  In a previous work [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 2D 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
$\sigma_{xy}=e^2\nu/h(1+\eta^2)$ by the light-matter coupling $\eta$. Most of
the aforementioned phenomena should be experimentally accessible and
corresponding implications are discussed.
