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.

Figures (6)

• Figure 1: The double-well potential $V(x)$ is drawn for $U=35$, $a=0$, $b=1$. The wavefunctions, $\varphi_i(x)$, of the first ten eigenstates are shown (not to scale), each zeroed on their eigenenergy $E_i$, for mass $m=1$.
• Figure 2: (a) The energy spectrum of the first 10 eigenstates of the model are plotted as a function of the coupling strength $\eta$ for $\omega=0.9 \omega_{12},U=35,b=1,m=1$. (b) Likewise, the spectrum for the first 14 eigenstates is plotted for $\omega=0.9 \omega_{13},U=150,b=1,m=1$ (with each line being nearly doubly degenerate in this case).
• Figure 3: The ground state expectation value of the photon number, $\langle n^{(g)}\rangle$, is calculated in each gauge $g$ for increasing coupling $\eta$, for frequencies (a) $\omega=0.9\omega_{12}$ and (b) $\omega=0.9\omega_{13}$. The insets show the full range of the dipole gauge photon number, $\langle n^{\mathrm{(dipole)}}\rangle$.
• Figure 4: The energy spectrum is plotted under the Peierls substitution, both in the absence of and including the self-polarization, for the two regimes: (a) $\omega=0.9\omega_{12}$ (intraband coupling) and (b) $\omega=0.9\omega_{13}$ (interband coupling). The true results from Fig. \ref{['fig:spectrum']} are plotted for comparison.
• Figure 5: The spectrum of the model is shown for material truncations performed in the Coulomb, dipole, and Peierls gauges, under increasing coupling $\eta$. The truncations are placed as: (a--c) $N_{\mathrm{el}}=2,4,12$ for the coupling $\omega=0.9\omega_{12}$ and (d--f) $N_{\mathrm{el}}=4,6,12$ for $\omega=0.9\omega_{13}$. The converged results from Fig. \ref{['fig:spectrum']} are plotted for comparison.