  We evaluate the microwave admittance of a one-dimensional chain of fluxonium
qubits coupled by shared inductors. Despite its simplicity, this system
exhibits a rich phase diagram. A critical applied magnetic flux separates a
homogeneous ground state from a phase with a ground state exhibiting
inhomogeneous persistent currents. Depending on the parameters of the array,
the phase transition may be a conventional continuous one, or of a
commensurate-incommensurate nature. Furthermore, quantum fluctuations affect
the transition and possibly lead to the presence of gapless "floating phases".
The signatures of the soft modes accompanying the transitions appear as a
characteristic frequency dependence of the dissipative part of admittance.
