Realizing a Symmetry Protected Topological Phase in a Superconducting Circuit

We propose a superconducting quantum circuit whose low-energy degrees of freedom are described by the sine-Gordon (SG) quantum field theory. For suitably chosen parameters, the circuit hosts a symmetry protected topological (SPT) phase protected by a discrete $\mathbb{Z}_2$ symmetry. The ground state of the system is twofold degenerate and exhibits local spontaneous symmetry breaking of the $\mathbb{Z}_2$ symmetry close to the edges of the circuit, leading to spontaneous localized edge supercurrents. The ground states host Majorana zero modes (MZM) at the edges of the circuit. On top of each of the two ground states, the system exhibits localized bound states at both edges, which are topologically protected against small disorder in the bulk. The spectrum of these boundary excitations should be observable in a circuit-QED experiment with feasible parameter choices.