  Recent experiments have revealed the tantalizing possibility of fabricating
lattice electronic systems strongly coupled to quantum fluctuations of
electromagnetic fields, e.g., by means of geometry confinement from a cavity or
artificial gauge fields in quantum simulators. In this work, we develop a
high-frequency expansion to construct the effective models for lattice
electrons strongly coupled to a continuum of off-resonant photon modes with
arbitrary dispersion. The theory is nonperturbative in the light-matter
coupling strength, and is therefore particularly suitable for the ultrastrong
light-matter coupling regime. Using the effective models, we demonstrate how
the dispersion and topology of the electronic energy bands can be tuned by the
cavity. In particular, quasi-one-dimensional physics can emerge in a
two-dimensional square lattice due to a spatially anisotropic band
renormalization, and a topologically nontrivial anomalous quantum Hall state
can be induced in a honeycomb lattice when the cavity setup breaks
time-reversal symmetry. We also demonstrate that the photon-mediated
interaction induces an unconventional superconducting paired phase distinct
from the pair-density-wave state discussed in models with truncated
light-matter coupling. Finally, we study a realistic setup of a Fabry-P\'{e}rot
cavity. Our work provides a systematic framework to explore the emergent
phenomena due to strong light-matter coupling and points out new directions of
engineering orders and topological states in solids.
