The role of polarization field terms in a model for a cavity quantum material

Abstract: Constructing models for cavity quantum materials requires a careful treatment of the light-matter coupling. In general, one must specify matrix elements constructed from the material wavefunctions, which are often unknown in a tight-binding framework. The Peierls substitution is often used to avoid introducing these additional parameters in the multi-center dipole (or Peiels) gauge, under the assumption that contributions from intraband and interband dipole moments can be neglected in the low-energy theory. We present the derivation of the Peierls gauge description in the passive view of canonical transformations. We construct a toy model for a multi-band system with two sites, which we couple to a uniform field in the Coulomb, dipole, and Peierls gauges. We find that the Peierls substitution can be justified as a low-energy, effectively single-band description in one dimension, but it misses both self-polarization corrections and the direct coupling needed to describe interband transitions in the full Peierls gauge theory. Moreover, the Coulomb, dipole, and Peierls gauges define distinct partitions of the composite system into the light and matter subsystems. We illustrate the implications of this subsystem relativity for physical observables and on the performance of orbital truncations in each gauge.