# How do phonons and electron-electron interactions break phase coherence of electrons? Also, do they break time reversal symmetry?

When considering electron scattering, I appreciate that phonons and electron-electron interactions need to be considered in a different light to scattering from a static, disordered potential.

With a static disordered potential one could, in principle, solve the Schrödinger equation for a given realisation of disorder. If the disorder is weak one gets diffusive transport and weak localisation due to coherent back scattering. The coherent back scattering arises due to time-reversal symmetry. Any path from one point back to itself is guaranteed to interfere constructively with the path that is identical, just in the opposite direction. However this is only true when we have phase coherence across the sample. If we didn't, we would not have such strong coherent back scattering.

It is commonly stated that phonons and electron-electron interactions, in which energy not just momentum is exchanged, break this phase coherence. My question is how do they break phase coherence and destroy weak localisation?

Secondly, do they break time reversal symmetry? Instructive to look at classical analogies. Imagine rolling a ball down a hill, if we videoed it and then reversed it you wouldn't be able to tell what video was the original (neglecting friction). If you imagine the classical analogy of a phonon and an electron collision as being between two classical bodies, this situation would not satisfy time reversal symmetry. Under time reversal, when considering electron states, we would not reverse the phonon's path. (Analogous to how we don't reverse a magnetic field when considering time reversal of electron states under an applied magnetic field, if we time reversed the whole system the magnetic field should be reversed too. If we don't reverse the magnetic field we see we don't have time reversal symmetry, hence allowing for things like the quantum hall effect.) So consider two classical bodies colliding, then consider reversing the path of one body. This new solution does not fit Newton's law, it does not satisfy time reversal.

If they do break time reversal symmetry this would explain why they break coherent back scattering.

Think about the exact (i.e. interacting) electron propagator $$\mathcal{G}(x,x') = -i \langle0|T \psi_\alpha(x) \bar{\psi}_\beta(x') |0 \rangle$$. If interactions are weak (i.e. in the perturbative regime), you can expand this as a series in powers of the coupling constant - to second order it is the noninteracting propagator, with some loop corrections from phonons and electron-electron $$a^\dagger a a^\dagger a$$ type vertices e.g. 