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Recent experiments have revealed the tantalizing possibility of fabricatinglattice electronic systems strongly coupled to quantum fluctuations ofelectromagnetic fields, e.g., by means of geometry confinement from a cavity orartificial gauge fields in quantum simulators. In this work, we develop ahigh-frequency expansion to construct the effective models for latticeelectrons strongly coupled to a continuum of off-resonant photon modes witharbitrary dispersion. The theory is nonperturbative in the light-mattercoupling strength, and is therefore particularly suitable for the ultrastronglight-matter coupling regime. Using the effective models, we demonstrate howthe dispersion and topology of the electronic energy bands can be tuned by thecavity. In particular, quasi-one-dimensional physics can emerge in atwo-dimensional square lattice due to a spatially anisotropic bandrenormalization, and a topologically nontrivial anomalous quantum Hall statecan be induced in a honeycomb lattice when the cavity setup breakstime-reversal symmetry. We also demonstrate that the photon-mediatedinteraction induces an unconventional superconducting paired phase distinctfrom the pair-density-wave state discussed in models with truncatedlight-matter coupling. Finally, we study a realistic setup of a Fabry-P\'{e}rotcavity. Our work provides a systematic framework to explore the emergentphenomena due to strong light-matter coupling and points out new directions ofengineering orders and topological states in solids.

Effective theory of lattice electrons strongly coupled to quantum electromagnetic fields