Why does the Schriefer-Wolff transformation works for phonons?

Question

One way to derive a Hamiltonian with attractive electron interactions is to start from the Hamiltonian with a part quadratic in electrons, quadratic in phonons, and a standard electron phono coupling and perform a Schriefer-Wolff transformation. After the transformation, the electron and phonon Hilbert space are decoupled. However, I was wondering why this is possible in the first place. I always understood the Schriefer-Wolff transformation as a way to decouple two parts of the Hilbert space that are separated by a large energy gap (As explained in Piers Coleman's book). But if we consider acoustic phonons with a linear dispersion that vanishes at zero momentum, the above condition is not satisfied. Why do people do it anyway?

Answer 1

Schrieffer-Wolff is an approximate way of diagonalizing electron-photon Hamiltonian. On the one hand it is inspired by by an exact polaron transformation (see, e.g., this thread), while on the other hand by the canonical application of SW to the Kondo problem. In more mundane terms it is equivalent to the second order perturbation theory.

Just like in the perturbation theory, the simplest test for applying SW transformation is that the perturbation is smaller than the spacing between the energy levels - the condition which is satisfied in the Kondo problem (with Pierce Coleman being a person who made significant contributions in this domain.) This condition is clearly not satisfied when Applying SW or the perturbation theory to a continuous spectrum, as in the case of electron-phonon coupling in solids or nearly free electron model, the basic band theory, perturbative treatment of electron gas (e.g., Fermi liquid), etc. Perhaps more mathematically incline people could add a discussion in terms of asymptotic series.. however, from a practical viewpoint, the applicability is justified by a presence of natural cutoffs - such as relaxation times, coherence length, temperature and others, which are always present/implied, although may not enter explicitly in the Hamiltonian, which focuses on the essential (like showing the attractive interaction and deriving superconductivity.)

Note that the application of SW transformation to Kondo problem also has a cutoff - since for some electron energies the condition $\frac{V}{\epsilon_k-\epsilon_0}$ breaks... but one assumes that the only electrons involved are those close to the Fermi surface, which allows limiting the electron energies to a layer of thickness $T$ near the Fermi surface, and the transformation works as long as $$ \frac{V}{\epsilon_F-\epsilon_0}\ll 1 $$

Note also that the ultimate confirmation for such an approach is provided by independent calculations using more mathematically precise methods. Thus, perturbative Fermi liquid picture can be shown to be true using, e.g., quantum field theory calculations applied to solid state... yet a simple perturbative calculation shows that it breaks in one dimension.