Electrons interacting with Goldstone modes and the rotating frame

TL;DR

This paper investigates how electrons interact with Goldstone modes in systems with broken continuous symmetry, revealing the necessity of a rotating frame for defining quasiparticles and analyzing spectral functions.

Contribution

It provides a comprehensive calculation of electron-Goldstone mode interactions across the Brillouin zone and introduces the concept of a rotating frame for quasiparticle definition.

Findings

Electrons couple to both the Goldstone field and its time derivative.

Quasiparticles are best described in a rotating frame following order parameter fluctuations.

Implications for spectral functions in symmetry-broken and disordered systems are discussed.

Abstract

We consider electronic systems with a spontaneously broken continuous symmetry. The scattering vertex between electrons and Goldstone modes is calculated over the entire Brillouin zone using the random phase approximation. This calculation reveals two things: (1) electrons always couple to both $\phi$ and $\partial_t \phi$, where $\phi$ is the Goldstone field, and (2) quasi-particles in a state with continuous symmetry breaking have to be defined in a rotating frame, which locally follows the fluctuations of the order parameter. The implications of these findings for electron spectral functions in both symmetry-broken and thermally disordered systems are discussed, and the examples of anti-ferromagnetism in the Hubbard model and spin spiral order in the three-band model are worked out in detail.