  A nonperturbative approach is developed to analyze superconducting circuits
coupled to quantized electromagnetic continuum within the framework of the
functional renormalization group. The formalism allows us to determine complete
physical pictures of equilibrium properties in the circuit quantum
electrodynamics (cQED) architectures with high-impedance waveguides, which have
recently become accessible in experiments. We point out that nonperturbative
effects can trigger breakdown of the supposedly effective descriptions, such as
the spin-boson and boundary sine-Gordon models, and lead to qualitatively new
phase diagrams. The origin of the failure of conventional understandings is
traced to strong renormalizations of circuit parameters at low-energy scales.
Our results indicate that a nonperturbative analysis is essential for a
comprehensive understanding of cQED platforms consisting of superconducting
circuits and long high-impedance transmission lines.
