Can electron-phonon coupling be extracted from the dielectric function?

Question

Computationally, there are various ways to calculate the electron-phonon coupling constants as a function of phonon index and momentum. My question is about the reverse process. Let us say that an experimentalist is (miraculously?) able to measure the effective Coulomb interaction in a material $V_{\textrm{eff}} (\mathbf{k}-\mathbf{k}', \nu)$ defined as:

$$V_{\textrm{eff}} (\mathbf{k}-\mathbf{k}', \nu) = \frac{V_0(\mathbf{k}-\mathbf{k}')}{\epsilon(\mathbf{k}-\mathbf{k}', \nu)}\tag{1}$$

where $V_0(\mathbf{k}-\mathbf{k}') = \frac{4\pi e^2}{\vert \mathbf{k}-\mathbf{k}' \vert^2}$ is the bare Coulomb interaction, and $\epsilon(\mathbf{k}-\mathbf{k}', \nu)$ is the dielectric function.

My question is the following: Is there a generic, model-independent way to extract out the electron-phonon coupling constants (or coupling functions, rather) or the $k$-resolved Eliashberg function from such a measurement of $V_{\textrm{eff}} (\mathbf{k}-\mathbf{k}', \nu)$ (or $\epsilon(\mathbf{k}-\mathbf{k}', \nu)$)?

Presumably, phonons should appear as poles in $\epsilon^{-1}(\mathbf{k}-\mathbf{k}', \nu)$, so clearly some information about phonons is contained in the dielectric function, but it is unclear to me what role electron-phonon coupling plays in the appearance of phonons in the dielectric function.

Edit: To be clear, I am not interested in comparing simulated dielectric functions (where coupling constants are inputs) to experiment dielectric functions. I am interested in whether the physics of electron-phonon coupling is fully contained and extractable from the dielectric function in principle.

Answer 1

Your equation for the effective Coulomb interaction is given as Eq. 7.203 this PDF of Mahan's "Many-Particle Physics" book, but he also writes the same quantity in terms of the screened ("screening" it discussed in, e.g. Section 5.4 of the same book) electron-phonon interaction (in his notation):

$$ \tag{1} V_\textrm{eff} = \frac{V_q}{\varepsilon_\textrm{total}} = \frac{V_q^{(\infty)}}{\varepsilon(q,\omega)} + V_{\textrm{sc-ph}}(q,\omega), $$

in which,

$$\tag{2} V_q^{(\infty)} = \frac{V_q}{\varepsilon_\infty}.$$

Therefore if you obtain $V_\textrm{eff}$ and the $\varepsilon (q,\omega)$ empirically, then the screened electron-phonon coupling function (or perhaps, the Fourier transform of it) could be obtained by:

$$\tag{4} V_{\textrm{sc-ph}}(q,\omega) = V_\textrm{eff} - \frac{V_q}{\varepsilon_\infty \varepsilon(q,\omega)}. $$

If empirically we have only $\varepsilon (q,\omega)$, then we can get the difference between $V_{\textrm{sc-ph}}$ and $V_\textrm{eff}$, but not $V_{\textrm{sc-ph}}$ alone unless we're able to express $V_\textrm{eff}$ in terms of known/obtainable quantities in another way (a bit like the "two equations for two unknowns" concept).

If empirically we have only $V_\textrm{eff}$, then to get $V_{\textrm{sc-ph}}$ alone we would need to express $\varepsilon (q,\omega)$ in terms of known/obtainable quantities, which is discussed a bit in Section 5.5 of the same book by Mahan. An "exact" expression is given as:

       

and a definition in terms of other objects (the impurity charge density $\rho_i(\mathbf{q})$ and screening charge density $\rho_s(\mathbf{q})$) is here:

and linear screening models simply assume:

The opening sentence of Section 5.5 of the same book by Mahan, says that even for the uniform electron gas, no one had derived an exact solution to Eq. 5.110 for the dielectric function. He presented four different approximate solutions: Thomas-Fermi (Section 5.5.1), Lindhard/RPA (Section 5.5.2), Hubbard (Section 5.5.3), Singwi-Sjoelander (Section 5.5.4) and that this is "only a small subset of the vast number which are available" (and that was decades ago).

The RPA is discussed again in Chapter 7, so I'll give the expression for $\varepsilon(q,\omega)$ from there:

For some cases the expressions can be spelled out more directly, for example if you have a polar solid insulator with conduction electrons, you would simply get:

which makes it easier to get your $V_{\textrm{sc-ph}}(q,\omega)$ function in terms from your empirical $V_\textrm{eff}$.