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.
We address the quantum-critical behavior of a two-dimensional itinerant ferromagnetic systems described by a spin-fermion model in which fermions interact with close to critical bosonic modes. We consider Heisenberg ferromagnets, Ising ferromagnets,
and the Ising nematic transition. Mean-field theory close to the quantum critical point predicts a superconducting gap with spin-triplet symmetry for the ferromagnetic systems and a singlet gap for the nematic scenario. Studying fluctuations in this ordered phase using a nonlinear sigma model, we find that these fluctuations are not suppressed by any small parameter. As a result, we find that a superconducting quasi-long-range order is still possible in the Ising-like models but long-range order is destroyed in Heisenberg ferromagnets.
