Fermi edge
condensed-matterelectronsfermionssolid-state-physicsterminology
Related Solutions
First, the Thomas-Fermi screening is a semiclassical static theory which assumes that the total potential $\phi(\mathbf{r})$ varies slowly in the scale of the Fermi length $l_{\text{F}}$, the chemical potential $\mu$ is constant and that $T$ is low. In principle, it does not rely on linear response theory.
The condition of slowly varying potential is a general condition of validity of semiclassical models. Physically, if the particle [electron] is represented by a wave packet, what is tellying us is that all the waves in the wavepacket will see the same potential and the particle will suffer [or enjoy!] a force as if it was point-like ["classical"] because such potentials gives rise to ordinary forces in the equation of motion describing the evolution of the position and wavevector of the packet. The wavepacket must have a well-defined wavevector on scale of the Brillouin zone [thus $\Delta k \simeq k_{\text{F}}$] and therefore can be spread in the real space over many primitive cells.
Mathematically, the assumption that your potential is a slowly varying function of the position implies that the theory is not valid for $|\mathbf{q}| \gg k_{\text{F}}$ [and therefore for $|\mathbf{r}| \ll l_{\text{F}}$].
On the other hand, the static Lindhard dielectric function is a fully quantum treatment of the problem and it is valid for all the ranges of $\mathbf{q}$. It includes, in the limit $\mathbf{q} \rightarrow 0$, the linearized Thomas-Fermi dielectric function. It only assumes linear response, that is, the induced density of charge is proportional to the total potential $\phi(\mathbf{r})$.
Note also that the Lindhard treatment is far more general than the Thomas-Fermi in the sense that it can describe both dynamic and static screening.
What exactly makes them different from ordinary insulators is the number of edge states. In a 2D topological insulator, you are guaranteed to have the Fermi energy at the edge of the sample cross the edge bands an odd number of times in half the edge Brillouin zone while in a trivial band insulator, if there are edge states, the Fermi energy will cross the edge bands an even number of times if at all in half the edge Brillouin zone. For instance, see below:
Now, suppose we were to dope the material with some holes. In this case, there Fermi energy would decrease. In image (a), it is possible for the Fermi energy to dip below the bottom of the surface band so that it is in both the bulk gap and the surface gap. In image (b), this is not possible. Reducing the Fermi energy merely changes which surface band you cross. Therefore, you cannot avoid crossing a surface band and the surface is guaranteed to be a metal. See below and notice the secondary red lines:
Therefore we say that the edge states in the topological insulator is "topologically protected" whereas the edge states in the trivial insulator are not. What is responsible for the edge states is a deeper question. The answer is that one can calculate a topological invariant, called the $Z_2$ invariant from the bulk band structure. Its origin can be thought of as an ill-defined phase of the wavefunction in the Brillouin zone though there are also other formulations. If you are familiar with the integer quantum hall effect, this should not be new to you. I hope this answers you question sufficiently, but feel free to comment if I have not answered your question fully.
Best Answer
As far as I know one usually speaks of a Fermi edge singularity when referring to a singularity occurring near the Fermi level threshold in a x-ray absorption/emission spectrum of a metal.
Below you will find some useful references:
One of the first papers on x-ray spectrum singularities for metals: Singularities in the X-ray absorption and emission of metals. III. One-body theory exact solution by P Nozieres, CT De Dominicis - Physical Review, 1969 - APS
The Fermi Edge Singularity and Boundary Condition Changing Operators Ian Affleck and Andreas W.W. Ludwig.
And a brief overview in these slides here.