Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance

Abstract

In a previous work [Rokaj et al., Phys. Rev. Lett. 123, 047202 (2019)] a translationally invariant framework called quantum-electrodynamical Bloch (QED-Bloch) theory was introduced for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized photon field in the optical limit. For such systems, we show that QED-Bloch theory predicts the existence of fractal polaritonic spectra as a function of the cavity coupling strength. In addition, for the energy spectrum as a function of the relative magnetic flux we find that a terahertz cavity can modify the standard Hofstadter butterfly. In the limit of no quantized photon field, QED-Bloch theory captures the well-known fractal spectrum of the Hofstadter butterfly and can be used for the description of two-dimensional materials in strong magnetic fields, which are of great experimental interest. As a further application, we consider Landau levels under cavity confinement and show that the cavity alters the quantized Hall conductance and that the Hall plateaus are modified as ${\ensuremath{\sigma}}_{xy}={e}^{2}\ensuremath{\nu}/h(1+{\ensuremath{\eta}}^{2})$ by the light-matter coupling $\ensuremath{\eta}$. Most of the aforementioned phenomena should be experimentally accessible, and corresponding implications are discussed.