  Strong coupling between matter and quantized electromagnetic fields in a
cavity has emerged as a possible route toward controlling the phase of matter
in the absence of an external drive. We develop a faithful and efficient
theoretical framework to analyze quantum geometry and topology in materials
ultrastrongly coupled to cavity electromagnetic fields in two dimensions. The
formalism allows us to accurately evaluate geometrical and topological
quantities, such as Berry phase and Chern number, in ultrastrong and deep
strong coupling regimes. We apply our general framework to analyze a concrete
model of massive Dirac fermions coupled to a circularly polarized cavity mode.
Surprisingly, in addition to an ordinary transition to the topological phase,
our analysis reveals a qualitatively new feature in deep strong coupling
regimes, namely, the emergence of reentrant transition to the topologically
trivial phase. We also present its intuitive understanding by showing the
unitary mapping between the low-energy effective theory of strongly coupled
light-matter systems and the Haldane honeycomb model.
